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All Textbook Solutions for Calculus Volume 1
For the following exercises, rise the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. 294. log6103For the following exercises, rise the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. 295. log0.5211For the following exercises, rise the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. 296. log2For the following exercises, rise the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. 297. log0.20.452Rewrite the following expressions in terms of exponentials and simplify. a.2cosh(Inx)b.cosh4x+sinh4xc.cosh2xsinh2xd.In(coshx+sinhx)+In(coshxsinhx)[T] The number of bacteria N in a culture after t days can be modeled by the function N(t)=1300(2)t/4 . Find the number of bacteria present after 15 days.[T] The demand D (in millions of barrels) for oil in ail oil-rich country is given by the function D(p)=150(2.7)0.25p where p is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.[T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by A(t)=100,000e0.055t . Find the amount A accumulated in 5 years.[T] An investment is compounded monthly, quarterly, or yearly and is given by the function A=P(1+jn)nt , where A is the value of the investment at time t, P is the initial principle that was invested, j is the annual interest rate, and n is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount A accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.[T] The concentration of hydrogen ions in a substance is denoted by [H+] , measured in moles per liter. The pH of a substance is defined by the logarithmic function pH=log[H+] . This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base. Find the pH of the following substances. Round answers to one digit. Determine whether the substance is an acid or a base. i.Eggs:[H+]=1.6108mol/Lii.Beer:[H+]=3.16103mol/Liii.TomatoJuice:[H+]=7.94105mol/L[T] Iodine-131 is a radioactive substance that decays according to the function Q(t)=Q0e0.08664t , where Q0 is the initial quantity of a sample of the substance and t is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.[T] According to the World Bank, at the end of 2013 (t = 0) the U.S. population was 316 million and was increasing according to the following model: P(t)=316e0.0074t , where P is measured in millions of people and t is measured in years after 2013. Based on this model, what will be the population of the United States in 2020? Determine when the U.S. population will be twice what it is in 2013.[T] The amount A accumulated after 1000 dollar’s is invested for t years at an interest rate of 4% is modeled by the function A(t)=1000(1.04)t . Find the amount accumulated after 5 years and 10 years. Determine how long it takes for the original investment to triple.[T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present. Use the exponential function Q=Q0ekt to determine the value k, which is the growth rate of the bacteria. Round to four decimal places. Determine approximately how long it takes for 200,000 bacteria to grow.[T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially. Use the exponential function P=P0at to determine the growth rate constant a. Round to four decimal places. Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.[T] The 1906 earthquake in Sail Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?True or False? Justify your answer with a proof or a counterexample. 310. A function is always one-to-one.True or False? Justify your answer with a proof or a counterexample. 311. fg=gf, assuming f and g are functions.True or False? Justify your answer with a proof or a counterexample. 312. A relation that passes the horizontal and vertical line tests is a one-to-one function.True or False? Justify your answer with a proof or a counterexample. 313. A relation passing the horizontal line test is a function.For the following problems, state the domain and range of the given functions: f=x2+2x3,g=In(x5),h=1x+4 314. hFor the following problems, state the domain and range of the given functions: f=x2+2x3,g=In(x5),h=1x+4 315. gFor the following problems, state the domain and range of the given functions: f=x2+2x3,g=In(x5),h=1x+4 316. hfFor the following problems, state the domain and range of the given functions: f=x2+2x3,g=In(x5),h=1x+4 317. gfFind the degree, y-intercept, and zeros for the following polynomial functions. 318. f(x)=2x2+9x5Find the degree, y-intercept, and zeros for the following polynomial functions. 319. f(x)=x3+2x22xSimplify the following trigonometric expressions. 320. tan2xsec2x+cos2xSimplify the following trigonometric expressions. 321. cos(2x)=sin2xSolve the following trigonometric equations on the interval =[2,2] exactly. 322. 6cos2x3=0Solve the following trigonometric equations on the interval =[2,2] exactly. 323. sec2x2secx+1=0Solve the following logarithmic equations. 324. 5x=16wSolve the following logarithmic equations. 325. log2(x+4)=3Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse f1(x) of the function. Justify your answer. 326. f(x)=x2+2x+1Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse f1(x) of the function. Justify your answer. 327. f(x)=1xFor the following problems, determine the largest domain oil which the function is one-to-one and find the inverse on that domain. 328. f(x)=9xFor the following problems, determine the largest domain oil which the function is one-to-one and find the inverse on that domain. 