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All Textbook Solutions for Calculus Volume 2

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 267. [T] x3sinxdx Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 268. [T] xx49dx Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 269. [T] x1+e x 2 dx Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 270. [T] 35x2xdx Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 271. [T] dxx x1 Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 272. [T] excos1(ex)dx Use a calculator or CAS to evaluate the following integrals. 273. [T] 0/4cos(2x)dx Use a calculator or CAS to evaluate the following integrals. 274. [T] 01xex2dx Use a calculator or CAS to evaluate the following integrals. 275. [T] 082x x 2 +36dx Use a calculator or CAS to evaluate the following integrals. 276. [T] 02/314+9x2dx Use a calculator or CAS to evaluate the following integrals. 277. [T] dxx2+4x+13 Use a calculator or CAS to evaluate the following integrals. .W4 273. |T] J ' cos(2x) 0 274. [T] J f1 -J-2 1x- e dx 0 275. [T] J dx 0 Vr2 + 36 276. [T] J -2/VJ 1—-dx 0 4 + 9.x2 277. [T] f dx J a2 + 4x + 13 278. [T] f dxJ 1 + sinx Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. 279.dxx‘ + 2x + 10dx- 6.1=dxcosx9sin x + 2sinxarctanb3)dxdxJ" ln|x|arcsin(ln|x|)^ * 280. 281. 282. 283. 284.Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. 279.dxx‘ + 2x + 10dx- 6.1=dxcosx9sin x + 2sinxarctanb3)dxdxJ" ln|x|arcsin(ln|x|)^ * 280. 281. 282. 283. 284.Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. 279.dxx‘ + 2x + 10dx- 6.1=dxcosx9sin x + 2sinxarctanb3)dxdxJ" ln|x|arcsin(ln|x|)^ * 280. 281. 282. 283. 284.Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. 279.dxx‘ + 2x + 10dx- 6.1=dxcosx9sin x + 2sinxarctanb3)dxdxJ" ln|x|arcsin(ln|x|)^ * 280. 281. 282. 283. 284.Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. 279.dxx‘ + 2x + 10dx- 6.1=dxcosx9sin x + 2sinxarctanb3)dxdxJ" ln|x|arcsin(ln|x|)^ * 280. 281. 282. 283. 284.Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. 279.dxx‘ + 2x + 10dx- 6.1=dxcosx9sin x + 2sinxarctanb3)dxdxJ" ln|x|arcsin(ln|x|)^ * 280. 281. 282. 283. 284.Use tables to perform the integration.Use tables to perform the integration.Use tables to perform the integration. 287. dx1cos( 4x)Use tables to perform the integration.Find the area y^4 4- 25x2) = 5, x — 0, y — 0, and a integrals or a CAS.The region bounded between the curve V = .1 =. 0.3 < x < 1.1, and the x-axis is Vl+cosx revolved about the x-axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)Use substitution and a table of integrals to find the area of the surface generated by revolving the curve y = c \ 0 < v < 3. about the x-axis. (Round the answer to two decimal places.)[T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve r2 . L y —0 < x < 1, about the x-axis. (Round the answerto two decimal places.)(T] Use a CAS or tables to find the area of the surface generated by revolving the curve y = cosx, 0 < x < about the x axis. (Round the answer to two decimal places.)Find the length of the curve y = q- over [0, 8].Find the length of the curve y = exover [0, ln(2)].Find the area of the surface formed by revolving the graph of y = 2Vx over the interval [0, 91 about the x-axis.Find the average value of the function /(x) =___ _ over the interval f—3, 3], x2+ 1 298. Approximate the arc length of the curve y — tan(jrx) decimal places.) Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)Approximate the integral to three decimal places using the indicated rule. 307. j sin2(jrx)dx; trapezoidal rule; n = 6 308. / —1trapezoidal rule; n — 6 J0 1+J? rJx; Simpson’s rule; n ~ 3 Z-° S -X2 ? J e dx\ trapezoidal rule; n — 4 r08 .2 / e x dx; Simpson’s rule; n = 4 J 0 -0.4 o / sin(x")£/x; trapezoidal rule; u—4 70 -0.4 J sinfx^t/x; Simpson’s rule; n — 4 -0.5 / CQyXZ/x; trapezoidal rule; n — 4 J o.i 1 -0.5 / Xt/x; Simpson’s rule; n = 4 J 0.1 A Approximate the integral to three decimal places using the indicated rule.Approximate the integral to three decimal places using the indicated rule.Approximate the integral to three decimal places using the indicated rule.Approximate the integral to three decimal places using the indicated rule.Approximate the integral to three decimal places using the indicated rule.Approximate the integral to three decimal places using the indicated rule.Approximate the integral to three decimal places using the indicated rule.Approximate the integral to three decimal places using the indicated rule.316. Evaluate / —-7 exactly and show that the result Joi+x2 y is jt/4. Then, find the approximate value of the integral using the trapezoidal rule with n — 4 subdivisions. Use the result to approximate the value of x. Approximate using the midpoint rule with four subdivisions to four decimal places.318. Approximate J" US^1S the trapezoidal rule with eight subdivisions to four decimal places.Use the trapezoidal rule with four subdivisions to -0.8 estimate / x3dx to four decimal places. J 0 Use the trapezoidal rule with four subdivisions to ,0.8 1 estimate / x^dx. Compare this value with the exact J 0 value and find the error estimate.Using Simpson’s rule with four subdivisions, find -tttl I cas(x)dx.Show that the exact value of / xe A dx = 1 — Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.Given J xe x dx = 1 — use the trapezoidal rule with 16 subdivisions to approximate the integral and find die absolute error.Find an upper bound for the error in estimating / (5x + 4)dx using the trapezoidal rule with six steps.Find an upper bound for the error in estimating subdivisions.Find an upper bound for the error in estimating 10 steps.Find an upper bound for the error in estimating Find an upper bound for the error in estimating I 2xcos(x)dx