Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample a. ddx(e5)=5e4 b. The Quotient Rule must be used to evaluate ddx(x2+3x+2x) c. ddx(1x5)=15x4 d. ddx(x3ex)=3x2ezHigher-order derivatives Find f(x), f(x), and f(x). 62. f(x)=1xHigher-order derivatives Find f(x),f(x), f(x), and f(x). 71.f(x)=x2(2+x3)First and second derivatives Find f(x) and f(x). 64. f(x)=xx+2First and second derivatives Find f(x) and f(x). 65. f(x)=x27xx+1Tangent lines Suppose f(2) = 2 and f(2) = 3. Let g(x)=x2f(x) and h(x)=f(x)x3. a. Find an equation of the line tangent to y = g(x) at x = 2. b. Find an equation of the line tangent to y = h(x) at x = 2.The Witch of Agnesi The graph of y=a3x2+a2, where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi). a. Let a = 3 and find an equation of the line tangent to y=27x2+9 at x = 2. b. Plot the function and the tangent line found in part (a).Derivatives from a table Use the following table to find the given derivatives. 74. ddx(f(x)g(x))|x=1Derivatives from a table Use the following table to find the given derivatives. 75.ddx(f(x)g(x))|x=2Derivatives from a table Use the following table to find the given derivatives. 76. ddx(xf(x))|x=3Derivatives from a table Use the following table to find the given derivatives. 77. ddx(f(x)x+2)|x=4Derivatives from a table Use the following table to find the given derivatives. 78. ddx(xf(x)g(x))|x=4Derivatives from a table Use the following table to find the given derivatives. 79. ddx(f(x)g(x)x)|x=4Flight formula for Indian spotted owlets The following table shows the average body mass m(t)(in g) and average wing chord length w(t) (in mm), along with the derivatives m(t) and w(t), of f-week-old Indian spotted owlets. The flight formula functionf(t)w(t)/m(t), which is the ratio of wing chord length to mass, is used to predict when these fledglings are old enough to fly. The values of fare less than 1, but approach 1 as t increases. When f is close to 1. the fledglings are capable of flying, which is important for determining when rescued fledglings can be released back into the wild. 82.State the units associated with the following derivatives and state the physical meaning of each derivative a. m'(t) b. w(t) c. f(t)Flight formula for Indian spotted owlets The following table shows the average body mass m(t) (in g) and average wing chord length w(t) (in mm), along with the derivatives m(t) andw(t), of t-week-old Indian spotted owlets. The flight formula functionf(t)=w(t)/m(t),. which is the ratio of wing chord length to mass, is used to predict when these fledglings are old enough to fly. The values of fare less than 1, but approach 1 as f increases. When f is close to 1. the fledglings are capable of flying, which is important for determining when rescued fledglings can be released back into the wild. 83.Complete the following step to examine the behavior of the flight formula. a. Sketch an approximate graph of y=f(t) by plotting and connecting the points (1,f(l)),(1.5,/(1.5)),...,(6.5,f(6.5)) with a smooth curve. b. For what value of t does f appear to be changing most rapidly? Round f to the nearest whole number. c. For the value of f found in part (b). use the table and the Quotient Rule to find f(t) Describe what is happening to the bird at this stage in its life. d Use your graph of f to predict what happens to f(t) as f grows larger and confirm your prediction by evaluating f(6.5) using the Quotient Rule Describe what is happening in the physical development of the fledglings as f grows larger.84EAssume both the graphs of f and g pass through the point (3,2),f(3)=5, and g(3)=10.Ifp(x)=f(x)g(x) and q(x)=f(x)/g(x), find the following derivatives. 85.q(3)86EDerivatives from graphs Use the figure to find the following derivatives. 87.ddx(f(x)g(x))|x=488EDerivatives from graphs Use the figure to find the following derivatives. 89.ddx(xg(x))|x=2Derivatives from graphs Use the figure to find the following derivatives. 90.ddx(x2f(x))|x=291EDerivatives from tangent lines Suppose the line tangent to the graph of f at x = 2 is y = 4x + l and suppose y = 3x 2 is the line tangent to the graph of g at x = 2. Find an equation of the line tangent to the following curves at x = 2 a. y = f(x)g(x) b. y=f(x)g(x)Explorations and Challenges Avoiding tedious work Given that q(x)5x8+6x5+5x4+3x2+20x+10010x10+8x9+6x5+6x2+4x+2 find q(0) without computing q(x).(Hint: Evaluate f(0),f(0),g(0), and g(0) where f is the numerator of q and g is the denominator of q.)Given that p(x)=(5ex+10x5+20x3+100x2+5x+20)(10x5+40x3+20x2+4x+10), find p'(0) without computing p(x).Means and tangents Suppose f is differentiable on an interval containing a and b, and let p(a,f(a)) and Q(b,f(b)) be distinct points on the graph of f. Let c be the x-coordinate of the point at which the lines tangent to the curve at P and Q intersect, assuming the tangent lines are not parallel (see figure). a. If f(x)=x2, show that c=(a+b)/2, the arithmetic mean of a and b, for real numbers a and b. b. If f(x)=x, show that c=ab, the geometric mean of a and b, for a0 and b0. c. If f(x)=1/x, show that c=2ab/(a+b), the harmonic mean of a and b. for a0, and b0. d.Find an expression for c in terms of a and b for any (differentiable) function f whenever c exists.Proof of the Quotient Rule Let F = f/g be the quotient of two functions that are differentiable at x. a. Use the definition of F to show that ddx(f(x)g(x))=limh0f(x+h)g(x)f(x)g(x+h)hg(x+h)g(x). b. Now add f(x)g(x) + f(x)g(x) (which equals 0) to the numerator in the preceding limit to obtain limh0f(x+h)g(x)f(x)g(x)+f(x)g(x)f(x)g(x+h)hg(x+h)g(x) Use this limit to obtain the Quotient Rule. c. Explain why F = (f/g) exists, whenever g(x) 0.Product Rule for the second derivative Assuming the first and second derivatives of f and g exist at x, find a formula for d2dx2(f(x)g(x)).One of the Leibniz Rules One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let (fg)(n) denote the nth derivative of the product fg, for n 1. a. Prove that (fg)(2) = fg + 2 fg + fg. Prove that, in general. (fg)(n)=k=0n(nk)f(k)g(nk), b. where (nk)=n!k!