CALCULUS BC



CALCULUS AB

LAST AP REVIEW

|Reminders: 1. Put new batteries in your calculator. Make sure your calculator is in RADIAN mode. |

|2. Get a good night’s sleep Tuesday night. |

|3. Eat breakfast Wednesday morning. |

|4. Bring: TI-89, pencils, eraser, pen, watch, snacks, bottle of water, jacket. |

Work these on notebook paper. Use your calculator only if the problem is labeled “calculator.”

1. [pic]= 2. [pic]= 3. [pic]

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4. [pic] 5. If [pic]

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6. Find a and b so that f is differentiable everywhere. [pic]

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8. Let R be the region bounded by the graphs of [pic] [pic]

(a) Find the area of R.

(b) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

rotated about the horizontal line [pic]

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

rotated about the horizontal line [pic]

(d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

rotated about the vertical line x = 5.

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9. Let R be the region bounded by the graphs of [pic]

(a) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are

rectangles whose heights are three times the lengths of the bases. Write, but do not evaluate, an integral

expression for the volume of this solid.

(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are

semicircles. Write, but do not evaluate, an integral expression that could be used to find the volume

of this solid.

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10. If [pic]

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Evaluate.

11. [pic] 12. [pic] 13. [pic] 14. [pic]

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15. If [pic] increases at a constant rate of 3 rad/min, at what rate is x

increasing in units/min when x = 3 units? 5 x

[pic]

16.

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17.

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18. Let [pic] be a differentiable function such that [pic] The

function g is differentiable and [pic] for all x. What is the value of [pic]

19. (Calculator) The first derivative of the function f is defined by [pic] for [pic] On what

intervals is f increasing?

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20. (Calculator) The derivative of the function f is given by [pic] How many points of inflection

does the graph of f have on the open interval [pic]

(A) One (B) Two (C) Three (D) Four (E) Five

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21. (Calculator) A particle moves along the x-axis so that at any time t > 0, its acceleration is given by

[pic] If the velocity of the particle is 2 at time t = 1, then the velocity of the particle at time t = 2 is

(A) 0.462 (B) 1.609 (C) 2.555 (D) 2.886 (E) 3.346

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22. (Calculator) A particle moves along the x-axis so that at any time [pic], its velocity is given by

[pic] The total distance traveled by the particle from t = 0 to t = 2 is

(A) 0.667 (B) 0.704 (C) 1.540 (D) 2.667 (E) 2.901

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23. (Calculator) The velocity, in ft/sec, of a particle moving along the x-axis is given by the

function [pic]. What is the average velocity of the particle from time t = 0 to t = 3?

(A) 20.086 ft/sec (B) 26.447 ft/sec (C) 32.809 ft/sec (D) 40.671 ft/sec (E) 79.342 ft/sec

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24. (Calculator) A particle moves along the y-axis so that its velocity at time [pic] is given by

[pic]. At time t = 0, the particle is at y = [pic]. (Note: [pic])

(a) Find the acceleration of the particle at time t = 2.

(b) Is the speed of the particle increasing or decreasing at time t = 2? Give a reason for your answer.

(c) Find the time [pic] at which the particle reaches its highest point. Justify your answer.

(d) Find the position of the particle at time t = 2. Is the particle moving toward the origin or away from the

origin at time t = 2? Justify your answer.

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25. (Calculator) A particle moves along the x-axis so that its velocity at time t, for [pic] is given by

[pic] The particle is at position x = 8 at time t = 0.

(a) Find all times t in the open interval [pic] at which the particle changes direction. During which time

intervals, for [pic] does the particle travel to the left?

(b) Find the average speed of the particle over the interval [pic]

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26. If [pic], which of the following is true?

(A) [pic] (B) [pic] (C) [pic]

(D) [pic] (E) [pic]

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27. Which of the following is the solution to the differential equation [pic], where [pic]?