using Simpson’s rule with four steps.Estimate the minimum number of subintervals needed to approximate the integral y (Sx'" + Sjt/x with an error magnitude of less than 0.0001 using the trapezoidal rule.Determine a value of n such that the trapezoidal rule will approximate [ v 1 + jc~ c/x with an error of no more J 0 than 0.01.Estimate the minimum number of subintervals + 4xVx with an error of magnitude less than 0.0001 using the trapezoidal rule.332. Estimate the minimum number of subintervals f4 1 needed to approximate the integral / —~dx with an J3(x-1)2 error magnitude of less than 0.0001 using the trapezoidal rule. 333. Use Simpson’s rule with four subdivisions to approximate the area under the probability density function 1 -r2/2 Use Simpsoifs rule with n — 14 to approximate (to three decimal places) the area of the region bounded by the graphs of y = 0, x = 0, and x = jr/2. The length of the ellipse x = cicgs(Z), y = £>sin(O> 0 < t < 2jt is given by eccentricity of the ellipse. Use Simpson’s rule with n — 6 subdivisions to estimate the length of the ellipse when a — 2 and e — 1/3.Estimate the area of the surface generated by revolving the curve y — cos(2x), 0 < jc < about the x axis. Use the trapezoidal rule with six subdivisions.Estimate the area of the surface generated by • 2 revolving the curve y = 2r. 0 < a < 3 about the x-axis. Use Simpson’s rule with n — 6.The growth rate of a certain tree (in feet) is given by —=— ie , where t is time in years. Estimate the / + 1 ’ J growth of the tree through the end of the second year by using Simpson’s rule, using two subintervals. (Round the answer to the nearest hundredth.) [T] Use a calculator to approximate J sm(/rA'k/-v using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.[T] Given j (3a2 — 2jrpjr = 100, approximate the value of this integral using the midpoint rule with 16 subdivisions and determine the absolute error.Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?The table represents the coordinates (x, y) that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot X y X y 0 125 600 95 100 125 700 88 200 120 800 75 300 112 900 35 400 90 1000 0 500 90 Choose the correct answer. When Simpson’s rule is used to approximate the definite integral, it is necessary that the number of partitions be an even number odd number either an even or an odd number a multiple of 4 The “Simpson” sum is based on the area under a The error formula for Simpson’s rule depends on_____ /(x) /'(x) /(4,w the number of steps Laplace Transforms In the last few chapters, we have looked at several ways to use integration for solving real-world problems. For this next project, we are going to explore a more advanced application of integration: integral transforms. Specifically, we describe the Laplace transform and some of its properties. The Laplace transform is used in engineering and physics to simplify the computations needed to solve some problems. It takes functions expressed in terms of time and trons/orms them to functions expressed in terms of frequency. It turns out that, in many cases, the computations needed to solve problems in the frequency domain are much simpler than those required in the time domain. The Laplace transform is defined in terms of an integral as 7 e"" ft Note that the input to a Laplace transform is a function of time, /(/), and the output is a function of frequency, F(j), Although many real-world examples require the use of complex numbers (involving the imaginary number i = V—1), in this project we limit ourselves to functions of real numbers. Let's stan with a simple example. Here we calculate the Laplace transform of /(f) = t. We have This is an improper integral, so we express it in terms of a limit, which gives Now we use integration by pans to evaluate the integral. Note that we are integrating with respect to t, so we treat the variable s as a constant. We have u—tdv— dt du=dt v —— ye_ir. Then we obtain = + + =~K + °1 - = JinL[[-i,-]-±[e--lj] - c+ c = 0-0 + -L s“ _x 2* s 1. Calculate the Laplace transform of /(/) = 1. 3.Calculate the Laplace transform of /(/) = : (Note, you will have to integrate by parts twice.) Laplace transforms are often used to solve differential equations. Differential equations are not covered in detail until later in this book; but, for now, let’s look at the relationship between the Laplace transform of a function and the Laplace transform of its derivative. Let’s start with the definition of the Laplace transform. We have WW! =r™r' = lim / e~st fifth. 4.Use integration by parts to evaluate Jjm^ e~sl fifth. (Let « = /{/) and dv — e '!dt.) After integrating by parts and evaluating the limit, you should see that Then, Thus, differentiation in the time domain simplifies to multiplication by s in the frequency domain. The final thing we look at in this project is how the Laplace transforms of fit] and its antiderivative are related. Let g(r) — f(u}dii. Then, ¦'o lim / :-* caj"5.Use integration by parts to evaluate hrn^y e ’ g(t)dl. (Let u = gif) and dv = e dt. Note, by the way, that we have defined gif, du — fifth.) As you might expect, you should see that L|^(r)| = |-L[/(/)i. Integration in the time domain simplifies to division by s in ±e frequency domain.Laplace Transforms In the last few chapters, we have looked at several ways to use integration for solving real-world problems. For this next project, we are going to explore a more advanced application of integration: integral transforms. Specifically, we describe the Laplace transform and some of its properties. The Laplace transform is used in engineering and physics to simplify the computations needed to solve some problems. It takes functions expressed in terms of time and trons/orms them to