For problems 21-23, find the horizontal, vertical, and oblique asymptotes. 21. f(x) = 22. f(x) =...

       

Transcribed Image Text:

For problems 21-23, find the horizontal, vertical, and oblique asymptotes. 21. f(x) = 22. f(x) = 23. f(x) = 24. Determine the domain such that the function f(x)=√x-2 + xe* is continuous over its domain. 25. x³-1 (3e-4) (Se*+2) In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. [√2+1, m> -1 (a²-cz≤-1 f(x) = [2²+1,z > c (2x, 2 Sc 26. 27. A particle moving along a line has a displacement according to the function x(t) = t² - 2t+4, where x is measured in meters and t is measured in seconds. Find the average velocity over the time period t = [0,2]. 28. From the previous exercise, estimate the instantaneous velocity at t = 2 by checking the average velocity within t = 0.01 sec. True or False? In problems 29-32, justify your answer with an explanation or a counterexample. 29. Every function has a derivative. 30. A continuous function has a continuous derivative. 31. A continuous function has a derivative. 32. If a function is differentiable, it is continuous. Use the limit definition to find the derivatives of the functions in problems 33 and 34. 33. f(x) = 34. f(x) = √√x +4 = x In problems 35-38, find the derivatives of the given functions. 4 35. f(x) = 3x³ - x2 36. f(x) = (4x²)² 37. f(x) = ex sin x 38. f(x) = x²cos x + √x tan (x) In problems 39-41, find the specified-order derivative of each function. 39. First derivative of y = xe* cos x 40. Third derivative of y = (3x + 2)² 41. Second derivative of y=x+ + x² sin (x) In 42 and 43, find the equation of the tangent line to the following equations at the specified point. 42. y = cos x + xatx = 0 43. y = x + e*at x = 1 44. Determine where the graph of y = x³ + x²-x-1 has a horizontal tangent line. In 45 and 46, draw the derivative of the given graphs. 45. 46. میں -2/-1 2 + 2-1 1 2 47. For the following graph, a. determine for which values of x = a the limf (x) exists but f is not continuous at x = a, and X b. determine for which values of x = a the function is continuous but not differentiable at x = a. + + 1 2 - لا 3 12 V -2 -1 1 -1 -2 AX 48. Use the graph to evaluate a. f'(-0.5), b. f'(0), c. f'(1), d. f'(2), and e. f'(3), if it exists. 5 -4 -2 -1 NO st 3 2 1 0 -1 + -2 + 1 2 3 4 5 49. A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after t seconds is given by s(t) = 16t² + 100t + 85. a. Find the velocity of the potato after 0.5 s and 5.75 s. b. Find the speed of the potato at 0.5 s and 5.75 s. c. Determine when the potato reaches its maximum height. d. Find the acceleration of the potato at 0.5 s and 1.5 s. e. Determine how long the potato is in the air. f. Determine the velocity of the potato upon hitting the ground. g. Determine the total distance traveled by the potato. For problems 21-23, find the horizontal, vertical, and oblique asymptotes. 21. f(x) = 22. f(x) = 23. f(x) = 24. Determine the domain such that the function f(x)=√x-2 + xe* is continuous over its domain. 25. x³-1 (3e-4) (Se*+2) In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. [√2+1, m> -1 (a²-cz≤-1 f(x) = [2²+1,z > c (2x, 2 Sc 26. 27. A particle moving along a line has a displacement according to the function x(t) = t² - 2t+4, where x is measured in meters and t is measured in seconds. Find the average velocity over the time period t = [0,2]. 28. From the previous exercise, estimate the instantaneous velocity at t = 2 by checking the average velocity within t = 0.01 sec. True or False? In problems 29-32, justify your answer with an explanation or a counterexample. 29. Every function has a derivative. 30. A continuous function has a continuous derivative. 31. A continuous function has a derivative. 32. If a function is differentiable, it is continuous. Use the limit definition to find the derivatives of the functions in problems 33 and 34. 33. f(x) = 34. f(x) = √√x +4 = x In problems 35-38, find the derivatives of the given functions. 4 35. f(x) = 3x³ - x2 36. f(x) = (4x²)² 37. f(x) = ex sin x 38. f(x) = x²cos x + √x tan (x) In problems 39-41, find the specified-order derivative of each function. 39. First derivative of y = xe* cos x 40. Third derivative of y = (3x + 2)² 41. Second derivative of y=x+ + x² sin (x) In 42 and 43, find the equation of the tangent line to the following equations at the specified point. 42. y = cos x + xatx = 0 43. y = x + e*at x = 1 44. Determine where the graph of y = x³ + x²-x-1 has a horizontal tangent line. In 45 and 46, draw the derivative of the given graphs. 45. 46. میں -2/-1 2 + 2-1 1 2 47. For the following graph, a. determine for which values of x = a the limf (x) exists but f is not continuous at x = a, and X b. determine for which values of x = a the function is continuous but not differentiable at x = a. + + 1 2 - لا 3 12 V -2 -1 1 -1 -2 AX 48. Use the graph to evaluate a. f'(-0.5), b. f'(0), c. f'(1), d. f'(2), and e. f'(3), if it exists. 5 -4 -2 -1 NO st 3 2 1 0 -1 + -2 + 1 2 3 4 5 49. A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after t seconds is given by s(t) = 16t² + 100t + 85. a. Find the velocity of the potato after 0.5 s and 5.75 s. b. Find the speed of the potato at 0.5 s and 5.75 s. c. Determine when the potato reaches its maximum height. d. Find the acceleration of the potato at 0.5 s and 1.5 s. e. Determine how long the potato is in the air. f. Determine the velocity of the potato upon hitting the ground. g. Determine the total distance traveled by the potato.