Study Guide 1.2 - 5.6

Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Student: Jalen Wilson Date: 07/12/24 Assignment: Study Guide: 1.2 - 5.6 1. Find the domain and range of the function. ​f(x) = − 6 + x A. ​D: ​[0,∞​), ​R: ​( − ∞​,∞​) B. ​D: ​( − ∞​,0], ​R: ​( − ∞​, − 6​] C. ​D: ​[0,∞​), ​R: ​[ − 6​,∞​) D. ​D: ​( − ∞​,∞​), ​R: ​[ − 6​,∞​) 2. Find the average rate of change of the function over the given interval. 2 y = x + 1​x, ​[3​, 9​] A. 10 B. 13 C. 15 D. 26 3 3. Find the average rate of change of the function over the given interval. y= 3 x−2 ​, ​[4, 7] A. 7 B. 2 C. 1 3 D. − 3 10 4. Use the graph to evaluate the limit. A. − 1 lim f(x) B. 2 x →0 C. − 2 5 y D. The limit does not exist. 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 x 1 2 3 4 5 5. Find the limit if it exists. 2 lim 9x − 8x − 10 x →6 A. 362 B. 266 C. 286 D. 382 E. The limit does not exist. 6. Find the limit if it exists. lim 2x + 87 x→ − 3 A. 81 B. − 9 C. − 81 D. 9 E. The limit does not exist. 7. Find the​limit, if it exists. 2 x −9 lim x−3 x →3 A. 3 B. 1 C. 6 D. The limit does not exist. 8. Find the​limit, if it exists. 2 lim x + 17x + 70 x→ − 7 x+7 A. 17 B. 238 C. 3 D. The limit does not exist. 9. Find the​limit, if it exists. 2 x − 25 lim 2 x→5 x − 6x + 5 A. 0 B. C. 5 4 5 2 D. The limit does not exist. 10. 2 The inequalities 1 − x 2 < sin x x < 1 hold when x is measured in radians and x < 1. Find lim x →0 sin x x ​, if it exists. A. 0 B. 1 C. 0.0007 D. The limit does not exist. 11. Find all points where the function is discontinuous. 5 y B. x = 2 4 C. x = − 2 and x = 0 3 D. x = − ​2, x = ​0, and x = 2 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 12. Find the limit. 7 lim −4 x x →∞ A. 4 B. − 4 C. 3 D. − 11 A. x = 0 and x = 2 x 1 2 3 4 5 E. There are no points where the function is discontinuous. 13. Find the limit. 2 x − 6x + 18 lim 3 2 x→∞ x − 2x + 5 A. ∞ B. 0 18 C. 5 D. 1 14. Find the limit. 2 − 10x − 8x + 15 lim 2 x→∞ − 17x + 8x + 19 A. 1 B. ∞ 10 C. 17 15 D. 19 15. Find an equation for the tangent to the curve at the given point. 2 y = x + 2​, ​( − 4​,18​) A. y = − 8x − 30 B. y = − 8x − 14 C. y = − 4x − 14 D. y = − 8x − 28 16. Find an equation for the tangent to the curve at the given point. 3 ​h(t) = t − 16t + 5​, ​(4​,5​) A. y = 37t − 123 B. y = 32t − 123 C. y = 5 D. y = 32t + 5 17. Find the indicated derivative. dy dx 3 if y = 9x A. 27x B. 27x2 C. 3x2 D. 27x3 18. Find the indicated derivative. dt dx x if t = A. 3 − B. 4x − 3 4x − 3 8x − 3 (4x − 3) C. 2 3 − (4x − 3) D. 2 3x − (4x − 3) 2 19. Find the second derivative. 