MAT 131 Calculus 1 Final Review PDF

MAT 131 Review for Final Exam Disclaimer: The following problems are provided for you to focus your studying efforts. There is no guarantee that these are the only kind of questions that will be on the exam. Every effort has been made to provide accurate ans...

MAT 131 Review for Final Exam Disclaimer: The following problems are provided for you to focus your studying efforts. There is no guarantee that these are the only kind of questions that will be on the exam. Every effort has been made to provide accurate answers, but mistakes are possible. Please inform me if you find an error. Evaluate the limits in #1-6. 10. At the point (a, f(a)), what are the signs of f′ (x) and f ″ (x) ? x+3 1. lim x →2 x −1 11. Is f continuous at a? At b? x 2 − 3x + 2 dy Find for # 12 – 20. 2. lim x →1 x −1 dx 12. y = (2x2 – 1)(x3 + 2) 2 x2 − 5x + 1 3. lim x →∞ 3x 2 + 7 x2 + 1 13. y= x2 + 1 x 4. lim x →∞ 3 x − 2 14. y = 3x5 − 4 x 2 sin(2θ ) 5. lim θ →0 θ sin x 15. y= 1 − cos x x 6. lim− x →0 x 16. y = esin x Use the graph of y = f(x) below to answer #7-11. 17. y = ln(cos 2 x) 18. y = (ln x)3 2 19. y = x 3e 4 − x 20. 8y – xy –x2 = 2 Integrate #21 – 25. 7. lim− f ( x) = x →b 1 1 21. ∫ (2 x − 3 + − e)dx x x 8. lim f ( x) = x→b + ∫x x 4 + 5 dx 3 22. 9. lim f ( x) = x →b  36. A ball is tossed into the air from a bridge, ∫t ⋅e t 2 +1 23. dt and its height s(t) in feet above the ground t seconds later is given by ∫ sin θ cosθ dθ 6 24. s(t) = -16t2 + 50 t + 36. a. What is the instantaneous velocity of the cos x ball when t = 1? 25. ∫ 1 + sin x dx b. What is the maximum height reached by the ball? Evaluate the definite integrals in #26 – 30. 37. Find the area enclosed by the curve π y = x2 + 3x + 2 and the x-axis. 4 26. ∫ tan θ dθ 0 38. Find the average value of y = x2 over [-1, 5]. 39. An ice cube is melting so that the edge 3 dx decreases in length by 0.10 cm/min. How fast is 27. ∫1 5 − x the volume changing when the edge is 4.50 cm? 1 40. Use a left-endpoint Riemann sum to 4 ∫−1 ( x + 2)3 dx 5 28. approximate ∫ f ( x)dx from the given table. 0 e x 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ln x 2 29. ∫1 x dx f(x) 12 14 15 16 15 11 10 8 5 4 2 1 3x d ⎡ x ⎤ 30. ∫0 ( x 2 + 1)2 dx 41. 3t ⎢ ∫ 2 dt ⎥ = dx ⎣ 2 t + 1 ⎦ 31. Write an equation of the tangent line to y = 2x6 – 6x2 at the point where x = 1. 32. Write an equation of the tangent line to f(x) = ln (x2 – 1) at the point where x = 2. 33. For f(x) = x2 ex, find the local maxima, minima, and inflection points. 34. You are planning to make an open box from an 8 ″ by 15 ″ piece of cardboard by cutting squares from the corners and folding up the sides. What are the dimensions of the box having maximum volume? 35. Given f (x) = 2x3 – 15x2 + 36x + 10. For what values of x is f (x) increasing? Concave up? ANSWERS 33. max at (-2,.5413), min at (0, 0), inflection points at x = −2 ± 2 : (-.586,.191) and 1. 5 (-3.414,.384) 2. –1 34. x = 5/3, Dimensions: 14/3 by 35/3 by 5/3 3. 2/3 35. Increasing on (-∞, 2) ∪ (3, ∞). 4. ∞ Concave up on (5/2, ∞). 5. 2 36. a. 18 ft/sec b. 75.06 ft 6. –1 37. 1/6 7. d 38. 7 8. c 39. –6.075cm3/min 9. does not exist 40. 55 10. 0, + 3x 11. yes; no 41. 12. 10x4 – 3x2 + 8x x2 + 1 3 12 1 − 32 3x 2 − 1 13. x − x = 3 2 2 2x 2 15 x − 8 x 4 14. 2 3x5 − 4 x 2 1 15. cos x − 1 sin x 16. e cos x 17. –2 tan x 3(ln x)2 18. x 2 2 19. −2 x4e4− x + 3x2e4− x 2x + y 20. 8− x 3 2 21. x 2 − x 3 + ln x − ex + C 2 3 1 4 22. ( x + 5) 2 + C 6 1 t 2 +1 23. e +C 2 sin 7 θ 24. +C 7 25. ln | 1 + sin x | + C 26. ½ ln 2 ≈.3466 27. ln 2 ≈.693 28. 16/9 29. 1 30. ¾ 31. y = -4 32. y = 4/3 (x – 2) + ln 3