midterm review calc ab

SKip 35,5¢ Calculus 1st Semester Final Review 1. Use the graph to find lim f (x) (if it exists). 9. Determine the value of ¢ so that f(x) is continuous on the s « HS3 entire real line if f(x)= el xx>3 10. Find the limit: lim — 2 3% =6x+9 11. Find all vertical asymptotes of the graph of x=0 J 34 1 + ==X 1 2 . L K 4x-2 2. Find the limit: lim ———. x=1 .X—3 X 44, 121 3. Find the limit: lim /' (x) if f(x)= b 5 2 X" +3x-1 fy="rm x+7 2 . . x -x-2 12. Find the vertical asymptote(s) of f(x)= 3 X +x-6 13. At each point indicated on the graph, determine whether the value of the derivative is positive, negative, zero, or if the function has no derivative. 7 14. Use the definition of a derivative to calculate the derivative of S =x2+2. 15. Find an equation of the tangent line to the graph of f{x) = 3x> +2xwhenx=1. 2 F=1 ok o Ptx-2 4. Find the limit: lim ¢ x—1 xX— 1 2 1-42x" -1 5. Find the limit: Tim fax x>l x—1 4 1, <0 6. Let f(x)= . Find each limit (if it exists). 2x-3, x>0 a. lim f(x) x>0 b. lim f(x) x—0" ¢ lim f(x) x—0" : . x-2 7. Find the values of x for which f(x) = 3 is x -4 discontinuous and label these discontinuities as removable or nonremovable. 16. Find the values of x for all points on the graph of f(x) = x3 - 2x2 + 5x — 16 at which the slope of the tangent line is 4. 17. Find all points at which the graph of f(x) = x> — 3x has horizontal tangent lines. 8. Letflx) = = and g(x) =x*. x-1 a. Find flg(x)). b. Find all values of x for which f{g(x)) is discontinuous. 18. The position function for an object is given by s(7) = 612 + 2401, where s is measured in feet and 7 is measured in seconds. Find the velocity of the object when 7 = 2 seconds. 2x 19. Differentiate: y — (1-3x7) 2 d 20, Caleulate 2 if y=—". dx 2-x 1-x

36. Determine whether the Mean Value Theorem applies to f(x) = i s i 2 21. Find the derivative of y=Vx +x. 3x — x2 on the interval [2, 3]. If the Mean Value Theorem can be applied, find all value(s) of ¢ in the interval such that 22. Find the derivative of y = o2 +2x+ 5)6. , f3)-f(2) f(¢) ==—————. If the Mean Value Theorem does not P o 23. Find LA for y=+2x+ 1(x3). apply, state why. dx . ; 7 Loz . 24. The position equation for the movement of a particle is given 37. Find the open intervals on which f (x)=— is increasing or bys= (13 + 1)2 where s is measured in feet and f is measured d . * in seconds. Find the acceleration of this particle at 1 second. cereasing. dy 38. Find the open intervals on which fix) = X3 =322 is increasing 25. Find — if y= . or decreasing. dx x+y 39, Use the graph to identify the open intervals on which the dy function is increasing or decreasing. 26. Use implicit differentiation to find — for X2 4xy+ y2 =5. y a j o1 ; 4_ 2 e 61 ' 27. Find the slope of the curve y* — xy* = x at the point | —, 1. 3 A 2 ko 1 1 ettt 28. The radius of a circle is increasing at the rate of 5 inches per 13 4 minute. At what rate is the area increasing when the radius is =37 : 10 inches? -6+ ' 1 i A . 04 ] 29. Airis being pumped into a spherical balloon at a rate of 28 ' cubic feet per minute. At what rate is the radius changing when the radius is 3 feet? 2x 4 40. Find all relative extrema of y = e [V == nrsj (x+4) 3 41. Find the relative minimum and relative maximum for f{x) = . —x+4x-3 2x3 +3x% - 12x. 30. A point moves along the curve y =———— so that the y . . ; 10 . 42, Use the first derivative test to find the x-values that give value is decreasing at a rate of 3 units per second. Find the y VR instantaneous rate of change of x with respect to time at the relative extrema for flx) = 2% + 2x, point on the curve where x = 5. 43. Let f(x)= . Show that f has no critical numbers. 31. Find all critical numbers for the function: f(x) = xy2x+1. 1-x 32. Find all critical numbers for the function: Ax) = 354 — 4x3. 44. A differentiable function f has only one critical number: x =— 3. Identify the relative extrema of fat (=3, {-3)) if 33. Find th.e minimum and maximum values of f(x) = K2-2x+1 (4= l and f(=2)=~-1. on the interval [0, 3]. 2 34. Consider f(x)= . 45. Find the intervals on which the graph of the function fix) = x4 (= 3)? — 4x3 + 2 is concave upward or downward. Then find all a. Sketch the graph of f(x). points of inflection for the function. b. Calculate {2) and f(4). c. State why Rolle’s Theorem does not apply to fon the 46. Find all points of inflection of the graph of the function fx) = interval (2, 4]. 2x(x — 4)3. 35. Decide whether Rolle’s Theorem can be applied to f(x) = - 47. Find all points of inflection of the graph of the function f(x) = 4x3 + 4x2 + 1 on the interval [1, 3]. If Rolle’s Theorem can B3 -x+7. be applied, find all value(s), ¢, in the interval such that f’(c)=0. If Rolle’s Theorem cannot be applied, state why. 48. Let f{x) =x3 — x2 + 3. Use the Second Derivative Test to determine which critical numbers, if any, give relative extrema.

