Calculus AB (AP-aligned)

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  1. No calculator

    f(x)=x2−36x−6f(x) = \dfrac{x^2 - 36}{x - 6} is not defined at x=6x = 6. What value does f(x)f(x) approach as x→6x \to 6?

  2. No calculator

    A ball's position is s(t)=4t2+9ts(t) = 4t^2 + 9t meters after tt seconds. Find its average velocity on [0,6][0, 6], in m/s.

  3. No calculator

    f(x)=8x5+3xf(x) = 8x^{ 5 } + 3x. Find f′(x)f'(x).

  4. No calculator

    x2+6y2=63x^2 + 6y^2 = 63. Find dydx\dfrac{dy}{dx} in terms of xx and yy.

  5. No calculator

    Find lim⁡x→0e5x−1x\displaystyle\lim_{x \to 0} \dfrac{e^{ 5x} - 1}{x}.

  6. No calculator

    A car drives at 30 mph for 2 hours, then at 57 mph until hour 5. How far does it travel, in miles?

  7. No calculator

    Find FF with F′(x)=2xcos⁡(x2)+4F'(x) = 2x\cos(x^2) + 4 and F(0)=0F(0) = 0.

  8. No calculator

    Which function is a solution of dydx=5y\dfrac{dy}{dx} = 5y?
    1. y=3e5xy = 3e^{ 5x}
    2. y=e3xy = e^{3x}
    3. y=5xyy = 5xy
    4. y=ex+5y = e^{x} + 5

  9. No calculator

    A particle has velocity v(t)=2t+7v(t) = 2t + 7 and starts at s(0)=4s(0) = 4. Find its position at t=5t = 5.

  10. No calculator

    Find lim⁡x→7x2−15x+56x−7\displaystyle\lim_{x \to 7} \dfrac{x^2 - 15x + 56}{x - 7}.

  11. No calculator

    f(x)=x2+4xf(x) = x^2 + 4x. Use f′(4)=lim⁡h→0f(4+h)−f(4)hf'(4) = \displaystyle\lim_{h \to 0} \dfrac{f(4 + h) - f(4)}{h} to find f′(4)f'(4).

  12. No calculator

    f(x)=8ex+5x2f(x) = 8e^x + 5x^2. Find f′(x)f'(x).

  13. No calculator

    Find the slope of the tangent line to the circle x2+y2=25x^2 + y^2 = 25 at the point (4,3)(4, 3).

  14. No calculator

    f(x)=x3−12xf(x) = x^3 - 12x. At which xx does ff have a local maximum?

  15. No calculator

    Estimate ∫05x2 dx\displaystyle\int_0^{ 5 } x^2\,dx with a left Riemann sum of 5 subintervals of width 1.

  16. No calculator

    Evaluate ∫012x(x2+1)4 dx\displaystyle\int_0^1 2x(x^2 + 1)^{ 4 }\,dx.

  17. No calculator

    In the slope field of dydx=x+3y\dfrac{dy}{dx} = x + 3y, what is the slope of the segment at (−3,1)(-3, 1)?

  18. No calculator

    Find the area of the region between y=5xy = 5x and y=x2y = x^2.

  19. No calculator

    f(x)={x+6if x<52xif x≥5f(x) = \begin{cases} x + 6 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}. What is lim⁡x→5f(x)\displaystyle\lim_{x \to 5} f(x)?
    1. It does not exist: the left and right limits differ
    2. 1111
    3. 1010

  20. No calculator

    For f(x)=x2f(x) = x^2, f′(x)=2xf'(x) = 2x. Draw the tangent line to the graph at x=1x = 1: place two points on it.

  21. No calculator

    f(x)=3sin⁡x+5cos⁡xf(x) = 3\sin x + 5\cos x. Find f′(x)f'(x).

  22. No calculator

    f(x)=x3+8xf(x) = x^3 + 8x has an inverse gg. Note f(1)=9f(1) = 9. Find g′(9)g'(9).

  23. No calculator

    f(x)=x3−3x2f(x) = x^3 - 3x^2. At which xx does ff have an inflection point?

  24. No calculator

    Use geometry: find ∫−6636−x2 dx\displaystyle\int_{-6}^{ 6 } \sqrt{ 36 - x^2}\,dx.

  25. No calculator

    Evaluate ∫012xx2+2 dx\displaystyle\int_0^1 \dfrac{2x}{x^2 + 2}\,dx.

  26. No calculator

    Solve dydx=xy\dfrac{dy}{dx} = \dfrac{x}{y} with y(0)=8y(0) = 8 (so y>0y > 0). Give yy in terms of xx.