329. f(x)=x2+3x+4A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts. 331. a. Find the equation C=f(x) that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts. 332. a. Find the inverse function x=f1(C) and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has $8000 to spend. For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts. 333. Tire population can be modeled by P(t)=82.567.5cos[(/6)t]+t , where f is time in months (t = 0 represents January 1) and P is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts. 334. In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as P(t)=82.567.5cos[(/6)t]+t, where t is time in months ( t = 0 represents January 1) and P is population (in thousands). When is the first time the population reaches 200,000?For the following problems, consider radioactive dating. A human skeleton is found in ail archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation y=ert , where y is the percentage of radiocarbon still present in the material, t is the number of years passed, and r = -0.0001210 is the decay rate of radiocarbon. 335. If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?For the following problems, consider radioactive dating. A human skeleton is found in ail archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation y=ert , where y is the percentage of radiocarbon still present in the material, t is the number of years passed, and r = -0.0001210 is the decay rate of radiocarbon. 336. Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?For the following exercises, points P(l, 2) and Q(x, y) are on the graph of the function f(x)=x2+1 . 1. [T] Complete the following table with the appropriate values: y-coordinate of Q, the point Q(x, y), and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. x y Q(x,y) msec 1.1 a. e. i. 1.01 b. f. j. 1.001 c. g. k. 1.0001 d. h. l.For the following exercises, points P(l, 2) and Q(x, y) are on the graph of the function f(x)=x2+1 . 2. Use the values in the right column of the table in the preceding exercise to guess the value of die slope of die Line tangent to f at x = 1.For the following exercises, points P(l, 2) and Q(x, y) are on the graph of the function f(x)=x2+1 3. Use the value in the preceding exercise to find the equation of the tangent line at point P. Graph f(x) and the tangent line.For the following exercises, points P(l, 1) and Q(x, y) are on the graph of the function f(x)=x3 . 4.[T] Complete the following table with the appropriate values: y-coordinate of Q, the point Q(x, y), and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. X y Q(x,y) msec 1.1 a. e. i. 1.01 b. f. j. 1.001 c. g. k. 1.0001 d. h. l.For the following exercises, points P(l, 1) and Q(x, y) are on the graph of the function f(x)=x3 . 5. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at x = 1,For the following exercises, points P(l, 1) and Q(x, y) are on the graph of the function f(x)=x3 . 6. Use the value in the preceding exercise to find the equation of the tangent line at point P. Graph f(x) and the tangent line.For the following exercises, points P(4, 2) and Q(x, y) are on the graph of the function f(x)=x . 7.[T] Complete the following table with the appropriate values: y-coordinate of Q, the point Q(x, y), and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. x y Q(x,y) msec 4.1 a. e. i. 4.01 b. f. j. 4.001 c. g. k. 4.0001 d. h. l.For the following exercises, points P(4, 2) and Q(x, y) are on the graph of the function f(x)=x . 8. Use the values in the tight column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at x = 4.For the following exercises, points P(4, 2) and Q(x, y) are on the graph of the function f(x)=x . 9. Use the value in tire preceding exercise to find the equation of the tangent line at point P.For the following exercises, points P(l.5, 0) and Q( , y) are on the graph of the function f()=cos() . 10.[T] Complete the following table with the appropriate values: y-coordinate of Q, the point Q(x, y), and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. X y Q( ,y) msec 1.4 a. e. i. 1.49 b. f. j. 1.499 c. g. k. 1.4999 d. h. l.For the following exercises, points P( 1.5, 0) and Q( , y) are on the graph of the function f()=cos() . 11. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at x = 4.For the following exercises, points P( 1.5, 0) and Q( , y) are on the graph of the function f()=cos() . 12. Use the value in the preceding exercise to find the equation of the tangent line at point P.For the following exercises, points P(-1, -1) and Q(x, y) are on the graph of the function f(x)=1x . 10.[T] Complete the following table with the appropriate values: y-coordinate of Q, the point Q(x, y), and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. x y Q(x, y) msec -1.05 a. e. i. -1.01 b. f. j. -1.005 c. g. k. -1.001 d. h. l.For the following exercises, points P(-1,-1) and Q(x, y) are on the graph of the function f(x)=1x . 14. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to f at x = - 1.For the following exercises, points P(-1, - 1) and Q(x, y) are on the graph of the function f(x)=1x . 15. Use the value in the preceding exercise to find the equation of the tangent line at point P.For the following exercises, the position function of a ball dropped from the top of a 200-meter tall building is given by s(t) = 200 - 4.9t2 , where position s is measured in meters and time t is measured in seconds. Round your answer to eight significant digits. 16. [T] Compute the average velocity of the ball over the given time intervals. [4.99, 5] [5, 5.01] [4.999, 5] [5,5.001]For the following exercises, the position function of a ball dropped from the top of a 200-meter tall building is given by s(t)=2004.9t2 , where position s is measured in meters and time t is measured in seconds. Round your answer to eight significant digits. 17. Use the preceding exercise to guess the instantaneous velocity of the ball at t = 5 sec.For the following exercises, consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. Its height in meters at time t seconds is h(t)=15t4.9t2 . 