functions expressed in terms of frequency. It turns out that, in many cases, the computations needed to solve problems in the frequency domain are much simpler than those required in the time domain. The Laplace transform is defined in terms of an integral as 7 e"" ft Note that the input to a Laplace transform is a function of time, /(/), and the output is a function of frequency, F(j), Although many real-world examples require the use of complex numbers (involving the imaginary number i = V—1), in this project we limit ourselves to functions of real numbers. Let's stan with a simple example. Here we calculate the Laplace transform of /(f) = t. We have This is an improper integral, so we express it in terms of a limit, which gives Now we use integration by pans to evaluate the integral. Note that we are integrating with respect to t, so we treat the variable s as a constant. We have u—tdv— dt du=dt v —— ye_ir. Then we obtain = + + =~K + °1 - = JinL[[-i,-]-±[e--lj] - c+ c = 0-0 + -L s“ _x 2* s 1. Calculate the Laplace transform of /(/) = 1. 3.Calculate the Laplace transform of /(/) = : (Note, you will have to integrate by parts twice.) Laplace transforms are often used to solve differential equations. Differential equations are not covered in detail until later in this book; but, for now, let’s look at the relationship between the Laplace transform of a function and the Laplace transform of its derivative. Let’s start with the definition of the Laplace transform. We have WW! =r™r' = lim / e~st fifth. 4.Use integration by parts to evaluate Jjm^ e~sl fifth. (Let « = /{/) and dv — e '!dt.) After integrating by parts and evaluating the limit, you should see that Then, Thus, differentiation in the time domain simplifies to multiplication by s in the frequency domain. The final thing we look at in this project is how the Laplace transforms of fit] and its antiderivative are related. Let g(r) — f(u}dii. Then, ¦'o lim / ;-* caj"5.Use integration by parts to evaluate hrn^y e ’ g(t)dl. (Let u = gif) and dv = e dt. Note, by the way, that we have defined gif, du — fifth.) As you might expect, you should see that L|^(r)| = |-L[/(/)i. Integration in the time domain simplifies to division by s in ±e frequency domain.Laplace Transforms In the last few chapters, we have looked at several ways to use integration for solving real-world problems. For this next project, we are going to explore a more advanced application of integration: integral transforms. Specifically, we describe the Laplace transform and some of its properties. The Laplace transform is used in engineering and physics to simplify the computations needed to solve some problems. It takes functions expressed in terms of time and trons/orms them to functions expressed in terms of frequency. It turns out that, in many cases, the computations needed to solve problems in the frequency domain are much simpler than those required in the time domain. The Laplace transform is defined in terms of an integral as 7 e"" ft Note that the input to a Laplace transform is a function of time, /(/), and the output is a function of frequency, F(j), Although many real-world examples require the use of complex numbers (involving the imaginary number i = V—1), in this project we limit ourselves to functions of real numbers. Let's stan with a simple example. Here we calculate the Laplace transform of /(f) = t. We have This is an improper integral, so we express it in terms of a limit, which gives Now we use integration by pans to evaluate the integral. Note that we are integrating with respect to t, so we treat the variable s as a constant. We have u—tdv— dt du=dt v —— ye_ir. Then we obtain = + + =~K + °1 - = JinL[[-i,-]-±[e--lj] - c+ c = 0-0 + -L s“ _x 2* s 1. Calculate the Laplace transform of /(/) = 1. 3.Calculate the Laplace transform of /(/) = : (Note, you will have to integrate by parts twice.) Laplace transforms are often used to solve differential equations. Differential equations are not covered in detail until later in this book; but, for now, let’s look at the relationship between the Laplace transform of a function and the Laplace transform of its derivative. Let’s start with the definition of the Laplace transform. We have WW! =r™r' = lim / e~st fifth. 4.Use integration by parts to evaluate Jjm^ e~sl fifth. (Let « = /{/) and dv — e '!dt.) After integrating by parts and evaluating the limit, you should see that Then, Thus, differentiation in the time domain simplifies to multiplication by s in the frequency domain. The final thing we look at in this project is how the Laplace transforms of fit] and its antiderivative are related. Let g(r) — f(u}dii. Then, ¦'o lim / ;-* caj"5.Use integration by parts to evaluate hrn^y e ’ g(t)dl. (Let u = gif) and dv = e dt. Note, by the way, that we have defined gif, du — fifth.) As you might expect, you should see that L|^(r)| = |-L[/(/)i. Integration in the time domain simplifies to division by s in ±e frequency domain.Laplace Transforms In the last few chapters, we have looked at several ways to use integration for solving real-world problems. For this next project, we are going to explore a more advanced application of integration: integral transforms. Specifically, we describe the Laplace transform and some of its properties. The Laplace transform is used in engineering and physics to simplify the computations needed to solve some problems. It takes functions expressed in terms of time and trons/orms them to functions expressed in terms of frequency. It turns out that, in many cases, the computations needed to solve problems in the frequency domain are much simpler than those required in the time domain. The Laplace transform is defined in terms of an integral as 7 e"" ft Note that the input to a Laplace transform is a function of time, /(/), and the output is a function of frequency, F(j), Although many real-world examples require the use of complex numbers (involving the imaginary number i = V—1), in this project we limit ourselves to functions of real numbers. Let's stan with a simple example. Here we calculate the Laplace transform of /(f) = t. We have This is an improper integral, so we express it in terms of a limit, which gives Now we use integration by pans to evaluate the integral. Note that we are integrating with respect to t, so we treat the variable s as a constant. We have u—tdv— dt du=dt v —— ye_ir. Then we obtain = + + =~K + °1 - = JinL[[-i,-]-±[e--lj] - c+ c = 0-0 + -L s“ _x 2* s 1. Calculate the Laplace transform of /(/) = 1. 