(A) 0,2 (B) 1,5 (C) 2,7 (D) None of These
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1) Consider the function f (x) x 2 . On what intervals are the hypotheses of the Mean Value Theorem satisfied?
(a) [0,2] (b) [1,5] (c) [2,7] (d) None of these
2) Consider the following graph of f (x) xsin x on the domain [-4,4]. How many values of c in (-4, 4) appear to satisfy the Mean Value Theorem equation?
(a) None (b) One (c) Two (d) Three (e) Four or more
3) The diagonal of a square is increasing at a rate of 3 inches per minute. When the area of the square is 18 square inches, how fast (in inches per minute) is the perimeter increasing?
3 2 (a) 18 (b) 3 2 (c) (d) 6 (e) 6 2 2
4) A spherical snowball with diameter 4 inches is removed from the freezer in June and begins melting uniformly such that it is shrinking 2 cubic inches per minute. How fast (in square inches per minute) is its surface area decreasing when the radius is 1 inch?
1 1 (a) (b) (c) 1 (d) 4 (e) 4 2 2 5) Which is the slope of the line tangent to y 2 xy x 2 11 at (2, 3)?
5 1 4 9 (a) (b) 0 (c) (d) (e) 2 8 7 7
dy 6) If xy 2 y 3 x 2 5 , then dx
y 2 2x y 2 2x 5 2x 5 (a) (b) (c) 3y 2 2xy 3y 2 2xy 2y 3y 2 2x x y 2 (d) (e) 2y 3y 2 xy
9 7) The distance of a particle from its initial position is given by s(t) t 5 , (t 1) where s is feet and t is minutes. Find the velocity at t = 1 minute in feet per minute.
5 13 1 9 7 (a) (b) (c) (d) (e) 4 4 2 4 4
8) What is the slope of the graph of y sin x at x ? 3
3 1 1 3 (a) (b) (c) 0 (d) (e) 2 2 2 2
9) The area of a circle is A r 2 . How does the instantaneous rate of change of the area with respect to the radius when r = 2 compare to the average rate of change of the area as the radius changes from r = 1 to r = 3?
(a) The instantaneous rate of change is twice the average rate of change. (b) The instantaneous rate of change is equal to the average rate of change. (c) The instantaneous rate of change is half the average rate of change. (d) The instantaneous rate of change is three times the average rate of change. (e) Their relationship cannot be determined from the given information. x 3 10) Find the equation of the line tangent to y at x = 1. x 2 1
5 9 5 1 1 3 (a) y x (b) y x (c) y x 2 2 2 2 2 2 3 1 3 7 (d) y x (e) y x 2 2 2 2
Use the table below for #11-12.
X f (x) g(x) f (x) g(x) 1 4 2 5 ½ 3 3 7 -4 -1 2
d 11) The value of ( f g) at x = 3 is dx
5 3 21 (a) (b) (c) -13 (d) 12 (e) 2 2 2
d f 12) The value of at x = 1 is dx g
(a) 2 (b) 3 (c) 5 (d) 6 (e) 10
In #13-14, use the table below to find the value of the first derivative of the given functions for the given value of x. X f (x) g(x) f (x) g(x) 1 3 2 0 ¾ 1 2 7 -4 -1 3
13) [ f (x)]2 at x = 2
2 14 (a) 0 (b) (c) (d) 6 (e) None of these 3 3
14) f (g(x)) at x = 1
1 1 9 21 (a) (b) 0 (c) (d) (e) 3 4 4 4 Free Response Questions (all non-calculator)
1984 AB2
x sin x Let f be the function defined by f (x) for x . cos x 2 2
(a) State whether f is an even function or an odd function. Justify your answer.. (b) Find f (x). (c) Write an equation for the line tangent to the graph of f at the point (0, f (0)).
1990 AB4
The radius r of a sphere is increasing at a constant rate of 0.04 centimeters per second. 4 (Note: The volume of a sphere with radius r is V r 3 .) 3
(a) At the time when the radius of the sphere is 10 centimeters, what is the rate of increase of the volume? (b) At the time when the volume of the sphere is 36 cubic centimeters, what is the rate of increase of the area of a cross section through the center of the sphere? (c) At the time when the volume and the radius of the sphere are increasing at the same numerical rate, what is the radius?
1994 AB3
Consider the curve defined by x 2 xy y 2 27 .
(a) Write an expression for the slope of the curve at any point (x, y). (b) Determine whether the lines tangent to the curve at the x-intercepts of the curve are parallel. Show the analysis that leads to your conclusion. (c) Find the points on the curve where the lines tangent to the curve are vertical. Derivatives Test B
Name ______