18. [T] Compute the average velocity of the stone over the given time intervals. [1, 1.05] [1, 1.01] [1, 1.005] [1, 1.001]For the following exercises, consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. Its height in meters at time t seconds is h(t)=15t4.9t2 . 19. Use the preceding exercise to guess the instantaneous velocity of the stone at t = 1 sec.For the following exercises, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by h(t)=600+78.4t4.9t2 , where t is measured in seconds. 20.[T] Compute the average velocity of the rocket over the given time intervals. [9, 9.01] [8.99, 9] [9, 9.001] [8.999, 9]For the following exercises, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by h(t)=600+78.4t4.9t2 , where t is measured in seconds. 21. Use the preceding exercise to guess the instantaneous velocity of the rocket at t = 9 sec.For the following exercises, consider an athlete running a 40-m dash. The position of the athlete is given by d(t)=t36+4t , where d is the position in meters and t is the time elapsed, measured in seconds. 22. [T] Compute the average velocity of the runner over the given time intervals. [1.95, 2.05] [1.995, 2.005] [1.9995, 2.0005] [2, 2.00001]For the following exercises, consider an athlete running a 40-m dash. The position of the athlete is given by d(t)=t36+4t , where d is the position in meters and t is the time elapsed, measured in seconds. 23. Use the preceding exercise to guess the instantaneous velocity of the runner at t = 2 sec. For the following exercises, consider the function f(x)=|x| .For the following exercises, consider the function.
24. Sketch the graph of f over the interval [-1, 2] and shade the region above the x-axis.
For the following exercises, consider the function f(x)=|x| . 25. Use the preceding exercise to find the exact value of the area between the x-axis and the graph of f over the interval [-1, 2] using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer.For the following exercises, consider the function f(x)=1x2 . (Hint: This is the upper half of the circle of radius 1 positioned at (0,0).) 26. Sketch the graph of f over the interval [-1, 1],For the following exercises, consider the function. (Hint: This is the upper half of the circle of radius 1 positioned at (0,0).)
27. Use the preceding exercise to find the exact area between the x-axis and the graph of f over the interval [-1, 1] using rectangles. For the rectangles, use squares 0.4 by 0.4 units, and approximate both above and below the lines. Use geometry to find the exact answer.
For the following exercises, consider the function f(x)=x2+1 . 28. Sketch the graph of f over the interval [-l, 1].For the following exercises, consider the function f(x)=x2+1 . 29. Approximate the area of the region between the x-axis and the graph of f over the interval [-1, 1].For the following exercises, consider the function f(x)=x21|x1| . 30. [T] Complete the following table for the function. Round your solutions to four decimal places. x f(x) x f(x) 0.9 a. 1.1 e. 0.99 b. 1.01 f. 0.999 c. 1.001 g. 0.999 d. 1.0001 h.For the following exercises, consider the function f(x)=x21|x1| . 31. What do your results in the preceding exercise indicate about the two-sided limx1f(x) ? Explain yourFor the following exercises, consider the function f(x)=(1+x)1/x . 32.[T] Make a table showing the values of f for x=0.01,0.001,0.0001,0.00001 and for x=0.01,0.001,0.0001,0.00001 . Round your solutions to five decimal places. x f(x) X f(x) -0.01 a. 0.01 e. -0.001 b. 0.001 f. -0.0001 c. 0.0001 g- -0.00001 d. 0.00001 h.For the following exercises, consider the function f(x)=(1+x)1/x . 33. What does the table of values in the preceding exercise indicate about the function f(x)=(1+x)1/x ?For the following exercises, consider the function f(x)=(1+x)1/x . 34. To which mathematical constant does the limit in the preceding exercise appear to be getting closer? In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.For the following exercises, consider the function f(x)=(1+x)1/x . 35. [T] limx0sin2xx;0.1,0.01,0.001,0.0001 x sin2xx x sin2xx -0.1 a. 0.1 e. -0.01 b. 0.01 f. -0.001 c. 0.001 g. -0.0001 d. 0.0001 h.For the following exercises, consider the function f(x)=(1+x)1/x . 36. [T] limx0sin3xx;0.1,0.01,0.001,0.0001 x sin3xx x sin3xx -0.1 a. 0.1 e. -0.01 b. 0.01 f. -0.001 c. 0.001 g. -0.0001 d. 0.0001 h.For the following exercises, consider the function f(x)=(1+x)1/x . 37. Use the preceding two exercises to conjecture (guess) the value of the following limit: limx0sinaxx for a, a positive real value.[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 38. limx2x24x2+x6 x x24x2+x6 x x24x2+x6 1.9 a. 2.1 e. 1.99 b. 2.01 f. 1.999 c. 2.001 g. 1.9999 d. 2.0001 h.[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 39. limx1(12x) x 12x x 12x 0.9 a. 1.1 e. 0.99 b. 1.01 f. 0.999 c. 1.001 g. 0.9999 d. 1.0001 h.[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 40.. limx051e1/x x 51e1/x x 51e1/x -0.1 a. 0.1 e. -0.01 b. 0.01 f. -0.001 c. 0.001 g. -0.0001 d. 0.0001 h.[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 41. limx0z1z2(z+3) z z1z2(z+3) z z1z2(z+3) -0.1 a. 0.1 e. -0.01 b. 0.01 f. -0.001 c. 0.001 g. -0.0001 d. 0.0001 h.[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 42. limt0+costt t costt 0.1 a. 0.01 b. 0.001 c. 0.0001 d.[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 43. limt212xx21 x 12xx21 x 12xx21 1.9 a. 2.1 e. 1.99 b. 2.01 f. 1.999 c. 2.001 g. 1.9999 d. 2.0001 h.[T] In the following exercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail? 44. lim0sin() sin() sin() -0.1 a. 0.1 e. -0.01 b. 0.01 f. -0.001 c. 0.001 g. -0.0001 d. 0.0001 f.