3.Calculate the Laplace transform of /(/) = : (Note, you will have to integrate by parts twice.) Laplace transforms are often used to solve differential equations. Differential equations are not covered in detail until later in this book; but, for now, let’s look at the relationship between the Laplace transform of a function and the Laplace transform of its derivative. Let’s start with the definition of the Laplace transform. We have WW! =r™r' = lim / e~st fifth. 4.Use integration by parts to evaluate Jjm^ e~sl fifth. (Let « = /{/) and dv — e '!dt.) After integrating by parts and evaluating the limit, you should see that Then, Thus, differentiation in the time domain simplifies to multiplication by s in the frequency domain. The final thing we look at in this project is how the Laplace transforms of fit] and its antiderivative are related. Let g(r) — f(u}dii. Then, ¦'o lim / ;-* caj"5.Use integration by parts to evaluate hrn^y e ’ g(t)dl. (Let u = gif) and dv = e dt. Note, by the way, that we have defined gif, du — fifth.) As you might expect, you should see that L|^(r)| = |-L[/(/)i. Integration in the time domain simplifies to division by s in ±e frequency domain.Laplace Transforms In the last few chapters, we have looked at several ways to use integration for solving real-world problems. For this next project, we are going to explore a more advanced application of integration: integral transforms. Specifically, we describe the Laplace transform and some of its properties. The Laplace transform is used in engineering and physics to simplify the computations needed to solve some problems. It takes functions expressed in terms of time and trons/orms them to functions expressed in terms of frequency. It turns out that, in many cases, the computations needed to solve problems in the frequency domain are much simpler than those required in the time domain. The Laplace transform is defined in terms of an integral as 7 e"" ft Note that the input to a Laplace transform is a function of time, /(/), and the output is a function of frequency, F(j), Although many real-world examples require the use of complex numbers (involving the imaginary number i = V—1), in this project we limit ourselves to functions of real numbers. Let's stan with a simple example. Here we calculate the Laplace transform of /(f) = t. We have This is an improper integral, so we express it in terms of a limit, which gives Now we use integration by pans to evaluate the integral. Note that we are integrating with respect to t, so we treat the variable s as a constant. We have u—tdv— dt du=dt v —— ye_ir. Then we obtain = + + =~K + °1 - = JinL[[-i,-]-±[e--lj] - c+ c = 0-0 + -L s“ _x 2* s 1. Calculate the Laplace transform of /(/) = 1. 3.Calculate the Laplace transform of /(/) = : (Note, you will have to integrate by parts twice.) Laplace transforms are often used to solve differential equations. Differential equations are not covered in detail until later in this book; but, for now, let’s look at the relationship between the Laplace transform of a function and the Laplace transform of its derivative. Let’s start with the definition of the Laplace transform. We have WW! =r™r' = lim / e~st fifth. 4.Use integration by parts to evaluate Jjm^ e~sl fifth. (Let « = /{/) and dv — e '!dt.) After integrating by parts and evaluating the limit, you should see that Then, Thus, differentiation in the time domain simplifies to multiplication by s in the frequency domain. The final thing we look at in this project is how the Laplace transforms of fit] and its antiderivative are related. Let g(r) — f(u}dii. Then, ¦'o lim / ;-* caj"5.Use integration by parts to evaluate hrn^y e ’ g(t)dl. (Let u = gif) and dv = e dt. Note, by the way, that we have defined gif, du — fifth.) As you might expect, you should see that L|^(r)| = |-L[/(/)i. Integration in the time domain simplifies to division by s in ±e frequency domain. Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.Evaluate the integrals. If the integral diverges, answer “diverges.” 374. 1dxxe Evaluate the integrals. If the integral diverges, answer “diverges.” 375. 01dxx Evaluate the integrals. If the integral diverges, answer “diverges.” 376. 01dx 1x Evaluate the integrals. If the integral diverges, answer “diverges.” 377. 01dx1x Evaluate the integrals. If the integral diverges, answer “diverges.” 378. 0dxx2+1 Evaluate the integrals. If the integral diverges, answer “diverges.” 379. 11dx 1 x 2 Evaluate the integrals. If the integral diverges, answer “diverges.” 380. 01Inxxdx Evaluate the integrals. If the integral diverges, answer “diverges.” 381. 0eIn(x)dx Evaluate the integrals. If the integral diverges, answer “diverges.” 382. 0xexdx Evaluate the integrals. If the integral diverges, answer “diverges.” 383. x ( x 2 +1 )2dx Evaluate the integrals. If the integral diverges, answer “diverges.” 384. 