4 2 y = 6x − 6x + 7 2 A. 24x − 12x 2 B. 24x − 12 2 C. 72x − 12 2 D. 72x − 12x 20. Find the second derivative. y= 1 2 1 + 3x 5x A. − 2 5x 6 B. 5x C. − 4 − 5x 5x 4 1 + 3 3x 2 3 3x 2 6 D. 4 3 + 1 − 2 3x 2 3 3x 21. At time​t, the position of a body moving along the​s-axis is s = t3 − 21t2 + 144t m. Find the​body's acceleration each time the velocity is zero. A. ​a(12​) = 72 ​m/sec2 ​, ​a(16​) = 12 ​m/sec2 B. ​a(6​) = − 6 ​m/sec2 ​, ​a(8​) = 6 ​m/sec2 C. ​a(6​) = 6 ​m/sec2 ​, ​a(8​) = − 6 ​m/sec2 D. ​a(6​) = 0 ​m/sec2 ​, ​a(8​) = 0 ​m/sec2 22. The function s = ​f(t) gives the position of a body moving on a coordinate​line, with s in meters and t in seconds. Find the​body's speed and acceleration at the end of the time interval. 3 2 s = − t + 4t − 4​t, 0 ≤ t ≤ 4 2 A. 20 ​m/sec, − 16 ​m/sec B. 20 ​m/sec, − 4 ​m/sec C. 4 ​m/sec, 0 ​m/sec D. − 20 ​m/sec, − 16 ​m/sec 2 2 2 23. Find the derivative. y= sin x 7x + 7x sin x dy x cos x + sin x 7 sin x + 7x cos x + A. dx = 2 2 7x sin x dy x cos x − sin x 7 sin x − 7x cos x + B. dx = 2 2 7x sin x C. dy dx = cos x 7 + 7 cos x dy sin x − x cos x 7x cos x − 7 sin x + D. dx = 2 2 49x sin x 24. Find the derivative. 7 r = 2 − θ cos θ dr A. dθ B. dθ C. D. dr dr dθ dr dθ 6 7 = 7θ cos θ − θ sin θ 6 7 = 7θ sin θ − θ cos θ 6 7 = − 7θ cos θ + θ sin θ 6 = = 7θ sin θ 25. Given y = ​f(u) and u = ​g(x), find dy dx = f′​(g(x))g′​(x). 2 y = ​u(u − ​1), u = x + x A. 2 2 4x 1 x + + B. 2 2 4x x + C. 4 3 6 2 2x x + x − D. 4 3 6 2 1 x + x − 26. Write the function in the form y = ​f(u) and u = ​g(x). Then find y = (2x + 11) 6 6 A. y = u ​; u = 2x + 11​; 6 B. y = u ​; u = 2x + 11​; 6 C. y = u ​; u = 2x + 11​; dy dx dy dx dy dx 6 dy D. y = 6u + 11​; u = x ​; dx = 12(2x + 11) = 6(2x + 11) = 2(2x + 11) 5 6 5 = 12x 27. Find the derivative of y with respect to x. y = ln (x − 2) 1 A. 2−x 1 B. x−2 1 C. x+2 D. − 1 x+2 28. Determine all critical points for the function. 3 2 ​f(x) = x − 3x + 4 A. x = 0 and x = 2 B. x = 0 C. x = 0 and x = 1 D. x = − 1 and x = 1 5 dy dx as a function of x. 29. Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′​(c) in the conclusion of the Mean Value Theorem for the function and interval. 2 ​f(x) = x + 5x + 2​, ​[ − 2​,3​] A. ​0, B. − C. 1 2 1 1 ​, 2 2 1 2 D. − 2​, 3 30. Determine the​x-value of all critical points for the following function. 2 f(x) = x x−3 2 The​x-value(s) of the critical​point(s) of f(x) = x x−3 ​(Use a comma to separate answers as​needed.) ​is/are x = 0,6 . 31. Using the derivative of​f(x) given​below, determine the critical points of​f(x). f′​(x) = ​(x + 2​)(x + 9​) A. ​0, − 9​, − 2 B. − 11 C. − 9​, − 2 D. 2​, 9 32. Using the derivative of​f(x) given​below, determine the critical points of​f(x). 