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Name: ______________________________________ Hour______
Getting Ready for AP Calculus
1. A particle moves along the x-axis so that its velocity at any time t is given by
v(t) = 3t2 – 18t + 24 and its position is given by x(t) = t3 – 9t2 + 24t + 4.
a. For what values of t is the particle at rest?
b. Find the total distance traveled by the particle from t = 1 to t = 3.
Hint: the particle is moving to the left when v(t) < 0 and to the right when v(t) > 0.
2. Let f(x) = and f ‘(x) = .
a. What is the domain of f?
b. Write an equation for the line tangent to the graph of f at x = 0.
3. Let R be the region enclosed by the graphs of y = (64x)1/4 and y = x. Graph and shade the
region described.
4. Let f be the function given by f(x) = 2 ln(x2 + 3) – x with domain -3 < x < 5. Use a graphing
utility to find each relative maximum point and each relative minimum point of f. Include a
sketch of the graph with your answer.
5. The trough shown in the figure is 5 feet long, and its vertical cross sections are inverted
isosceles triangles with base 2 feet and height 3 feet.
a. Find the volume of water in the trough when its is full.
b. Find the depth of the water when the trough is ¼ full by volume. (hint: use ratio of similar triangles)
c. Find the area of the surface of the water (shaded in the figure) when the trough
Is ¼ full by volume.
6. At any time t > 0, in days, y = y0ekt can be used as a model to determine the population of a
bacteria colony, where k is a constant and y is the number of bacteria present. The initial
population is 1,000 and the population triples during the first 5 days.
a. Write an expression for y at any time t > 0.
b. By what factor will the population have increased in the first 10 days?
c. At what time t, in days, will the population have increased by a factor of 6?
7. Consider the curve given by the equation y3 + 3x2y + 13 = 0 and whose derivative is given by
.
a. Evaluate dy/dx at the point (2, -1).
b. Write an equation for the line tangent to the curve at the point (2, -1)
8. Let R be the region enclosed by the graph of y = ln x, the line x = 3, and the x-axis.
a. Sketch the graph and shade the region described.
b. Using common geometric formulas, estimate the area of the enclosed region.
c. Is your estimate greater than or less than the actual area. Justify your answer.
9. Let an = (2x)n. Evaluate .
10. The position of a particle moving in the xy-plane at any time t, 0 < t < 2π, is given by the
parametric equations x = sin t and y = cos(2t). The velocity of a particle is given by the
vector .
a. Write an equation for the path of the particle in terms of x and y that does not involve
trigonometric functions. (hint: use y=cos(2t) = 2cos2t – 1)
b. Sketch the path of the particle in the xy-plane.
c. For what values of t is the particle at rest?
11. Let f be the function given by f(x) = .
a. Find the domain of f. (hint: set up inequality and solve by graphing)
b. Describe the symmetry, if any of the graph of f.
c. Is f(x) even, odd, or neither? Show the analysis that supports your answer.
12. Let f be the function defined by f(x) = 2xe –x for all real numbers x. Write an equation of the
horizontal asymptote for the graph of f. Recall: horizontal asymptotes are a description of
global behavior. So, evaluate . (hint: graph f(x))
13. Write the first 4 terms for the series .
14. Let f be the function given by f(x) = x3 – 7x + 6.
a. Find the zeros of f.
b. Write an equation of the line tangent to the graph of f at x = -1. Hint: The slope of
the tangent line is evaluated at the x-value of interest.
15. Let f be the function given by .
a. Find the domain of f.
b. Write an equation for each vertical asymptote to the graph of f.
c. Write an equation for each horizontal asymptote to the graph of f.
16. Let R be the region inside the graph of the polar curve r = 2 and outside the graph of the
polar curve r = 2(1 – sinθ).
a. Sketch the two polar curves and shade the region R. (put calculator in polar mode)
b. Find the points where the two curves intersect.
17. Let f(x) = sin (x2) for 0 < x < 1 bounded by the x-axis.
a. Use a graphing utility to sketch the graph of f(x) and shade the described region.
b. Use four trapezoids with equal “widths” to calculate the approximate area of the
region.
c. Is your approximation a good estimate of the actual area? Justify your answer.
18. The temperature, in degrees Celsius (oC), of the water in a pond is
a smooth, continuous function W of time t. The table shows the
water temperature as recorded every 3 days over a 15-day period.
a. Use data from the table to find an approximation for the
the rate of change for W(12). Show the computations
that lead to your answer. Indicate units of measure.
b. Approximate the average temperature, in degrees
Celsius, of the water over the time interval 0 < t < 15
days by using a trapezoidal approximation with
subintervals of length Δt = 3 days.
19. Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t > 0. After her
parachute opens, her velocity is v(t) = -34e-2t – 16.
a. Terminal velocity is defined as . Find the terminal velocity of the skydiver to
the nearest foot per second.
b. It is safe to land when her speed is 20 feet per second. At what time t does she
reach this speed? (hint: treat speed as negative value)
20. The graph of the velocity v(t), in ft/sec, of a car traveling on a straight road, for 0 < t < 50, is
shown. A table of values for v(t), at 5 second intervals of time t, is also given.
a. During what intervals of time is the acceleration of the car positive? Justify your
answer.
b. Find the average acceleration of the car, in ft/sec2, over the interval 0 < t < 50.
c. Find one approximation for the acceleration of the car, in ft/sec2, at t = 40. Show
the computations you used to arrive at your answer.
d. Approximate with a Riemann sum, using the midpoints of 5 subintervals of
equal length. Using correct units, explain the meaning of this integral.
21. If than is
A. ln 2 B. ln 8 C. ln 16 D. 4 E. nonexistent
22. The function f is given by f(x) = x4 + x2 – 2. On which of the following intervals is f
increasing? A. B.
C. D.
E.
23. The function f is continuous on the closed interval [0, 2] and has values that are given
in the table below. The equation f(x) = ½ must have at least two solutions in the interval
[0, 2] if k = ?
Hint: Look up the Intermediate Value Theorem.
A. 0 B. ½ C. 1 D. 2 E. 3
24. The graph of a function is shown. Which of the following statements about f is false?
A. f is continuous at x = a.
B. f has a relative maximum at x = a.
C. x = a is in the domain of f.
D.
E. exists
25. If a ≠ 0, then is
A. B. C. D. 0 E. nonexistent
26. The function f is continuous on the closed interval [2, 8] and has values that are given in
the table. Using the subintervals [2, 5], [5, 7], and [7, 8], what is the trapezoidal
approximation of the area enclosed f(x) and the x-axis.
A. 110 B. 130 C. 160
D. 190 E. 210
*Note how the table has unequal
intervals of x.