(nk)! are the binomial coefficients. c. Compare the result of (b) to the expansion of (a + b)n.Product Rule for three functions Assume that f, g, and h are differentiable at x. a. Use the Product Rule (twice) to find a formula for ddx(f(x)g(x)h(x)). b. Use the formula in (a) to find ddx(e2x(x1)(x+3)).Evaluate limx0tan2xxAt what point on the interval [0,2] does the graph of f(x)=sinx have tangent lines with positive slopes? At what points on the Interval [0,2] is cosx0? Explain the connectionThe formulas for ddx(cotx),ddx(secx), and ddx(cscx) can be determined using the Quotient Rule. Why?4QCWhy is it not possible to evaluate limx0sinxx by direct substitution?How is limx0sinxx used in this section?Explain why the Quotient Rule is used to determine the derivative of tan x and cot x.How can you use the derivatives ddx(sinx)=cosx, ddx(tanx)=sec2x, and ddx(secx)=secxtanxto remember the derivatives of cos x, cot x, and csc x?Let f(x) = sin x. What is the value of f()?Find the value of ddx(tanx)|x=3Find an equation of the line tangent to the curve y=sinx at x=0.Where does the graph of sin x have a horizontal tangent line? Where does cos x have a value of zero? Explain the connection between these two observations.Find d2dx2(sinx+cosx)Find d2dx2(secx).Trigonometric limits Use Theorem 3.11 to evaluate the following limits. 7. limx0sin3xxTrigonometric limits Use Theorem 3.11 to evaluate the following limits. 8. limx0sin5x3xTrigonometric limits Use Theorem 3.11 to evaluate the following limits. 9. limx0sin7xsin3xTrigonometric limits Use Theorem 3.11 to evaluate the following limits. 10. limx0sin3xtan4xTrigonometric limits Use Theorem 3.11 to evaluate the following limits. 11. limx0tan5xxTrigonometric limits Use Theorem 3.11 to evaluate the following limits. 12. lim0cos21Trigonometric limits Use Theorem 3.11 to evaluate the following limits. 13. limx0tan7xsinxTrigonometric limits Use Theorem 3.11 to evaluate the following limits. 14. lim0sec1Trigonometric limits Use Theorem 3.11 to evaluate the following limits. 15. limx2sin(x2)x24Trigonometric limits Use Theorem 3.11 to evaluate the following limits. 16. limx3sin(x+3)x2+8x+15Trigonometric limits Evaluate the following limits or state that they do not exist. 51. limx0sinaxsinbx, where a and b are constants with b 0Trigonometric limits Evaluate the following limits or state that they do not exist. 50. limx0sinaxbx, where a and b are constants with b 0Calculating derivatives Find dy/dx for the following functions. 17. y = sin x + cos xCalculating derivatives Find dy/dx for the following functions. 18. y = 5x2 + cos xCalculating derivatives Find dy/dx for the following functions. 19. y = ex sin x;Calculating derivatives Find dy/dx for the following functions. 20. y = sin x + 4e0.5xCalculating derivatives Find dy/dx for the following functions. 21. y = x sin xCalculating derivatives Find the derivative of the following functions. 28.y=ex(cosx+sinx)Calculating derivatives Find dy/dx for the following functions. 23. y=cosxsinx+1Calculating derivatives Find dy/dx for the following functions. 24. y=1sinx1+sinxCalculating derivatives Find dy/dx for the following functions. 25. y = sin x cos xCalculating derivatives Find the derivative of the following functions. 32.y=asinx+bcosxasinxbcosx a and b are nonzero constantsCalculating derivatives Find dy/dx for the following functions. 27. y = cos2 xCalculating derivatives Find dy/dx for the following functions. 28. y=xsinx1+cosxCalculating derivatives Find the derivative of the following functions. 35.y=w2sinw+2wcosw2sinwCalculating derivatives Find the derivative of the following functions. 36.y = x3 cos x + 3x2 sin x + 6x cos x 6 sin xCalculating derivatives Find dy/dx for the following functions. 57. y = cos x sin xCalculating derivatives Find dy/dx for the following functions. 58. y=12+sinxCalculating derivatives Find dy/dx for the following functions. 56. y=sinx1+cosxCalculating derivatives Find the derivative of the following functions. 40.y=sinx+cosxexCalculating derivatives Find dy/dx for the following functions. 61. y=1cosx1+cosxDerivatives involving other trigonometric functions Find the derivative of the following functions. 32. y = tan x + cot xDerivatives involving other trigonometric functions Find the derivative of the following functions. 33. y = sec x + csc xDerivatives involving other trigonometric functions Find the derivative of the following functions. 34. y = sec x tan xCalculating derivatives Find the derivative of the following functions. 45.y=excscxDerivatives involving other trigonometric functions Find the derivative of the following functions. 36. y=tanw1+tanwDerivatives involving other trigonometric functions Find the derivative of the following functions. 37. y=cotx1+cscxDerivatives involving other trigonometric functions Find the derivative of the following functions. 38. y=tant1+sectDerivatives involving other trigonometric functions Find the derivative of the following functions. 39. y=1seczcsczDerivatives involving other trigonometric functions Find the derivative of the following functions. 40. y = csc2 1Calculating derivatives Find the derivative of the following functions. 51.y=xcosxsinxDerivatives of other trigonometric functions Verify the following derivative formulas using the Quotient Rule. 29. ddx(cotx)=csc2xDerivatives of other trigonometric functions Verify the following derivative formulas using the Quotient Rule. 30. ddx(secx)=secxtanxDerivatives of other trigonometric functions Verify the following derivative formulas using the Quotient Rule. 31. ddx(cscx)=cscxcotxVelocity of an oscillator An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30 (sin t 1), where t 0 is measured in seconds and y is positive in the upward direction. a. Graph the position function, for 0 t 10. b. Find the velocity of the oscillator, v(t) = y(t). c. Graph the velocity function, for 0 t 10. d. At what times and positions is the velocity zero? e. At what times and positions is the velocity a maximum? f. The acceleration of the oscillator is a(t) = v(t). Find and graph the acceleration function.Damped sine wave The graph of f(t)=etsin t is an example of a damped sine wave; it is used in a variety of applications, such as modeling the vibrations of a shock absorber. a. Use a graphing utility to graph f and explain why this curve is called a damped sine wave. b. Compute f'(t) and use it to determine where the graph of f has a horizontal tangent. c. Evaluate limtet sin t by using the Squeeze Theorem. What does the result say about the oscillations of this damped sine wave?Second-order derivatives Find y for the following functions. 41. y = x sin xSecond derivatives Find y for the functions. 58.y=x2cosxSecond-order derivatives Find y for the following functions. 43. y = ex sin xSecond-order derivatives Find y for the following functions. 44. y=12excosxSecond-order derivatives Find y for the following functions. 45. y = cot xSecond-order derivatives Find y for the following functions. 46. y = tan xSecond-order derivatives Find y for the following functions. 