0exdx Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.Evaluate dx 5Vl -jv2 . (Be careful!) (Express your answer using three decimal places.)Evaluate (Express the answer in exact form.)Evaluate dx 2 (x2-l)3/2 Find the area of the region in the first quadrant between the curve y = and the x-axis.Find the area of the region bounded by the curve 7 y — the x-axis? and on the left by n — I.Find the area under the curve (X+1)3'2 bounded on the left by x = 3.Find the area under v = —-—~ 1 + x2in the first quadrant.Find the volume of the solid generated by revolving "3 about the x-axis the region under the curve y — from X = 1 to x = co.Find the volume of the solid generated by revolving about the y-axis the region under the curve v = 6e in the first quadrant.Find the volume of the solid generated by revolving about the x-axis the area under the curve y = 3e_A in the first quadrant.The Laplace transform of a continuous function over the defined by F(.v) — I e '1 J 0 (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F. f(x) = 1 f(x) = x /(a) = eos(2_v) f(x) = eax The Laplace transform of a continuous function over the defined by F(.v) — I e '1 J 0 (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F. f(x) = 1 f(x) = x /(a) = eos(2_v) f(x) = eax The Laplace transform of a continuous function over the defined by F(.v) — I e '1 J 0 (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F. f(x) = 1 f(x) = x /(a) = eos(2_v) f(x) = eax (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F. 401f(x) = 1 402402f(x) = x 403403/(a) = eos(2_v) 404f(x) = eax 405. Use the formula for arc length to show that the * *22* circumference of the circle x + y — 1 is 2k. A function is a probability density function if it satisfies / co = 1. The probability -oo‘ that a random variable x lies between a and b is given by P(a < je < Z>) = / J zt Show that /(jr) = r Oifx < 0 '.7e_7*ifx> 0 is a probability density function.Find the probability that x is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four- place decimal accuracy.For the fallowing exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. Je1 sin(x)e/x cannot be integrated by parts. / -—— Jx cannot be integrated using partial J x4 + 1 fractions. In numerical integration, increasing the number of points decreases the error. Integration by parts can always yield the integral. For the fallowing exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. Je1 sin(x)e/x cannot be integrated by parts. / -—— Jx cannot be integrated using partial J x4 + 1 fractions. In numerical integration, increasing the number of points decreases the error. Integration by parts can always yield the integral. For the fallowing exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. Je1 sin(x)e/x cannot be integrated by parts. / -—— Jx cannot be integrated using partial J x4 + 1 fractions. In numerical integration, increasing the number of points decreases the error. Integration by parts can always yield the integral. For the fallowing exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. Je1 sin(x)e/x cannot be integrated by parts. / -—— Jx cannot be integrated using partial J x4 + 1 fractions. In numerical integration, increasing the number of points decreases the error. Integration by parts can always yield the integral. For the following exercises, evaluate the integral using the specified method. 412- j sin(4x)Jx using integration by parts 414. y^Vxln(x)Jjv using integration by parts 415. /— dx using partial fractions ■ jr3 + 2jc“ - 5x - 6 a CAS For the following exercises, evaluate the integral using the specified method. 412- j sin(4x)Jx using integration by parts 414. y^Vxln(x)Jjv using integration by parts 415. /— dx using partial fractions ■ jr3 + 2jc“ - 5x - 6 a CAS For the following exercises, evaluate the integral using the specified method. 412- j sin(4x)Jx using integration by parts 414. y^Vxln(x)Jjv using integration by parts 415. /— dx using partial fractions ■ jr3 + 2jc“ - 5x - 6 a CAS For the following exercises, evaluate the integral using the specified method. 412- j sin(4x)Jx using integration by parts 414. y^Vxln(x)Jjv using integration by parts 415. /— dx using partial fractions ■ jr3 + 2jc“ - 5x - 6 a CAS For the following exercises, evaluate the integral using the specified method. 412- j sin(4x)Jx using integration by parts 414. y^Vxln(x)Jjv using integration by parts 415. /— dx using partial fractions ■ jr3 + 2jc“ - 5x - 6 a CAS For the following exercises, evaluate the integral using the specified method. 412- j sin(4x)Jx using integration by parts 414. y^Vxln(x)Jjv using integration by parts 415. /— dx using partial fractions ■ jr3 + 2jc“ - 5x - 6 a CAS For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals. 423IT] j + 2dx J 1 424[T] j e~^1}dx J0 425[T] f For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals. 423IT] j + 2dx J 1 424[T] j e~^1}dx J0 425[T] f For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals. 423.IT] j + 2dx J 1 424.[T] j e~^1}dx J0 425.[T] f For the following exercises, evaluate the integrals, if possible. 426, J -^dx, for what values of n does this integral J _A converge or diverge? oo dx For the following exercises, evaluate the integrals, if possible. 426, J -^dx, for what values of n does this integral J _A converge or diverge? oo dx For the following exercises, consider the gamma function e '>/;l dy. A 428. Show that r(a) — (a — l)r(a — 1). 429* Extend to show that T(cf) — (a — 1)!, assuming a is a positive integer The fastest car in the world, the Bugati Veyran, can reach a top speed of 408 km/h. The graph represents its velocity. 4 420- 370- 320-■ 270 - f 220170 ■■ 120 70 —i 1 1 1 1 • h*- 02:00 02:40 03:20 04:00 mm:ss [T] Use the graph to estimate the velocity every 20 sec and fit to a graph of the form v(f) = a exp l,A sin(cx) + d. {Hint: Consider the time units.)