2 f′​(x) = (x − 4) ​(x + 8​) A. − 4​, ​0, 8 B. − 4​, 8 C. − 8​, 4 D. − 8​, − 4​, 4 33. Using the derivative of​f(x) given​below, determine the intervals on which​f(x) is increasing or decreasing. 1 3 f′​(x) = x ​(x − 2​) A. Decreasing on​(0, 2​); increasing on ​(2​, ∞​) B. Decreasing on​(0, 2​); increasing on ​( − ∞​, ​0) ∪ ​(2​, ∞​) C. Increasing on​(0, ∞​) D. Decreasing on ​( − ∞​, ​0) ∪ ​(2​, ∞​); increasing on​(0, 2​) 34. Use​l'Hôpital's Rule to evaluate the limit. 2 lim x →5 x − 7x + 10 x−5 A. 10 B. 17 C. 3 D. − 2 35. Use​l'Hôpital's Rule to evaluate the limit. 3 2 x − 10x + 9 lim x →1 x−1 A. 23 B. 20 C. 13 D. − 17 36. Use​l'Hôpital's Rule to evaluate the limit. 2 15x − 3x + 5 lim 2 x→∞ 2x + 5x + 2 A. 1 B. 2 15 C. − D. 15 2 15 2 37. Find an antiderivative of the given function. − 12 4 x A. 3 5 x B. 4 3 x C. 4 4 x D. − 3 4 x 38. Find the most general antiderivative. ∫ 1 3 1 3 −x − A. 1 3x B. − C. 4 dx 3 x − x 4 4 +C x x − − +C 2 4 3 2 1 4x 1 3x 4 1 2x − x 1 − + +C 4 2 9 2 3 D. − 3x − 3x + C 39. Find the most general antiderivative. ∫ ( − 4 cos t)dt A. − B. − 4 sin t sin t 4 +C +C C. − 4 sin t + C D. − 4 cos t + C 40. Find the most general antiderivative. ∫ sin θ( cot θ + csc θ)dθ A. cos θ + C B. csc θ + cos θ + C C. sin θ + θ + C D. sin θ + C 41. Which of the graphs shows the solution of the given initial value​problem? dy dx = − 6​x, y = 0 when x = − 1 A. B. 10 y C. 10 x -10 10 -10 y D. 10 x -10 10 -10 y 10 x -10 10 -10 y x -10 10 -10 42. Solve the initial value problem. ds dt = cos t − sin t, s π = 12 2 A. s = sin t + cos t + 13 B. s = sin t − cos t + 11 C. s = 2 sin t + 10 D. s = sin t + cos t + 11 43. Suppose the velocity of a body moving along the​s-axis is t = 4 given that s = s0 when t = 0. A. 56.8 B. − 2 C. 52.8 D. Not enough information is given. 44. Evaluate the integral. π/ 4 ∫ 0 7 sec θ tan θ dθ A. 7 1 − B. 2 2 −1 C. 7 2 −1 D. 0 45. Evaluate the integral using the given substitution. ∫ x cos A. 2 2x 1 4 2 dx​, u = 2x 2 sin 2x 2 +C B. x 2 2 sin 2x 2 C. sin 2x D. 1 u +C sin u + C +C ds dt = 9.8t − 1. Find the​body's displacement over the time interval from t = 2 to 46. Evaluate the integral using the given substitution. dx ∫ ​, 7x + 10 u = 7x + 10 A. 2 7x + 10 + C 7 B. 2 1 7x + 10 2 C. +C 7x + 10 + C 7 1 D. 7(7x + 10) +C 3/2 47. Evaluate the integral. ∫ x2 3 x + 7 dx A. 2 x3 + 7 B. 2 3 C. − D. 3 x +7 2 3 9 3 x +7 +C 3/2 x +7 3 2 3/2 +C − 1/2 3/2 +C +C 48. Use the substitution formula to evaluate the integral. π ∫ (1 + cos 9t)2 sin 9t dt 0 A. B. C. D. 1 27 8 27 1 9 8 3 49. Find the area enclosed by the given curves. 2 y = 2x − x ​, y = 2x − 4 A. B. C. D. 37 3 31 3 34 3 32 3 50. Find the area enclosed by the given curves. 2 y = ​x, y = x A. B. C. D. 1 3 1 12 1 2 1 6