(A) [pic] for x > 0 (B) [pic] (C) [pic]

(D) [pic] (E) [pic]

28. [pic]

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Match the slope fields with their differential equations.

(A) (B)

(C) (D)

29. [pic] 30. [pic] 31. [pic] 32. [pic]

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33.

Shown above is a slope field for which of the following differential equations?

(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E)[pic]

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34. Consider the differential equation [pic]. Let [pic] be a particular solution to this differential

equation with the initial condition [pic].

(a) Write an equation for the tangent line that passes through the point [pic], and use it to

approximate the value of [pic].

(b) For [pic], [pic]. Is the approximation you found in (a) for [pic]an underestimate or

an overestimate? Explain.

(c) Find the particular solution [pic] to the differential equation with the initial condition [pic].

35. Consider the differential equation [pic].

(a) On the axes provided, sketch a slope field for the differential equation given at the nine points indicated.

(b) Find [pic] in terms of x and y.

(c) What is the particular solution [pic] to the differential equation with the initial condition [pic]?

(d) Is the point [pic] a relative maximum, relative minimum, or neither? Justify your answer.

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36.

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37. (Calculator) Oil is leaking from a tanker at the rate of [pic] gallons per hour,

where t is measured in hours. How much oil leaks out of the tanker from time t = 0 to t = 10?

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38. (Calculator) A pizza, heated to a temperature of 350 degrees Fahrenheit (°F), is taken out of an oven and placed

in a 75°F room at time t = 0 minutes. The temperature of the pizza is changing at a rate of [pic] degrees

Fahrenheit per minute. To the nearest degree, what is the temperature of the pizza at time t = 5 minutes?

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39. (Calculator) The density function in people per mile for the population of the small coastal town of Westport,

WA, is given by [pic]where x is the distance along a straight highway from the ocean and

[pic] is measured in people per mile. The town extends for two miles from the ocean so that [pic]

Find the population of Westport.

40. A city is built around a circular lake that has a radius of 1 mile. The population density of the city is

[pic] people per square mile, where r is the distance from the center of the lake, in miles.

Which of the following expressions gives the number of people who live within 1 mile of the lake?

(A) [pic] (B) [pic] (C) [pic]

(D) [pic] (E) [pic]

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41. (Calculator) A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the time

interval [pic] hours, water is pumped into the tank at the rate [pic] gallons per hour.

During the same time interval, water is removed from the tank at the rate [pic] gallons per hour.

(a) Is the amount of water in the tank increasing at time t = 15? Why or why not?

(b) To the nearest whole number, how many gallons of water are in the tank at time t = 18?

(c) At what time t, for [pic], is the amount of water in the tank at an absolute minimum? Show the work that

leads to your conclusion.

(d) For t > 18, no water is pumped into the tank, but water continues to be removed at the rate [pic] until

the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an

equation involving an integral expression that can be used to find the value of k.

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42. Consider the curve given by [pic]

(a) Show that [pic]

(b) Show that there is a point P with x-coordinate 3 at which the line tangent to the curve at P is horizontal.

Find the y-coordinate of P.

(c) Find the value of [pic] at the point P found in part (b). Does the curve have a local maximum, a local

minimum, or neither at the point P? Justify your answer.

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43.

|x |2 |3 |5 |8 |13 |

|[pic] |1 |4 |[pic] |3 |6 |

Let f be a function that is twice-differentiable for all real numbers. The table above gives values of f for

selected points in the closed interval [pic] Evaluate [pic]

____________________________________________________________________________________________44. Let R be the region in the first quadrant enclosed by the graphs of [pic]

and [pic] as shown in the figure on the right.

(a) Write an equation for the line tangent to the graph of f at [pic]

(b) Find the area of R.

(c) Write, but do not evaluate, an integral expression for the volume of the solid

generated when R is rotated about the horizontal line y = 1.

(d) The region R is the base of a solid. For this solid, the cross sections

perpendicular to the x-axis are squares. Write, but do not evaluate, an integral

expression for the volume of this solid.

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