[T] Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.Determine the order of the following differential equations. 1. y+y=3y2 Determine the order of the following differential equations. 2. (y)2=y+2y Determine the order of the following differential equations. y+yy=3x2 Determine the order of the following differential equations. y=y+3t2 Determine the order of the following differential equations. dydt=t Determine the order of the following differential equations. 6. dydx+d2ydx2=3x4 Determine the order of the following differential equations. 7. ( dy dx)2+8dydt+3y=4t Verify that the following functions are solutions to the given differential equation. 8. y=x33solvesy'=x2 Verify that the following functions are solutions to the given differential equation. 9. y=2ex+x1solvesy'=xy Verify that the following functions are solutions to the given differential equation. 10. y=e3xex2solvesy=3y+ex Verify that the following functions are solutions to the given differential equation. 11. y=11xsolvesy=y2 Verify that the following functions are solutions to the given differential equation. 12. y=ex2/2solvesxy=1 Verify that the following functions are solutions to the given differential equation. 13. y=4+Inxsolvesxy=1 Verify that the following functions are solutions to the given differential equation. 14. y=3x+xInxsolvesy'=Inx Verify that the following functions are solutions to the given differential equation. 15. y=2exx1solvesy=y+x Verify that the following functions are solutions to the given differential equation. 16. y=e3+sinx2cosx2solves y=cosx+y Verify that the following functions are solutions to the given differential equation. 17. y=ecosxsolvesy=ysinx Verify the following general solutions and find the particular solution. 18. Find the particular solution to the differential equation y' = 4x2that passes through (-3, -30), given that y=C+4x33is a general solution.Verify the following general solutions and find the particular solution. 19. Find the particular solution to the differential equation y' = 3x3that passes through (1, 4.75), given that y=C+3x44 is a general solution.Verify the following general solutions and find the particular solution. 20. Find the particular solution to the differential equation y' = 3x2y that passes through (0, 12), given that y=Cex3 is a general solution.Verify the following general solutions and find the particular solution. 21. Find the particular solution to the differential equation y' = 2xy that passes through (0,12) , given that y=Cex2 is a general solution.Verify the following general solutions and find the particular solution. 22. Find the particular solution to the differential equation y'= (2xy)2 that passes through (1,12) , given that y = 3c+4x3 is a general solution.Verify the following general solutions and find the particular solution. 23. Find the particular solution to the differential equation y' x2= y that passes through (1,2e) , given that y = Ce-1/x is a general solution.Verify the following general solutions and find the particular solution. 24. Find the particular solution to the differential equation 8dxdt = -2cos(2t) - cos(4t) that passes through ( , ), given that x = C 18 sin(2t) 132 sin(4t) is a general solution.Verify the following general solutions and find the particular solution. 25. Find the particular solution to the differential equation dudt=tanuthat passes through (1,2) , given that u=sin1(eC+t) is a general solution.Verify the following general solutions and find the particular solution. 26. Find the particular solution to the differential equation dydt=e(t+y)that passes through (1, 0) given that y = -In(C - et) is a general solution.Verify the following general solutions and find the particular solution. 27. Find the particular solution to the differential equation y' (1 - x2) = 1 + y that passes through (0, -2), given that y=Cx+11x1 is a general solution.For the following problems, find the general solution to the differential equation. 28. y=3x+ex For the following problems, find the general solution to the differential equation. 29. y=Inx+tanx For the following problems, find the general solution to the differential equation. y=sinxecosx For the following problems, find the general solution to the differential equation. 31. y=4x For the following problems, find the general solution to the differential equation. 32. y=sin1(2x)For the following problems, find the general solution to the differential equation. y=2tt2+16 For the following problems, find the general solution to the differential equation. 34. x=cotht+Int+3t2 For the following problems, find the general solution to the differential equation. 35. x=t4+t For the following problems, find the general solution to the differential equation. 36. y=y For the following problems, find the general solution to the differential equation. 37. y=yx Solve the following initial-value problems starting from y(t = 0) = 0 and y (t = 0) = 1. Draw both solutions on the same graph. 38. dydt=2t Solve the following initial-value problems starting from y(t = 0) = 0 and y (t = 0) = 1. Draw both solutions on the same graph. 39. dydt=t Solve the following initial-value problems starting from y(t = 0) = 0 and y (t = 0) = 1. Draw both solutions on the same graph. 40. dydt=2y Solve the following initial-value problems starting from y(t = 0) = 0 and y (t = 0) = 1. Draw both solutions on the same graph. 41. dydt=y Solve the following initial-value problems starting from y(t = 0) = 0 and y (t = 0) = 1. Draw both solutions on the same graph. 42. dydt=2 Solve the following initial-value problems starting from y0= 10. At what time does y increase to 100 or drop to 1? 43. dydt=4t Solve the following initial-value problems starting from y0= 10. At what time does y increase to 100 or drop to 1? 