[T] In the following exercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail? 45. lima01acos(a) a 1acos(a) 0.1 a. 0.01 b. 0.001 c. 0.0001 d.In the following exercises, consider the graph of the function y=f(x) shown here. Which of the statements about y=f(x) are true and which are false? Explain why a statement is false. 46. limx10f(x)=0In the following exercises, consider the graph of the function y=f(x) shown here. Which of the statements about y=f(x) are true and which are false? Explain why a statement is false. 47. limx2+f(x)=3In the following exercises, consider the graph of the function y=f(x) shown here. Which of the statements about y=f(x) are true and which are false? Explain why a statement is false. 48. limx8f(x)=f(8)In the following exercises, consider the graph of the function y=f(x) shown here. Which of the statements about y=f(x) are true and which are false? Explain why a statement is false. 49. limx6f(x)=5In the following exercises, use the following graph of function y=f(x) to find the values, if possible. Estimate when necessary. 50. limx1f(x)In the following exercises, use the following graph of function y=f(x) to find the values, if possible. Estimate when necessary. 51. limx1+f(x)In the following exercises, use the following graph of function y=f(x) to find the values, if possible. Estimate when necessary. 52. limx1f(x)In the following exercises, use the following graph of function y=f(x) to find the values, if possible. Estimate when necessary. 53. limx2f(x)In the following exercises, use the following graph of function y=f(x) to find the values, if possible. Estimate when necessary. 54. f(1)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 55. limx0f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 56. limx0+f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 57. limx0f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 58. limx2f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 59. limx2f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 60. limx2+f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 61. limx2f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 62. limx2f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 63. limx2+f(x)In the following exercises, use the graph of function y=f(x) shown here to find the value, if possible. Estimate when necessary. 64. limx2f(x)In the following exercises, use the graph of the function y=g(x) shown here to find the values, if possible. Estimate when necessary. 65. lim x 0 g( x )In the following exercises, use the graph of the function y=g(x) shown here to find the values, if possible. Estimate when necessary. 66. limx0+g(x)In the following exercises, use the graph of the function y=g(x) shown here to find the values, if possible. Estimate when necessary. 67. limx0g(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 68. limx0h(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 69. limx0+h(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 70. limx0h(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 71. limx0f(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 72. limx0+f(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 73. limx0f(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 74. limx1f(x)In the following exercises, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary. 75. limx2f(x)In the following exercises, sketch the graph of a function with the given properties. 76. limx2f(x)=1,limx4f(x)=3,limx4+f(x)=6,x=4 is not defined.In the following exercises, sketch the graph of a function with the given properties. 77. limxf(x)=0,limx1f(x)=,limx1+f(x)=,limx0f(x)=f(0),f(0)=1,limxf(x)=In the following exercises, sketch the graph of a function with the given properties. 78 limxf(x)=2,limx3f(x)=,limx3+f(x)=,limxf(x)=2,f(0)=13In the following exercises, sketch the graph of a function with the given properties. 79. limxf(x)=2,limx2f(x)=,limxf(x)=2,f(0)=0In the following exercises, sketch the graph of a function with the given properties. 80. limxf(x)=0,limx1f(x)=,limx1+f(x)=,f(0)=1,limx1f(x)=,limx1+f(x)=,limxf(x)=0Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, x, is shown here. We are mainly interested in the location of the front of the shock, labeled xSFin the diagram. a. Evaluate limxxSF+(x) b. Evaluate limxxSF(x) c. Evaluate limxxSF(x) . Explain the physical meanings behind your answer.A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where x is the position in meters of the runner and t is time in seconds. What is limt2x(t) What does it t->2 mean physically? t(sec) x(m) 1.75 4.5 1.95 6.1 1.99 6.42 2.01 6.58 2.05 6.9 2.25 8.5Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: 1. Express the height h and the base b of the isosceles triangle in Figure 2.31 in terms of and r.Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: 2. Using the expressions that you obtained in step 1, express the area of the isosceles triangle in terms of and r. (Substitute (l/2) sin for sin(/2)cos(/2) in your expression.)Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: 3. If an n-sided regular polygon is inscribed in a circle of radius r, find a relationship between and n. Solve this for n. Keep in mind there are 2 radians in a circle. (Use radians, not degrees.)Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: 4. Find an expression for the area of the n-sided polygon in terms of r and .Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: 5. To find a formula for the area of the circle, find the limit of the expression in step 4 as goes to zero. (Hint: lim0(sin)=1 .)In the following exercises, use the limit Laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s). 83. limx0(4x22x+3)In the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s). 84. limx1x3+3x2+547xIn the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s). 