47. y = sec x csc xSecond-order derivatives Find y for the following functions. 48. y = cos sinExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. ddx(sin2x)=cos2x. b. d2dx2(sinx)=sinx. c. d4dx4(cosx)=cosx. d. The function sec x is not differentiable at x = /2.Trigonometric limits Evaluate the following limits or state that they do not exist. 52. limx/2cosxx(/2)Trigonometric limits Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.) 67.limx/4tanx1x/4Trigonometric limits Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.) 68.limh0sin(6+h)12hTrigonometric limits Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.) 69.limh0cos(6+h)32h70ETrigonometric limits Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.) 71.limh0tan(56+h)+13hEquations of tangent lines a. Find an equation of the line tangent to the following curves at the given value of x. b. Use a graphing utility to plot the curves and the tangent line. 62. y=4sinxcosx;x=3Equations of tangent lines a. Find an equation of the line tangent to the following curves at the given value of x. b. Use a graphing utility to plot the curves and the tangent line. 63. y=1+2sinx;x=6Equations of tangent lines a. Find an equation of the line tangent to the following curves at the given value of x. b. Use a graphing utility to plot the curves and the tangent line. 64. y=cscx;x=4Equations of tangent lines a. Find an equation of the line tangent to the following curves at the given value of x. b. Use a graphing utility to plot the curves and the tangent line. 65. y=cosx1cosx;x=3Locations of tangent lines a. For what values of x does g(x) = x sin x have a horizontal tangent line? b. For what values of x does g(x) = x sin x have a slope of 1?Locations of horizontal tangent lines For what values of x does f(x) = x 2 cos x have a horizontal tangent line?Matching Match the graphs of the functions in ad with the graphs of their derivatives in AD.A differential equation A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y(t) + y(t) = 0 (see Chapter 8). a. Show that y = A sin t satisfies the equation for any constant A. b. Show that y = B cos t satisfies the equation for any constant B. c. Show that y = A sin t + B cos t satisfies the equation for any constants A and B.Using identities Use the identity sin 2x = 2 sin x cos x to find ddx(sin2x). Then use the identity cos 2x = cos2 x sin2 x to express the derivative of sin 2x in terms of cos 2x.81E82EProof of ddx(cosx)=sinx Use the definition of the derivative and the trigonometric identity cos(x+h)=cosxcoshsinxsinh to prove that ddx(cosx)=sinx.Continuity of a piecewise function Let f(x)={3sinxxifx0aifx=0. For what values of a is f continuous?Continuity of a piecewise function Let g(x)={1cosx2xifx0aifx=0 For what values of a is g continuous?86E87E88E89E90EDoes the speedometer in your car measure average or instantaneous velocity?For an object moving along a line, is it possible for its velocity to increase while its speed decreases? Is it possible for its velocity to decrease while its speed increases? Give an example to support your answers.Describe the velocity of an object that has a positive constant acceleration. Could an object have a positive acceleration and a decreasing speed?In Example 3, does the rock have a greater speed at t = 1 or t = 3? Example 3 Motion in a Gravitational Field Suppose a stone is thrown vertically upward with an initial velocity of 64 ft/s from a bridge 96 ft above a river. By Newtons laws of motion, the position of the stone (measured as the height above the river) after t seconds is s(t)=16t2+64t+96, where s = 0 is the level of the river (Figure 3.45a). The position function in Example 3 is derived in Section 6.1. Once again we mention that the graph of the position function is not the path of the stone. a. Find the velocity and acceleration functions. b. What is the highest point above the river reached by the stone? c. With what velocity will the stone strike the river?5QCIn Example 5, what happens to the average cost as the number of items produced increases from x = 1 to x = 100? Example 5 Average and Marginal Costs Suppose the cost of producing x items is given by the function (Figure 3.51) C(x)=0.2x2+50x+100, for 0 x 1000. a. Determine the average and marginal cost functions. b. Determine the average and marginal cost for the first 100 items and interpret these values. c. Determine the average and marginal cost for the first 900 items and interpret these values.Explain the difference between the average rate of change and the instantaneous rate of change of a function f.Complete the following statement. If dydx is large, then small changes in x result in relatively _____ changes in the value of y.Complete the following statement: If dydx is small, then small changes in x result in relatively _____ changes in the value of y.Suppose the function s(t) represents the position (in feet) of a stone t seconds after it is thrown directly upward from 6 feet above Earth's surface. a. Is the stones acceleration positive or negative when the stone is moving upward? Explain. b. What is the value of the instantaneous velocity v(t) when the stone reaches its highest point? Explain.Suppose w(t) is the weight (in pounds) of a golden retriever puppy t weeks after it is born. Interpret the meaning of w(15) = 1.75.What is the difference between the velocity and speed of an object moving in a straight line?Define the acceleration of an object moving in a straight line.An object moving along a line has a constant negative acceleration. Describe the velocity of the object.The speed of sound (in m/s) in dry air is approximated by the function v(T) = 331 + 0.6T, where T is the air temperature (in degrees Celsius). Evaluate v(T) and interpret its meaning.At noon, a city park manager starts filling a swimming pool. If V(t) is the volume of water (in ft3) in the swimming pool t hours after noon, then what does dV/dt represent?Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. a. Determine the average velocity of the car during the first 45 minutes of the trip. b. Find the average velocity of the car over the interval [0.25, 0.75]. Is the average velocity a good estimate of the velocity at 9:30 A.M.? c. Find the average velocity of the car over the interval [1.75, 2.25]. Estimate the velocity of the car at 11:00 A.M. and determine the direction in which the patrol car is moving. d. Describe the motion of the patrol car relative to the patrol station between 9:00 A.M. and noon.Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 t 1.5). b. Calculate the average velocity of the airliner between 1:30 P.M. and 2:30 P.M. (7.5 t 8.5). c. At what time(s) is the velocity 0? Give a plausible explanation. d. Determine the velocity of the airliner at noon (t = 6) and explain why the velocity is negative.13EExplain why a decreasing demand function has a negative elasticity function.Position, velocity, and acceleration Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t). where s is measured in feet, with s 0 corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at t = 1. d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? 11. f(t) = t2 4t; 0 t 5Position, velocity, and acceleration Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t). where s is measured in feet, with s 0 corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at t = 1. d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? 12. f(t) = t2 + 4t 3; 0 t 5Position, velocity, and acceleration Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t). where s is measured in feet, with s 0 corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at t = 1. d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? 13. f(t) = 2t2 9t + 12; 0 t 3Position, velocity, and acceleration Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t). where s is measured in feet, with s 0 corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at t = 1. d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? 14. f(t) = 18t 3t2; 0 t 8Position, velocity, and acceleration Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t). where s is measured in feet, with s 0 corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at t = 1. d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? 15. f(t) = 2t3 21t2 + 60t; 0 t 5Position, velocity, and acceleration Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t). where s is measured in feet, with s 0 corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at t = 1. d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? 16. f(t) = 6t3 + 36t2 54t; 0 t 4A dropped stone on Earth The height (in feet) of a stone dropped from a bridge 64 feet above a river at t = 0 seconds is given by s(t)=16t2+64. Find the velocity of the stone and its speed when it hits the water.A dropped stone on Mars A stone is dropped off the edge of a 54-ft cliff on Mars, where the acceleration due to gravity is about 12 ft/s2. The height (in feet) of the stone above the ground t seconds after it is dropped is s(t)=6t2+54. Find the velocity of the stone and its speed when it hits the ground.Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t)=16t2+32t+48. a. Determine the velocity v of the stone after t seconds b. When does the stone reach its highest point? c. What is the height of the stone at the highest point? d. When does the stone strike the ground? e. With what velocity does the stone strike the ground? f. On what intervals is the speed increasing?Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19 6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t)=4.9t2+19.6+24.5. a. Determine the velocity v of the stone after t seconds. b. When does the stone reach its highest point? c. What is the height of the stone at the highest point? d. When does the stone strike the ground? e. With what velocity does the stone strike the ground? f. On what intervals is the speed increasing?A stone thrown vertically on Mars Suppose a stone is thrown vertically upward from the edge of a cliff on Mars (where the acceleration due to gravity is only about 12 ft/s2) with an initial velocity of 64 ft/s from a height of 192 ft above the ground. The height s of the stone above the ground after t seconds is given by s = 6t2 + 64t + 192. a. Determine the velocity v of the stone after t seconds. b. When does the stone reach its highest point? c. What is the height of the stone at the highest point? d. When does the stone strike the ground? e. With what velocity does the stone strike the ground?Maximum height Suppose a baseball is thrown vertically upward from the ground with an initial velocity of v0ft/s. The approximate height of the ball (in feet) above the ground after t seconds is given by s(t)=16t2+v0t. a. What is the height of the ball at its highest point? b. With what velocity does the bail strike the ground?Initial velocity Suppose a baseball is thrown vertically upward from the ground with an initial velocity of v0ft/s. Its height above the ground after t seconds is given by s(t)=16t2+v0t. Determine the initial velocity of the ball if it reaches a high point of 128 ft.Population growth in Washington The population of the state of Washington (in millions) from 2010 (t = 0) to 2016 (t = 6) is modeled by the polynomial p(t)=0.0078t2+0.028t+6.73. a. Determine the average growth rate from 2010 to 2016. b. What was the growth rate for Washington in 2011 (t = 1) and 2015 (t = 5)? c. Use a graphing utility to graph p for 0 t 6. What does this graph tell you about population growth in Washington during the period of time from 2010 to 2016?Average and marginal cost Consider the following cost functions. a. Find the average cost and marginal cost functions. b. Determine the average and marginal cost when x = a. c. Interpret the values obtained in part (b). 21. C(x) = 1000 + 0.1x, 0 x 5000, a = 2000Average and marginal cost Consider the following cost functions. a. Find the average cost and marginal cost functions. b. Determine the average and marginal cost when x = a. c. Interpret the values obtained in part (b). 22. C(x) = 500 + 0.02x, 0 x 2000, a = 1000Average and marginal cost Consider the following cost functions. a. Find the average cost and marginal cost functions. b. Determine the average and marginal cost when x = a. c. Interpret the values obtained in part (b). 23. C(x) = 0.01x2 + 40x + 100, 0 x 1500, a = 1000Average and marginal cost Consider the following cost functions. a. Find the average cost and marginal cost functions. b. Determine the average and marginal cost when x = a. c. Interpret the values obtained in part (b). 24. C(x) = 0.04x2 + 100x + 800, 0 x 1000, a = 500Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars. a. According to the model, how many DVDs can be sold in a day at a price of 10? b. According to the model, what is the maximum price that can be charged (above which no DVDs can be sold)? c. Find the elasticity function for this demand function. d. For what prices is the demand elastic? Inelastic? e. If the price of DVDs is raised from 10.00 to 10.25, what is the exact percentage decrease in demand (using the demand function)? f. If the price of DVDs is raised from 10.00 to 10.25, what is the approximate percentage decrease in demand (using the elasticity function)?Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p)=1800p40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p. a. Recalling that the demand must be positive, what is the domain of this function? b. According to the model, how many tires can be sold in a day at a price of 60 per tire? c. Find the elasticity function on the domain of the demand function. d. For what prices is the demand elastic? Inelastic? e. If the price of tires is raised from 60 to 62, what is the approximate percentage decrease in demand (using the elasticity function)?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the acceleration of an object remains constant, then its velocity is constant. b. If the acceleration of an object moving along a line is always 0, then its velocity is constant. c. It is impossible for the instantaneous velocity at all times a t b to equal the average velocity over the interval a t b. d. A moving object can have negative acceleration and increasing speed.A feather dropped on the moon On the moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of 40 m above the surface of the moon. Then its height s (in meters) above the ground after t seconds is s = 40 0.8t2. Determine the velocity and acceleration of the feather the moment it strikes the surface of the moon.Comparing velocities A stone is thrown vertically into the air at an initial velocity of 96 ft/s. On Mars, the height s (in feet) of the stone above the ground after t seconds is s = 96t 6t2 and on Earth it is s = 96t 16t2. How much higher will the stone travel on Mars than on Earth?Comparing velocities Two stones are thrown vertically upward with matching initial velocities of 48 ft/s at time t = 0. One stone is thrown from the edge of a bridge that is 32 ft above the ground and the other stone is thrown from ground level. The height of the stone thrown from the bridge after t seconds is f(t) = 16t2 + 48t + 32, and the height of the stone thrown from the ground after t seconds is g(t) = 16t2 + 48t. a. Show that the stones reach their high points at the same time. b. How much higher does the stone thrown from the bridge go than the stone thrown from the ground? c. When do the stones strike the ground and with what velocities?Matching heights A stone is thrown from the edge of a bridge that is 48 ft above the ground with an initial velocity of 32 ft/s. The height of this stone above the ground t seconds after it is thrown is f(t) = 16t2 + 32t + 48. If a second stone is thrown from the ground, then its height above the ground after t seconds is given by g(t) = 16t2 + 0 t, where 0 is the initial velocity of the second stone. Determine the value of 0 such that both stones reach the same high point.Velocity of a car The graph shows the position s = f(t) of a car t hours after 5:00 P.M. relative to its starting point s = 0, where s is measured in miles. a. Describe the velocity of the car. Specifically, when is it speeding up and when is it slowing down? b. At approximately what time is the car traveling the fastest? The slowest? c. What is the approximate maximum velocity of the car? The approximate minimum velocity?Velocity from position The graph of s = f(t) represents the position of an object moving along a Line at time t 0. a. Assume the velocity of the object is 0 when t = 0. For what other values of t is the velocity of the object zero? b. When is the object moving in the positive direction and when is it moving in the negative direction? c. Sketch a graph of the velocity function. d. On what intervals is the speed increasing?Fish length Assume the length L (in cm) of a particular species of fish after t years is modeled by the following graph. a. What does dL/dt represent and what happens to this derivative as t increases? b. What does the derivative tell you about how this species of fish grows? c. Sketch a graph of L and L.Average and marginal profit Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that x items have already been sold. Consider the following cost functions C and price functions p. a. Find the profit function P. b. Find the average profit function and marginal profit function. c. Find the average profit and marginal profit if x = a units are sold. d. Interpret the meaning of the values obtained in part (c). 37. C(x) = 0.02x2 + 50x + 100, p(x) = 100, a = 500Average and marginal profit Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that x items have already been sold. Consider the following cost functions C and price functions p. a. Find the profit function P. b. Find the average profit function and marginal profit function. c. Find the average profit and marginal profit if x = a units are sold. d. Interpret the meaning of the values obtained in part (c). 38. C(x) = 0.02x2 + 50x + 100, p(x) = 100 0.1x, a = 500Average and marginal profit Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that x items have already been sold. Consider the following cost functions C and price functions p. a. Find the profit function P. b. Find the average profit function and marginal profit function. c. Find the average profit and marginal profit if x = a units are sold. d. Interpret the meaning of the values obtained in part (c). 39. C(x) = 0.04x2 + 100x + 800, p(x) = 200, a = 1000Average and marginal profit Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that x items have already been sold. Consider the following cost functions C and price functions p. a. Find the profit function P. b. Find the average profit function and marginal profit function. c. Find the average profit and marginal profit if x = a units are sold. d. Interpret the meaning of the values obtained in part (c). 40. C(x) = 0.04x2 + 100x + 800, p(x) = 200 0.1x, a = 1000U.S. population growth The population p(t) (in millions) of the United States t years after the year 1900 is shown in the figure. Approximately when (in what year) was the U.S. population growing most slowly between 1925 and 2020? Estimate the growth rate in that year.Average and marginal production Economists use production functions to describe how the output of a system varies with respect to another variable such as labor or capital. For example, the production function P(L) = 200L + 10L2 L3 gives the output of a system as a function of the number of laborers L The average product A(L) is the average output per laborer when L laborers are working; that is A(L) = P(L)/L. The marginal product M(L) is the approximate change in output when one additional laborer is added to L laborers; that is, M(L)dPdL. a. For the given production function, compute and graph P, A, and M. b. Suppose the peak of the average product curve occurs at L = L0, so that A (L0) = 0. Show that for a general production function, M(L0) = A(L0).Velocity of a marble The position (in meters) of a marble rolling up a long incline is given by s=100tt+1, where t is measured in seconds and s = 0 is the starting point. a. Graph the position function. b. Find the velocity function for the marble. c. Graph the velocity function and give a description of the motion of the marble. d. At what time is the marble 80 m from its starting point? e. At what time is the velocity 50 m/s?Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67 + 0.70b + 0.0067b2. a. Graph the height function. b. Plot and interpret the meaning of dhdb.51EDiminishing returns A cost function of the form C(x)=12x2 reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.Revenue function A store manager estimates that the demand for an energy drink decreases with increasing price according to the function d(p)=100p2+1, which means that at price p (in dollars), d(p) units can be sold. The revenue generated at price p is R(p) = p d(p) (price multiplied by number of units). a. Find and graph the revenue function. b. Find and graph the marginal revenue R(p). c. From the graphs of R and R, estimate the price that should be charged to maximize the revenue.Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g 25.8g2 + 12.5g3 1.6g4, for 0 g 4. a. Graph and interpret the mileage function. b. Graph and interpret the gas mileage m(g)/g. c. Graph and interpret dm/dg.Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10 sin t 10 cos t, where x is positive when the mass is above the equilibrium position. a. Graph and interpret this function. b. Find dxdt and interpret the meaning of this derivative. c. At what times is the velocity of the mass zero? d. The function given here is a model for the motion of an object on a spring. In what ways is this model unrealistic?Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal = 4184 J. One hour of walking consumes roughly 106 J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W = 1 J/s. Other useful units of power are kilowatts (l kW = 103 W) and megawatts (l MW = 106 W). If energy is used at a rate of l kW for one hour, the total amount of energy used is 1 kilowatt-hour (l kWh = 3.6 106 J). Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t+4t2t39kWh, where t = 0 corresponds to midnight. a. Graph the energy function. b. The power is the rate of energy consumption; that is, P(t) = E(t). Find the power over the interval 0 t 24. c. Graph the power function and interpret the graph. What are the units of power in this case?A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions (t) and (t), respectively, where 0 t 4 and t is measured in minutes (see figure). These angles are measured in radians, where = = 0 represent the starting position and = = 2 represent the finish position. The angular velocities of the runners are (t) and (t). a. Compare in words the angular velocity of the two runners and the progress of the race. b. Which runner has the greater average angular velocity? c. Who wins the race? d. Jeans position is given by (t) = t2/8. What is her angular velocity at t = 2 and at what time is her angular velocity the greatest? e. Juans position is given by (t) = t(8 t)/8. What is his angular velocity at t = 2 and at what time is his angular velocity the greatest?Flow from a tank A cylindrical tank is full at time t = 0 when a valve in the bottom of the tank is opened. By Torricellis Law, the volume of water in the tank after t hours is V = 100(200 t)2, measured in cubic meters. a. Graph the volume function. What is the volume of water in the tank before the valve is opened? b. How long does it take the tank to empty? c. Find the rate at which water flows from the tank and plot the flow rate function. d. At what time is the magnitude of the flow rate a minimum? A maximum?Bungee jumper A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1 + et cos t), for t 0. a. Determine her velocity at t = 1 and t = 3. b. Use a graphing utility to determine when she is moving downward and when she is moving upward during the first 10 s. c. Use a graphing utility to estimate the maximum upward velocity.Spring runoff The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by V(t)={45t2if0t4545(t2180t+4050)if45t90, where V is measured in cubic feet and t is measured in days, with t = 0 corresponding to May 1. a. Graph the volume function. b. Find the flow rate function V(t) and graph it. What are the units of the flow rate? c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?Temperature distribution A thin copper rod, 4 meters in length, is heated at its midpoint, and the ends are held at a constant temperature of 0. When the temperature reaches equilibrium, the temperature profile is given by T(x) = 40x(4 x), where 0 x 4 is the position along the rod. The heat flux at a point on the rod equals kT(x), where k 0 is a constant. If the heat flux is positive at a point, heat moves in the positive x-direction at that point, and if the heat flux is negative, heat moves in the negative x-direction. a. With k = 1, what is the heat flux at x = 1? At x = 3? b. For what values of r is the heat flux negative? Positive? c. Explain the statement that heat flows out of the rod at its ends.Explain why it is not practical to calculate ddx(5x+4)100 by first expanding (5x+4)100.Identify an inner function (call it g) of y = (5x + 4)3. Let u = g(x) and express the outer function f in terms of u.Let y = tan10 (x6). Find f, g, and h such that y = f(u), where u = g(v) and v = h(x).Two equivalent forms of the Chain Rule for calculating the derivative of y = f(g(x)) are presented in this section. State both forms.Identify the inner and outer functions in the composition (x2 + 10)5.Identify an inner function u = g(x) and an outer function y = f(u) of y=(x3 + x + l)4. Then calculate using dydx using dydx=dydududx.Identify an inner function u = g(x) and an outer function y = f(u) of y=ex3+2x. Then calculate dydx using dydx=dydududx.The two composite functions y = cos3 x and y = cos x3 look similar, but in fact are quite different. For each function, identify the inner function u = q(x) and the outer function y = f(u); then evaluate dydx using the Chain Rule.Let h(x) = f(g(x)), where f and g are differentiable on their domains. If g(1) = 3 and g(1) = 5, what else do you need to know to calculate h(1)?Fill in the blanks. The derivative of f(g(x)) equals f evaluated _____ at multiplied by g evaluated at _____.Evaluate the derivative of y=(x2+2x+1)2 using d/dx(f(g(x))=f(g(x))g(x)).Evaluate the derivative of y=4x+1 using d/dx(f(g(x))=f(g(x))g(x).Express Q(x) = cos4 (x2 + 1) as the composition of three functions; that is, identify f, g, and h so that Q(x) = f(g(h(x))).Given that h(x)=f(g(x)), find h(3) if g(3)=4,g(3)=5,f(4)=9, and f(4)=10.Given that h(x) = f(g(x)), use the graphs of f and g to find h'(4).What is the derivative of y=ekx?Find f(x) if f(x)=15e3x.Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dydx. 7. y = (3x + 7)10For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx. 16.y=(5x2+11x)4/3Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dydx. 9. y = sin5 xFor each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx. 18.y=sinx5Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dydx. 13. y=x2+1Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dydx. 12. y=7x1For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx. 21.y=e4x2+1Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dydx. 14. y=exVersion 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dydx. 15. y = tan 5x2Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dydx. 16. y=sinx4Chain Rule using a table Let h(x)= f(g(x)) and p(x) = g(f(x)). Use the table to compute the following derivatives. a. h(3) b. h(2) c. p(4) d. p(2) e. h(5)26EVersion 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 19. y = (3x2 + 7x)10Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 20. y = (x2 + 2x + 7)8Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 21. y=10x+1Calculate the derivative of the following functions. 30.y=x2+93Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 23. y = 5(7x3 + 1)3Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 24. y = cos 5tVersion 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 25. y = sec (3x + 1)Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 26. y = csc exVersion 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 27. y = tan exVersion 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 28. y = etan tVersion 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 29. y = sin (4x3 + 3x + 1)Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 30. y = csc (t2 + t)Calculate the derivative of the following functions. 39.y=(5x+1)2/3Calculate the derivative of the following functions. 40.y=x(x+1)1/3Calculate the derivative of the following functions. 41.y=2x4x34Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 32. y = cos4 + sin4Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 33. y = (sec x + tan x)5Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. 34. y = sin (4 cos z)Chain Rule for powers Use the Chain Rule to find the derivative of the following functions. 41. y = (2x6 3x3 + 3)25Chain Rule for powers Use the Chain Rule to find the derivative of the following functions. 42. y = (cos x + 2 sin x)8Calculate the derivative of the following functions. 47.y=(1+2tanu)4.5Chain Rule for powers Use the Chain Rule to find the derivative of the following functions. 44. y = (1 ex)4Repeated use of the Chain Rule Calculate the derivative of the following functions. 45. y=1+cot2xCalculate the derivative of the following functions. 50.g(x)=xe3xCalculate the derivative of the following functions. 51.y=2ex+3ex3Calculate the derivative of the following functions. 52.f(x)=xe7xRepeated use of the Chain Rule Calculate the derivative of the following functions. 47. y = sin (sin (ex))Repeated use of the Chain Rule Calculate the derivative of the following functions. 48. y = sin2 (e3x + 1)Repeated use of the Chain Rule Calculate the derivative of the following functions. 49. y = sin5 (cos 3x)Calculate the derivative of the following functions. 56.y=cos7/4(4x3)Calculate the derivative of the following functions. 57.y=e2t1+e2tRepeated use of the Chain Rule Calculate the derivative of the following functions. 52. y = (1 e0.05x)1Repeated use of the Chain Rule Calculate the derivative of the following functions. 53. y=x+xRepeated use of the Chain Rule Calculate the derivative of the following functions. 54. y=x+x+xRepeated use of the Chain Rule Calculate the derivative of the following functions. 55. y = f(g(x2)), where f and g are differentiable for all real numbersRepeated use of the Chain Rule Calculate the derivative of the following functions. 56. y = (f(g(xm)))n, where f and g are differentiable for all real numbers, and m and n are integersCombining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 57. y=(xx+1)5Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 58. y=(exx+1)8Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 59. y=ex2+1sinx3Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 60. y = tan (x ex)Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 61. y = 2 sec 5Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 62. y=(3x4x+2)5Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 63. y = ((x + 2)(x2 + 1))4Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 64. y = e2x(2x 7)5Calculate the derivative of the following functions. 71.y=x4+cos2x5Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 66. y=tett+1Calculate the derivative of the following functions. 73.y=(p+3)2sinp2Calculate the derivative of the following functions. 74.y=(2z+5)1.75tanzSquare root derivatives Find the derivative of the following functions. 75. y=f(x), where f is differentiable and nonnegative at xCalculate the derivative of the following functions. 76.y=f(x)g(x)5, where f and g are differentiable at xExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function x sin x can be differentiated without using the Chain Rule. b. The function ex+1 should be differentiated using the Chain Rule. c. The derivative of a product is not the product of the derivatives, but the derivative of a composition is a product of derivatives. d. dydxP(Q(x))=P(x)Q(X)Smartphones From 2007 to 2014, there was a dramatic increase in smartphone sales The number of smartphones (in millions) sold to end users from 2007 to 2014 (see figure) is modeled by the function c(t) = 114.9e0.345t, where t represents the number of years after 2007. a.Determine the average growth rate in smartphone sales between the years 2007 and 2009 and between 2012 and 2014. During which of these two time intervals was the growth rate greater? b.Find the Instantaneous growth rate in smartphone sales at t = 1 (2008) and t = 6 (2013)? At which of these times was the instantaneous growth rate greater c.Use a graphing utility to graph the growth rate, for 0 t 7. What does the graph tell you about growth of smartphone sales to end users from 2007 to 2014?Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.Mass of Juvenile desert tortoises A study conducted at the University of New Mexico found that the mass m(t) (In grams) of a juvenile desert tortoise t days after a switch to a particular diet is described by the function m(t) = m0e0 004t, where m0 is the mass of the tortoise at the time of the diet switch. If m0 = 64, evaluate m(65) and interpret he nearing of this result.Cell population The population of a culture of cells after t days is approximated by the function P(t)=16001+7e0.02t,fort0 a. Graph the population function. b. What is the average growth rate during the first 10 days? c. Looking at the graph, when does the growth rate appear to be a maximum? d. Differentiate the population function to determine the growth rate function P(t). e. Graph the growth rate When is it a maximum and what is the population at the time that the growth rate is a maximum?Bank account A 200 investment in a savings account grows according to A(t) = 200e0.0398t, for t 0. where t is measured in years. a. Find the balance of the account after 10 years. b. How fast is the account growing (in dollars/year) at t = 10? c. Use your answers to parts (a) and (b) to write the equation of the line tangent to the curve A = 200e0.0398t at the point (10, A(10)).Pressure and altitude Earths atmospheric pressure decreases with altitude from a sea level pressure of 1000 millibars (a unit of pressure used by meteorologists). Letting z be the height above Earths surface (sea level) in kilometers, the atmospheric pressure is modeled by p(z) = 1000ez/10. a. Compute the pressure at the summit of Mt. Everest, which has an elevation of roughly 10 km. Compare the pressure on Mt. Everest to the pressure at sea level. b. Compute the average change in pressure in the first 5 km above Earths surface. c. Compute the rate of change of the pressure at an elevation of 5 km. d. Does p(z) increase or decrease with z? Explain. e. What is the meaning of limzp(z)=0?Finding slope locations Let f(x) = xe2x. a. Find the values of x for which the slope of the curve y = f(x) is 0. b. Explain the meaning of your answer to part (a) in terms of the graph of f.Second derivatives Find d2ydx2 for the following functions. 70. y = x cos x2Second derivatives Find d2ydx2 for the following functions. 71. y = sin x2Second derivatives Find d2ydx2 for the following functions. 72. y=x2+2Second derivatives Find d2ydx2 for the following functions. 73. y=e2x290ETangent lines Determine an equation of the line tangent to the graph of y=(x21)2x36x1 at the point (0, 1).Tangent lines Determine equations of the lines tangent to the graph of y=x5x2 at the points (1, 2) and (2, 2). Graph the function and the tangent lines.Tangent lines Assume f and g are differentiable on their domains with h(x) = f(g(x)). Suppose the equation of the line tangent to the graph of g at the point (4, 7) is y = 3x 5 and the equation of the line tangent to the graph of f at (7, 9) is y = 2x + 23. a. Calculate h(4) and h(4). b. Determine an equation of the line tangent to the graph of h at the point on the graph where x = 4.94ETangent lines Find the equation of the line tangent to y = e2x at x=12ln3. Graph the function and the tangent line.96EComposition containing sin x Suppose f is differentiable for all real numbers with f(0) = 3, f(1) = 3, f(0) = 3, and f(1) = 5. Let g(x) = sin (f(x)). Evaluate the following expressions. a. g(0) b. g(1)Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y = 0 when the mass hangs at rest. Suppose you push the mass to a position y0, units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is y=y0(tkm),(4) where k 0 is a constant measuring the stiffness of the spring (the larger the value of k, the stiffer the spring) and y is positive in the upward direction. 98.Use equation (4) to answer the following questions a. Find dy/dt, the velocity of the mass. Assume k and m are constant. b. How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring? c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness (k is increased by a factor of 4)? d. Assume y has units of meters, t has units of seconds, m has units of kg and k has units of kg /s2. Show that the units of the velocity in part (a) are cons stent.Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y = 0 when the mass hangs at rest. Suppose you push the mass to a position y0, units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is y=y0cos(tkm),(4) where k 0 is a constant measuring the stiffness of the spring (the larger the value of k, the suffer the spring) and y is positive in the upward direction. 99.Use equation (4) to answer the following questions. a.Find the second derivative d2ydt2. b.Verify that d2ydt2=kmy.Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y = 0 when the mass hangs at rest. Suppose you push the mass to a position y0 units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is y=y0cos(tkm), where k 0 is a constant measuring the stiffness of the spring (the larger the value of k, the stiffer the spring) and y is positive in the upward direction. 100.Use equation (4) to answer the following questions. a.The period T is the time required by the mass to complete one oscillation. Show that T=2mk. b.Assume k is constant and calculate dTdm c.Give a physical explanation of why dTdm is positive.A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by y=10et/2cost8. a. Graph the displacement function. b. Compute and graph the velocity of the mass, v(t) = y(t). c. Verify that the velocity is zero when the mass reaches the high and low points of its oscillation.Oscillator equation A mechanical oscillator (such as a mass on a spring or a pendulum) subject to frictional forces satisfies the equation (called a differential equation) y(t)+2y(t)+5y(t)=0, where y is the displacement of the oscillator from its equilibrium position. Verify by substitution that the function y(t)= et (sin 2t 2 cos 2t) satisfies this equation.103E104E105EDeriving trigonometric identities a. Differentiate both sides of the identity cos 2t = cos2 t sin2 t to prove that sin 2t = 2 sin t cos t. b. Verify that you obtain the same identity for sin 2t as in part (a) if you differentiate the identity cos 2t = 2 cos2 t 1. c. Differentiate both sides of the identity sin 2t = 2 sin t cos t to prove that cos 2t = cos2 sin2 t.