44. dydt=4y Solve the following initial-value problems starting from y0= 10. At what time does y increase to 100 or drop to 1? 45. dydt=2y Solve the following initial-value problems starting from y0= 10. At what time does y increase to 100 or drop to 1? 46. dydt=e4t Solve the following initial-value problems starting from y0= 10. At what time does y increase to 100 or drop to 1? 47. dydt=e4t Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 48. [T] y ' = y(x)Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 49. [T] xy' = y Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 50. [T] y' = t3 Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 51. [T] y' = x + y (Hint: y = Cex x 1 is the general solution)Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 52. [T] y' = x In x + sin x Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 53. Find the general solution to describe the velocity of a ball of mass 1 1b that is thrown upward at a rate a ft/sec.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 54. In the preceding problem, if the initial velocity of the ball thrown into the air is a = 25 ft/s, write the particular solution to the velocity of the ball. Solve to find the time when the ball hits the ground.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 55. You throw two objects with differing masses m1and m2upward into the air with the same initial velocity a ft/s. What is the difference in their velocity after 1 second?Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 56. [T] You throw a ball of mass 1 kilogram upward with a velocity of a = 25 m/s on Mars, where the force of gravity is g = -3.711 m/s2. Use your calculator to approximate how much longer the ball is in the air on Mars.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 57. [T] For the previous problem, use your calculator to approximate how much higher the ball went on Mars.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 58. [T] A car on the freeway accelerates according to a = 15 cos( t), where t is measured in hours. Set up and solve the differential equation to determine the velocity of the car if it has an initial speed of 51 mph. After 40 minutes of driving, what is the driver’s velocity?Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 59. [T] For the car in the preceding problem, find the expression for the distance the car has traveled in time t, assuming an initial distance of 0. How long does it take the car to travel 100 miles? Round your answer to hours and minutes.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 60. [T] For the previous problem, find the total distance traveled in the first hour.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 61. Substitute y = Be3tinto y'— y = 8e3tto find a particular solution.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 62. Substitute y = a cos(2t) + b sin(2t) into y' + y = 4sin(2t) to find a particular solution.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 63. Substitute y=a+bt+ct2into y'+y= 1 +t2to find a particular solution.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 64. Substitute y = aetcost + betsin t into y' = 2etcos t to find a particular solution.Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from y(t = 0) = -10 to y(t = 0) = 10 increasing by 2. Is there some critical point where the behavior of the solution begins to change? 65. Solve y' = ektwith the initial condition y(0) = 0 and solve y’ = 1 with the same initial condition. As k approaches 0, what do you notice?For the following problems, use the direction field below from the differential equation y' = 2y. Sketch the graph of the solution for the given initial conditions. 66. y(0) = 1 For the following problems, use the direction field below from the differential equation y' = 2y. Sketch the graph of the solution for the given initial conditions. 67. y(0) = 0 For the following problems, use the direction field below from the differential equation y' = 2y. Sketch the graph of the solution for the given initial conditions. 68. y (0) = 1 For the following problems, use the direction field below from the differential equation y' = 2y. Sketch the graph of the solution for the given initial conditions. 69. Are there any equilibria? What are their stabilities?For the following problems, use the direction field below from the differential equation y'= y 2-2y. Sketch the graph of the solution for the given initial conditions. 70. y(0) = 3 For the following problems, use the direction field below from the differential equation y'= y 2-2y. Sketch the graph of the solution for the given initial conditions. 71. y(0) = 1 For the following problems, use the direction field below from the differential equation y'= y 2-2y. Sketch the graph of the solution for the given initial conditions. 72. y(0) = 1 For the following problems, use the direction field below from the differential equation y'= y 2-2y. Sketch the graph of the solution for the given initial conditions. 73. Are there any equilibria? What are their stabilities?Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field? 74. y=t3 Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field? 75. y=et Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field? 76. dydx=x2cosx Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field? 77. dydt=tet Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field? 78. dxdt=cosh(t)Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have? 79. y=y21 Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have? 80. y=yx Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have? 81. y=1y2x2 Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have? 82. y=t2siny Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have? 83. y=3y+xy Match the direction field with the given differential equations. Explain your selections. 