  27. No calculator

    Find the area of the region between x=y2x = y^2 and x=9x = 9, integrating with respect to yy.

  28. No calculator

    Find lim⁡x→0sin⁡(6x)x\displaystyle\lim_{x \to 0} \dfrac{\sin(6x)}{x}.

  29. No calculator

    Here is the graph of y=f(x)y = f(x). Tap the point of the graph where f′(x)=0f'(x) = 0.

  30. No calculator

    f(x)=x2sin⁡xf(x) = x^{ 2 }\sin x. Find f′(x)f'(x).
  31. Find lim⁡x→∞4x2+15x2+4\displaystyle\lim_{x \to \infty} \dfrac{ 4x^2 + 1}{ 5x^2 + 4}.
  32. V(t)V(t) is the volume of water in a tank, in liters, tt minutes after a tap is opened. V′(2)=8V'(2) = 8. What does this mean?
    1. At t=2t = 2 min, the volume is increasing at about 8 liters per minute.
    2. After 2 minutes, the tank holds 8 liters.
    3. In the first 2 minutes, 8 liters flowed in.
    4. It takes 8 minutes to add 2 liters.
  33. f(x)=xx+6f(x) = \dfrac{x}{x + 6}. Find f′(x)f'(x).
  34. A circular ripple's radius grows at 5 cm/s. How fast is its area growing when the radius is 11 cm? (cm²/s, exact)
  35. Find the maximum value of f(x)=x2−6xf(x) = x^2 - 6x on [0,8][0, 8].
  36. Find the antiderivative FF of f(x)=4x3+9f(x) = 4x^{ 3 } + 9 with F(0)=0F(0) = 0.
  37. Find FF with F′(x)=1x2+9F'(x) = \dfrac{1}{x^2 + 9} and F(0)=0F(0) = 0.
  38. A population grows with dPdt=kP\dfrac{dP}{dt} = kP. It starts at 104 and doubles every 9 years. What is it after 27 years?
  39. A solid's base is the region under y=xy = \sqrt{x} for 0≤x≤30 \le x \le 3. Cross-sections perpendicular to the xx-axis are squares. Find its volume.
  40. What is lim⁡x→6+7x−6\displaystyle\lim_{x \to 6^+} \dfrac{ 7 }{x - 6}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  41. f(x)=7tan⁡xf(x) = 7\tan x. Find f′(x)f'(x).
  42. A 15-ft ladder leans against a wall. Its foot slides away from the wall at 4 ft/s. How fast is the top moving when the foot is 9 ft from the wall? (ft/s; negative means down)
  43. f(x)=x2f(x) = x^2 on [2,5][2, 5]. Find the cc given by the Mean Value Theorem.
  44. Evaluate ∫05x2 dx\displaystyle\int_0^{ 5 } x^2\,dx.

Answers

  1. 1212
  2. 3333
  3. 40x4+340 x^{4} + 3
  4. −x6y- \frac{x}{6 y}
  5. 55
  6. 231231
  7. 4x+sin⁡(x2)4 x + \sin{\left(x^{2} \right)}
  8. y=3e5xy = 3e^{ 5x}
  9. 6464
  10. −1-1
  11. 1212
  12. 10x+8ex10 x + 8 e^{x}
  13. −43- \frac{4}{3}
  14. −2-2
  15. 3030
  16. 315\frac{31}{5}
  17. 00
  18. 1256\frac{125}{6}
  19. It does not exist: the left and right limits differ
  20. y=2x−1y = 2 x - 1
  21. −5sin⁡(x)+3cos⁡(x)- 5 \sin{\left(x \right)} + 3 \cos{\left(x \right)}
  22. 111\frac{1}{11}
  23. 11
  24. 18π18 \pi
  25. ln⁡(32)\ln{\left(\frac{3}{2} \right)}
  26. x2+64\sqrt{x^{2} + 64}
  27. 3636
  28. 66
  29. (−1,1)(-1, 1)
  30. x2cos⁡(x)+2xsin⁡(x)x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}
  31. 45\frac{4}{5}
  32. At t=2t = 2 min, the volume is increasing at about 8 liters per minute.
  33. 6(x+6)2\frac{6}{\left(x + 6\right)^{2}}
  34. 110π110 \pi
  35. 1616
  36. x4+9xx^{4} + 9 x
  37. tan⁡−1(x3)3\frac{\tan^{-1}{\left(\frac{x}{3} \right)}}{3}
  38. 832832
  39. 92\frac{9}{2}
  40. +∞+\infty
  41. 7sec⁡2(x)7 \sec^{2}{\left(x \right)}
  42. −3-3
  43. 72\frac{7}{2}
  44. 1253\frac{125}{3}