85. limx2x26x+3In the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s). 86. limx1(9x+1)2In the following exercises, use direct substitution to evaluate each limit. 87. limx7x2In the following exercises, use direct substitution to evaluate each limit. 88. limx2(4x21)In the following exercises, use direct substitution to evaluate each limit. 89. limx011+sinxIn the following exercises, use direct substitution to evaluate each limit. 90. limx2e2xx2In the following exercises, use direct substitution to evaluate each limit. 91. limx127xx+6In the following exercises, use direct substitution to evaluate each limit. 92. limx3Ine3xIn the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 93. limx4x216x4In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 94. limx2x2x22xIn the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 95. limx63x182x12In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 96. limh0( 1+h)21hIn the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 97. limt9t9t3In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 98. limh01a+h1ah , where a is a real-valued constantIn the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0, Then, evaluate the limit. 99. limsintanIn the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 100. limx1x31x21In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 101. limx1/22x2+3x22x1]In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit. 102. limx3x+41x+3In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example 2.23 to simplify the function to help determine the limit. 103. limx22x2+7x4x2+x2In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example 2.23 to simplify the function to help determine the limit. 104. limx2+2x2+7x4x2+x2In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example 2.23 to simplify the function to help determine the limit. 105. limx12x2+7x4x2+x2In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example 2.23 to simplify the function to help determine the limit. 106. limx1+2x2+7x4x2+x2In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 107. limx62f(x)g(x)In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 108. limx6g(x)1f(x)In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 109. limx6(f(x)+13g(x))In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 110. limx6( h( x ))32In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 111. limx6g(x)f(x)In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 112. limx6xh(x)In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 113. limx6[(x+1)f(x)]In the following exercises, assume that limx6f(x)=4,limx6g(x)=9,andlimx6h(x)=6 . Use these three facts and the limit laws to evaluate each limit. 114. limx6(f(x)g(x)h(x))[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits. 115. f(x)={ x 2,x3x+4.x3}a.limx3f(x)b.limx3+f(x)[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits. 116. g(x)={ x 31,x01,x0}a.limx0g(x)b.limx0+g(x)[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits. 117. h(x)={ x 22x+1,x23x,x2}a.limx2h(x)b.limx2+h(x)In the following exercises, use the following graphs and the limits laws to evaluate each limit. y=f(x) y=g(x) 118. limx3+(f(x)+g(x))In the following exercises, use the following graphs and the limit laws to evaluate each limit. y = fWIn the following exercises, use the following graphs and the limits laws to evaluate each limit. y=f(x) y=g(x) 120. limx0f(x)g(x)3yIn the following exercises, use the following graphs and the limits laws to evaluate each limit. y=f(x) y=g(x) 121. lim x5 2+g( x ) f( x )In the following exercises, use the following graphs and the limits laws to evaluate each limit. y=f(x) y=g(x) 122. limx1(f(x))2In the following exercises, use the following graphs and the limits laws to evaluate each limit. y=f(x) y=g(x) 123. limx1f(x)g(x)In the following exercises, use the following graphs and the limits laws to evaluate each limit. y=f(x) y=g(x) 124. limx7(xg(x))In the following exercises, use the following graphs and the limits laws to evaluate each limit. y=f(x) y=g(x) 125. limx9[xf(x)+2g(x)]For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions f(x), g(x), and h(x) when possible. 126. [T] True or False? If 2x1g(x)x22x+3 , then limx2g(x)=0For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions f(x), g(x), and h(x) when possible. 127. [T] lim02cos(1)For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions f(x), g(x), and h(x) when possible. 128. limx0f(x),wheref(x)={0,xrationalx2,xirrational[T] In physics, the magnitude of an electric field generated by a point charge at a distance r in vacuum is governed by Coulomb’s law: E(r)=q40r2 , where E represents the magnitude of the electric field, q is the charge of the particle, r is the distance between the particle and where the strength of the field is measured, and 140 is Coulomb's constant: 8.988109Nm2/C2 . Use a graphing calculator to graph E(r) given that the charge of the panicle is q=1010 . Evaluate limr0+E(r). What is the physical meaning of this quantity7? Is it physically relevant? Why are you evaluating from the right?[T] The density of an object is given by its mass divided by its volume: =m/V. Use a calculator to plot the volume as a function of density (V=m/), assuming you are examining something of mass 8 kg ( m = 8). Evaluate lim0+V() and explain the physical meaning.For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 131. f(x)=1xFor the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 132. f(x)=2x2+1For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 133. f(x)=xx2xFor the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 134. g(t)=t1+1For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 135. f(x)=5ex2For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 136. f(x)=|x2|x2For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 137. H(x)=tan2xFor the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 138. f(t)=t+3t2+5t+6For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 139. 