84. y=3y Match the direction field with the given differential equations. Explain your selections. 85. y=3t Match the direction field with the given differential equations. Explain your selections. 86. y=et Match the direction field with the given differential equations. Explain your selections. 87. y=12y+t Match the direction field with the given differential equations. Explain your selections. 89. y=ty Match the direction field with the given differential equations. Explain your selections. 89. y=tsiny Match the direction field with the given differential equations. Explain your selections. 90. y=tcosy Match the direction field with the given differential equations. Explain your selections. 91. y=ttany Match the direction field with the given differential equations. Explain your selections. 92. y=sin2y Match the direction field with the given differential equations. Explain your selections. 93 y=y2t3 Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 94. y' = -3y, y(0) = 1 Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 95. y' = t2 WEstimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 96. y' = 3t - y, y(0) = 1. Exact solution is y = 3t + 4e-t 3 Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 97. y' = y + t2, y(0) = 3. Exact solution is y = 5et2 t2 2t Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 98. y' = 2t, y(0) = 0 Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 99. [T] y' = e(x+y), + y(0) = 1. Exact solution is y = In(e + 1 ex)Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 100. y' = y2In(x + 1), y(0) = 1. Exact solution is y=1(x+1)(In(x+1)-1)Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 101. y' = 2x, y(0) = 0, Exact solution is y=2x1In(2)Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 102. y' = y, y(0) = 1. Exact solution is y = ex.Estimate the following solutions using Euler’s method with n = 5 steps over the interval t = [0, 1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? 103. y' = 5t, y(0) = 2. Exact solution is y=52t22 Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 104. Show that, by our assumption that the total population size is constant (S +I = N), you can reduce the system to a single differential equation in I: I' = c(N-I)I-rI.Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 105. Assuming the parameters are c = 0.5, N = 5, and r = 0.5, draw the resulting directional field.Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 106. [T] Use computational software or a calculator to compute the solution to the initial-value problem y'= ty, y(0) = 2 using Euler’s Method with the given step size h. Find the solution at t = 1. For a hint, here is “pseudo-code” for how to write a computer program to perform Euler’s Method for y' = f(t, y). y(0) = 2: Create function f(t, y) Define parameters y( 1) =y0, t(0) = 0, step size N, and total number of steps, N Write a for loop: for k = 1 to N fn = f(t(k), y(k)) y(k+ 1) = y(k) + h*fn t(k+1) = t(k)+ h Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 107. Solve the initial-value problem for the exact solution.Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 108. Draw the directional field Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 109. h=1 Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 110. [T] h = 10 Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 111. [T] h = 100 Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 112. [T] h = 1000 Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are S' = rI cSI and I' =cSI rI. Here c represents the contact rate and r is the recovery rate. 113. [T] Evaluate the exact solution at t = 1. Make a table of errors for the relative error between the Euler’s method solution and the exact solution. How much does the error change? Can you explain?Consider the initial-value problem y' = 2y, y(0) = 2. 114. Show that y = 2e-2x solves this initial-value problem.Consider the initial-value problem y' = 2y, y(0) = 2. 115. Draw the directional field of this differential equation.Consider the initial-value problem y' = 2y, y(0) = 2. 116. [T] By hand or by calculator or computer, approximate the solution using Euler’s Method at I = 10 using h = 5.Consider the initial-value problem y' = 2y, y(0) = 2. 117. [T] By calculator or computer, approximate the solution using Euler’s Method at t = 10 using h = 100.Consider the initial-value problem y' = 2y, y(0) = 2. 118. [T] Plot exact answer and each Euler approximation (for h = 5 and h = 100) at each h on the directional field. What do you notice?Solve the following initial-value problems with the initial condition y0= 0 and graph the solution. 119. dydt=y+1 Solve the following initial-value problems with the initial condition y0= 0 and graph the solution. 120. dydt=y1 Solve the following initial-value problems with the initial condition y0= 0 and graph the solution. 121. dydt=y+1 Solve the following initial-value problems with the initial condition y0= 0 and graph the solution. 122. dydt=y1 Find the general solution to the differential equation. 123. x2y=(x+1)y Find the general solution to the differential equation. 124. y=tan(y)x Find the general solution to the differential equation. 125. y=2xy2 Find the general solution to the differential equation. 126. dydt=ycos(3t+2)Find the general solution to the differential equation. 127. 2xdydx=y2 Find the general solution to the differential equation. 128. y=eyx2 Find the general solution to the differential equation. 129. (1+x)y=(x+2)(y1)Find the general solution to the differential equation. 130. dxdt=3t2(x2+4)

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