Calculus 1st Semester Final Review Reference: [6.6] [1] The limit does not exist. Reference: [11.23] [15] y=1Ix-6 Reference: [7.8] 21 0 Reference: [7.14] 31 5 Reference: [7.39] [41 3 Reference: [7.46] [51 =2 Reference: [8.10] a1 b. -3 [6] c. The limit does not exist. Reference: [8.20] [7] x=2, removable; x = —2, nonremovable Reference: [8.23] 5 xt -1 (8] b.-1,1 a. Reference: [8.26] 91 27 Reference: [9.8] [10] oo Reference: [9.23] [11] x=-7 Reference: [9.25] [12] x=-3 Reference: [10.5] a. no derivative b. negative . zero d. positive [13] e. zero Reference: [10.11] [14] —2x Reference: [11.26] 1 | [16] 3 Reference: [11.28] (171 (1,-2),(-1,2) Reference: [11.40] [18] 264 ft/sec Reference: [12.2] 6x° +2 (9] (1-3x°)° Reference: [12.25] -2 o) - Reference: [13.2] 2x+1 [21] 3(x2 + x)2/3 Reference: [13.3] [22] 120+ D2 +2x+5)° Reference: [13.11] X (7x+3) (23] V2x+1 Reference: [13.34] [24] 42 fusec? Reference: [14.3] . [251 (x+ y)2 +x Reference: [14.10] 2x-y (26] x+2y

Reference: [14.29] Reference: [18.4] 2 [38] Increasing (—eo, 0) and (2, 0); decreasing (0, 2) [27] ; Reference: [18.10] [39] Decreasing (—eo, 2) and (2, ) Reference: [15.5] [28] 100m in.2/min Reference: [18.16] Reference: [15.14] (2 i [40] J , relative maximum 54 7 — ft/min 29] 97 Reference: [18.17] Reference: [15.19] [41] Relative maximum: (-2, 20); relative minimum: (1, —7) [30] 5 units/sec Reference: [18.22] Reference: [16.3] . 3 Relative maximum at x =—. & o [42] 2 B 3 2 Reference: [18.27] Reference: [16.4] i [32] x=0,1 ! ()6)=(1_X)2 #0forall x# 1. 431 f "(x) is undefined at x = 1, a vertical asymptote. Reference: [16.17] [33] Minimum at (1, 0); Maximum at (3, 4) Reference: [18.28] [44] Relative maximum Reference: [17.3] a J Reference: [19.4] Concave upward: (—eo, 0), (2, ) Concave downward: (0, 2) [45] Points of inflection: (0, 2) and (2, —14) Reference: [19.14] [46] (4.0), (2,-32) Reference: [19.18] x [47] (1,4) 1 2 4 5 b f2) =fid) = 1 . [34] c.fis not continuous on [2, 4] Reference: [19.21] 2 x =0, relative maximum; x =—, relative minimum Reference: [17.7] [48] 3 [35] Rolle’s Theorem applies; ¢ =0, 1, and 2. Reference: [17.14] 5 The Mean Value Theorem applies; ¢ =—. [36] 2 Reference: [18.3] [37] Increasing (—e, 0); decreasing (0, co)