2x25x+3x1atx=1For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 140. h()=sincostanat=For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 141. g(u)={6 u 2+u22u1ifu1272ifu=12,atu=12For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 142. f(y)=sin(y)tan(y),aty=1For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 143. f(x)={ x 2 e xifx0x1ifx0,atx=0For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 144. f(x)={xsin(x)ifxxtan(x)ifx,atx=In the following exercises, find the value(s) of k that makes each function continuous over the given interval. 145. f(x)={3x+2xk2x3kx8In the following exercises, find the value(s) of k that makes each function continuous over the given interval. 146. f()={sin02cos(+k),2In the following exercises, find the value(s) of k that makes each function continuous over the given interval. 147. f(x)={ x 2+3x+2x+2x2k,x2In the following exercises, find the value(s) of k that makes each function continuous over the given interval. 148. f(x)={ekx,0x4x+3,4x8In the following exercises, find the value(s) of k that makes each function continuous over the given interval. 149. f(x)={kx,0x3x+1,3x10In the following exercises, use the Intermediate Value Theorem (IVT). 150. Let h(x)={3x24,x25+4x,x2 over the interval [0, 4], there is no value of x such that h(x) = 10, although h(0) < 10 and h(4) > 10. Explain why this does not contradict the IVT.In the following exercises, use the Intermediate Value Theorem (IVT). 151. A particle moving along a line has at each time t a position function s(t), which is continuous. Assume s(2) = 5 and s(5) = 2. Another particle moves such that its position is given by h(t)=s(t)t. Explain why there must be a value c for 2 < c < 5 such that h(c) = 0.In the following exercises, use the Intermediate Value Theorem (IVT). 152. [T] Use the statement “The cosine of t is equal to t cubed.” Write a mathematical equation of the statement. Prove that the equation in part a. has at least one real solution. Use a calculator to find an interval of length 0.01 that contains a solution.In the following exercises, use the Intermediate Value Theorem (IVT). 153. Apply the IVT to determine whether 2x=x3 has a solution in one of the intervals [1.25, 1.375] or [1.375, 1.5]. Briefly explain your response for each interval.Consider the graph of the function y=f(x) shown in the following graph. Find all values for which the function is discontinuous. For each value in part a., state why the formal definition of continuity does not apply. Classify each discontinuity as either jump, removable, or infinite.Let f(x)={3x,x1x3,x1 . Sketch the graph of f. Is it possible to find a value k such that f(l) = k, which makes f(x) continuous for all real numbers? Briefly explain.Let f(x)=x41x21forx1,1 . a. Sketch the graph of f. b. Is it possible to find values k1and k2such that f(1)=k and f(1)=k2 , and that makes f(x) continuous for all real numbers? Briefly explain.Sketch the graph of the function y=f(x) with properties i. through vii. i. The domain of f is (,+) . ii.f has an infinite discontinuity at x = -6. iii.f(-6) = 3 iv.limx3f(x)=limx3+f(x)=2 v.f(3)=3 vi.f is left continuous but not right continuous at x = 3. vii.limxf(x)= and limx+f(x)=+Sketch the graph of the function y=f(x) with properties i. through iv. i.The domain of f is [0, 5]. ii.limx1+f(x) and limx1f(x) exist and are equal. iii. f(x) is left continuous but not continuous at x = 2, and right continuous but not continuous at x= 3. f(x) has a removable discontinuity at x = 1, a jump discontinuity at x = 2, and the following limits hold: limx3f(x)= and limx3+f(x)=2 .In the following exercises, suppose y=f(x) is defined for all x. For each description, sketch a graph with the indicated property. 159. Discontinuous at x = 1 limx1f(x)=1 and limx2f(x)=4In the following exercises, suppose y=f(x) is defined for all x. For each description, sketch a graph with the indicated property. 160. Discontinuous at x = 2 but continuous elsewhere with limx0f(x)=12Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 161. f(t)=2etet is continuous everywhere.Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 162. If the left- and right-hand limits of f(x) as xa exist and are equal, then f cannot be discontinuous at x = a.Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 163. If a function is not continuous at a point, then it is not defined at that point.Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 164. According to the IVT, cosxsinxx=2 has a solution over the interval [-1, 1].Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 165. If f(x) is continuous such that f(a) and f(b) have opposite signs, then f(x) = 0 has exactly one solution in [a, b].Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 166. The function f(x)=x24x+3x21 is continuous over the interval [0, 3].Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 167. If f(x) is continuous everywhere and f(a), f(b) >0, then there is no root of f(x) in the interval [a, b].[T] The following problems consider the scalar form of Coulomb’s law, which describes the electrostatic force between two point charges, such as elections. It is given by the equation F(r)=ke|q1q2|r2 where keis Coulomb's constant, qi are the magnitudes of the charges of the two particles, and r is the distance between the two particles. 