A
A. P. Calculus First Semester Practice Test Name ________
A calculator is allowed on this section of the Exam.
I. Multiple Choice
_____ 1. If [pic], then [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
_____ 2. If the graph of [pic] has a point of inflection at [pic], then the
value of a is:
(A) 2 (B) [pic] (C) [pic] (D) [pic] (E) None of these
_____ 3. The equation of the normal line to the curve [pic] at the point where
[pic] is
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
_____ 4. Given [pic] determine the value of [pic] when [pic]
(A) [pic] (B) 0 (C) 6 (D) 12 (E) None of these
_____ 5. The derivative of [pic] is
(A) x (B) 12 (C) 12x (D) 0 (E) None of these
_____ 6. The second derivative of [pic] is
(A) 8 (B) 8x (C) x (D) 0 (E) None of these
_____ 7. A bouncing ball loses [pic] of its previous height each time that it rebounds. If the ball
is thrown up to a height of 60 feet, how many feet will it travel before coming to rest
(including the 60 feet on the way up)?
(A) 480 feet (B) 240 feet (C) 160 feet (D) 120 feet (E) 80 feet
_____ 8. [pic]
(A) [pic] (B) [pic] (C) [pic]
(D) [pic] (E) [pic]
_____ 9. If [pic], where n is a constant, then [pic]=
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
_____ 10. Water flows at 8 cubic feet per minute into a cylinder with radius 4 feet.
How fast is the water level rising?
(A) 2 ft/min (B) [pic] ft/min (C) [pic] ft/min (D) [pic] ft/min
(E) None of the above
_____ 11. The slope of the line tangent to the graph of [pic] at [pic] is
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
_____ 12. With respect to the origin, point [pic] is symmetric to
A [pic] B [pic] C [pic] D [pic] E [pic]
_____ 13. If [pic], then [pic]
(A) [pic] (B) [pic]
(C) [pic] (D) [pic]
(E) None of these
_____ 14. Determine the equation of the tangent line to the graph of [pic] at the point
where [pic] if [pic] and [pic].
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
(E) It cannot be determined from this information.
_____ 15. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
(E) [pic]
_____ 16. If [pic]
(A) 12 (B) 10 (C) 6 (D) 4 (E) 3
_____ 17. Determine the absolute maximum value and the absolute minimum value of the
function [pic] over the interval [pic].
(A) [pic]
(B) [pic]
(C) [pic]
(D) [pic]
(E) [pic]
_____ 18. In proving that [pic], what is the largest value of [pic] corresponding to [pic]
such that [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
_____ 19. If [pic], then [pic]
(A) [pic] (B) [pic] (C) [pic]
(D) [pic] (E) [pic]
_____ 20. A function f is defined by [pic]
If f is continuous at [pic], what is the value of k?
(A) [pic] (B) 0 (C) 2 (D) 4 (E) [pic]
II. Free Response
21. Prove the following derivative formula:
Given: [pic]
Prove: [pic]
_______________ 22. Given L feet of fencing, what is the maximum number of square feet that
can be enclosed if the fencing is used to make three sides of a
rectangular pen, using an existing wall as the fourth side?
_______________ 23. Given [pic], [pic], and [pic]
Determine [pic]
_______________ 24. At what points on the graph of [pic] is the slope equal to [pic] ?
_______________ 25. Determine [pic] , given that [pic]
_______________ 26. Determine [pic] , given that [pic]
_______________ 27. Determine [pic] , given that [pic]
_______________ 28. Determine [pic] , given that [pic]
_______________ 29. Determine [pic] , given that [pic] and [pic]
_______________ 30. Determine the domain of [pic]
_______________ 31. Determine the equation of the tangent to the curve [pic]
at the point [pic].
_______________ 32. A particle projected vertically upward with an initial velocity
of 128 ft/sec reaches an elevation [pic] at the end
of t seconds. How high does the particle rise?
_______________ 33. Is the function f(x) continuous at x = 0 ?
(SHOW WORK
BELOW)
[pic]
PROVE your answer:
_______________ 34. Determine the value of x if [pic]
[pic].
_______________ 35. If [pic], determine the 100th derivative of y with respect to x.
a =____________ 36. The curve [pic] passes through the points [pic] and
b =____________ [pic]. The value of y is greatest when [pic]. Determine the
c=_____________ values of a, b, and c.
_______________ 37. If the surface area of a sphere is increasing at the rate of 12 square feet
per second, how fast is the radius increasing when it is 2 feet?
_______________ 38. Solve for y (to three decimal places): [pic]
39. Write out the complete definition for continuity.
40. Sketch a curve which satisfies the following conditions:
[pic]
[pic]
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