168. To simplify the calculation of a model with many interacting particles, after some threshold value r = R, we approximate F as zero. Explain the physical reasoning behind this assumption. What is the force equation? Evaluate the force F using both Coulomb’s law and our approximation, assuming two protons with a charge magnitude of 1.60221019 coulombs (C), and the Coulomb constant ke=8.988109Nm2/C2 are 1 m apart. Also, assume R < 1 m. How much inaccuracy does our approximation generate? Is our approximation reasonable? Is there any finite value of R for which this system remains continuous at R?[T] The following problems consider the scalar form of Coulomb’s law, which describes the electrostatic force between two point charges, such as elections. It is given by the equation F(r)=ke|q1q2|r2 where keis Coulomb's constant, qi are the magnitudes of the charges of the two particles, and r is the distance between the two particles. 169. Instead of making the force 0 at R, instead we let the force be 1020 for rR. Assume two protons, which have a magnitude of charge 1.60221019C , and the Coulomb constant ke=8.988109Nm2/C2 . Is there a value R that can make this system continuous? If so, find it. Recall the discussion on spacecraft from die chapter opener. The following problems consider a rocket launch from Earth's surface. The force of gravity on the rocket is given by F(d)=mk/d2, where m is the mass of the rocket, d is the distance of the rocket from the center of Earth, and k is a constant.[T] The following problems consider the scalar form of Coulomb’s law, which describes the electrostatic force between two point charges, such as elections. It is given by the equation F(r)=ke|q1q2|r2 where keis Coulomb's constant, qi are the magnitudes of the charges of the two particles, and r is the distance between the two particles. 170. [T] Determine the value and units of k given that the mass of the rocket on Earth is 3 million kg. (Hint: The distance from the center of Earth to its surface is 6378 km.)[T] After a certain distance D has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function F(d)={mk d 2ifdD10,000ifdD by. Find the necessary condition D such that the force function remains continuous.As the rocket travels away from Earth’s surface, there is a distance D where the rocket sheds some of its mass, since it no longer needs the excess fuel storage. We can write this function as F(d)={ m 1k d 2ifdD m 2k d 2ifdD . Is there a D value such that this function is continuous, assuming m1m2 ?wqProve the following functions are continuous everywhere. 173. f()=sinProve the following functions are continuous everywhere. 174. g(x)=|x|Prove the following functions are continuous everywhere. 175. Whereisf(x)={oifxisirrational1ifxisrationalcontinuous?In the following exercises, write the appropriate definition for each of the given statements. 176. limxaf(x)=NIn the following exercises, write the appropriate definition for each of the given statements. 177. limtbg(t)=MIn the following exercises, write the appropriate definition for each of the given statements. 178. limxch(x)=LIn the following exercises, write the appropriate definition for each of the given statements. 179. limxa(x)=AThe following graph of the function f satisfies limx2f(x)=2 . In the following exercises, determine a value of 0 that satisfies each statement. 180. if 0|x2|,then|f(x)2|1 .The following graph of the function f satisfies limx2f(x)=2 . In the following exercises, determine a value of 0 that satisfies each statement. 181. If 0|x2|,then|f(x)2|0.5The following graph of the function f satisfies limx3f(x)=1 . In the following exercises, determine a value of 0 that satisfies each statement. 182. if 0|x3| , then |f(x)+1|1 .The following graph of the function f satisfies limx3f(x)=1 . In the following exercises, determine a value of 0 that satisfies each statement. 183. If 0|x3|,then|f(x)+1|2The following graph of the function f satisfies limx3f(x)=2 . In the following exercises, for each value of , find a value of 0 such that the precise definition of limits holds true. 184. =1.5The following graph of the function f satisfies limx3f(x)=2 . In the following exercises, for each value of , find a value of 0 such that the precise definition of limits holds true. 185. =3[T] In the following exercises, use a graphing calculator to find a number such that the statements hold true. 186. |sin(2x)12|0.1,whenever|x12|[T] In the following exercises, use a graphing calculator to find a number such that the statements hold true. 187. |x42|0.1,whenever|x8|In the following exercises, use the precise definition of limit to prove the given limits. 188. limx2(5x+8)=18In the following exercises, use the precise definition of limit to prove the given limits. 189. limx3x29x3=6In the following exercises, use the precise definition of limit to prove the given limits. 190. limx22x23x2x2=5In the following exercises, use the precise definition of limit to prove the given limits. 190. limx0x4=0In the following exercises, use the precise definition of limit to prove the given limits. 192. limx2(x2+2x)=8In the following exercises, use the precise definition of limit to prove the given one-sided limits. 193. limx55x=0In the following exercises, use the precise definition of limit to prove the given one-sided limits. 194. limx0+f(x)=2,wheref(x)={8x3,ifx04x2,ifx0.In the following exercises, use the precise definition of limit to prove the given one-sided limits. 195. limx1f(x)=3,wheref(x)={5x2,ifx17x1,ifx1.In the following exercises, use the precise definition of limit to prove the given infinite limits. 196. limx01x2=In the following exercises, use the precise definition of limit to prove the given infinite limits. 197. limx13( x+1)2=In the following exercises, use the precise definition of limit to prove the given infinite limits. 198. limx21( x2)2=An engineer is using a machine to cut a flat square of Aerogel of area 144 cm2. If there is a maximum error tolerance in the area of 8 cm2, how accurately must the engineer cut oil the side, assuming all sides have the same length? How do these numbers relate to ,,aandL ?Use the precise definition of limit to prove that the following limit does not exist: limx1|x1|x1 .Using precise definitions of limits, prove that limx0f(x) does not exist, given that f(x) is the ceiling function. (Hint: Try any 1 .)Using precise definitions of limits, prove that limx0f(x) does not exist: f(x)={1ifxisrational0ifxisirrational (Hint: Think about how you can always choose a rational number 0 < r < d, but |f(r)0|=1 .)Using precise definitions of limits, prove that limx0f(x) does not exist: f(x)={xifxisrational0ifxisirrational (Hint: Break into two cases, x rational and x irrationalUsing the function from the previous exercise, use the precise definition of limits to show that limxaf(x) does not exist for a0 . For the following exercises, suppose that limxaf(x)=L and limxag(x)=M both exist. Use the precise definition of limits to prove the following limit laws:limxa(f(x)g(x))=LMlimxa[cf(x)]=cL for any real constant c (Hint. Consider two cases: c = 0 and c0 .)limxa[f(x)g(x)]=LM.(Hint:|f(x)g(x)LM|=|f(x)g(x)f(x)M+f(x)MLM||f(x)||g(x)M|+|M||f(x)L|).wTrue or False. In the following exercises, justify your answer with a proof or a counterexample. 208. A function has to be continuous at x = a if the limxaf(x) exists.True or False. In the following exercises, justify your answer with a proof or a counterexample. 209. You can use the quotient rule to evaluate limx0sinxx .True or False. In the following exercises, justify your answer with a proof or a counterexample. 210. If there is a vertical asymptote at x = a for the function f(x), then f is undefined at the point x = a.True or False. In the following exercises, justify your answer with a proof or a counterexample. 211. If limxaf(x) does not exist, then f is undefined at the point x = a.Using the graph, find each limit or explain why the limit does not exist. a. limx1f(x) b. limx1f(x) c. limx0+f(x) d. limx2f(x)In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 213. limx22x23x2x2In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 214. limx03x22x+4In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 215. limx3x32x213x2In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 216. limx/2cotxcosxwIn the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 217. limx5x2+25x+5In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 218. limx23x22x8x24In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 219. limx1x21x31In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 220. limx1x21x1In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 221. limx44xx2In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 222. limx41x2In the following exercises, use the squeeze theorem to prove the limit. 223. limx0x2cos(2x)=0In the following exercises, use the squeeze theorem to prove the limit. 224. limx0x3sin(x)=0In the following exercises, use the squeeze theorem to prove the limit. 225. Determine the domain such that the function f(x)=x2+xex is continuous over its domain.In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. 226. f(x)={x2+1,xc2x,xcIn the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. 227. f(x)={x+1,x1x2+c,x1In the following exercises, use the precise definition of limit to prove the limit. 228. limx1(8x+16)=24In the following exercises, use the precise definition of limit to prove the limit. 229.A ball is thrown into the air and the vertical position is given by x(t)=4.9t2+25t+5 . Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.A particle moving along a line has a displacement according to the function x(t)=t22t+4 , where x is measured in meters and t is measured in seconds. Find the average velocity over the time period t = [0, 2].From the previous exercises, estimate the instantaneous velocity at t = 2 by checking the average velocity within t = 0.01 sec.For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). f(x)=4x+7;x1=2;x2=5For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 2. f(x)=8x3;x1=1,x2=3For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 3. f(x)=x2+2x+1;x1=3,x2=3.5For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 4. f(x)=x2+x+2;x1=0.5,x2=1.5For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 5. f(x)=43x1;x1=1,x2=3For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 6. f(x)=x72x+1;x1=2,x2=0For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 7. f(x)=x;x1=1,x2=16For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 8. f(x)=x9;x1=10,x2=13For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 9. f(x)=x1/3+1;x1=0,x2=8For the following exercises, use Equation 3.3 to find the slope of the secant line between the values x1 and x2 for each function y=f(x). 10. f(x)=6x2/3+2x1/3;x1=1,x2=27For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 11. f(x)=34x,a=2For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 12. f(x)=x5+6,a=1For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 13. f(x)=x2+x,a=1For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 14. f(x)=1xx2,a=0For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 15. f(x)=7x,a=3For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 16. f(x)=x+8,a=1For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 17. f(x)=23x2,a=2For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 18. f(x)=3x1,a=4For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 19. f(x)=2x+3,a=4For the following functions, use Equation 3.4 to find the slope of the tangent line mtan=f(a), and find the equation of the tangent line to f at x = a. 20. f(x)=3x2,a=3