1)If then
(a)(b)(c)
(d)(e)
2)Determine for the curve defined by .
(a)(b)(c)
(d)(e)
3)A particle’s position is given by The average velocity of the particle over is
(a)(b)(c)0
(d)(e)
4)In the right triangle shown below, is increasing at a constant rate of radians per minute. At what rate is the area of the triangle increasing, in square units per minute, when h is 24 units?
(a)
(b)39
(c)
(d)182
(e)195
5)What is the instantaneous rate of change for at x = 2?
(a)-27(b)-6(c)6(d)9(e)27
6)What is the slope of the curve defined by at the point (1, 1)?
(a)(b)(c)0(d)(e) It is undefined
7)The radius of a sphere is increasing at a rate of 2 inches per minute. At what rate (in cubic inches per minute) is the volume increasing when the surface area of the sphere is square inches?
(a) 2(b) (c) (d)(e)
8)What is the average rate of change of over [-1, 4]?
(a) (b) 3(c) 5(d) 10(e) 25
9)Let f, g, and their derivatives be defined by the table below. If then for what value, c, is
x / 1 / 2 / 3 / 4/ 3 / 0 / 1 / 2
/ 2 / 3 / 4 / 1
/ -1 / -1 / 0 / 2
/ -2 / 2 / 1 / -1
(a)1
(b)2
(c)3
(d)4
(e)None of the above
10)The normal line to the curve at the point (2, 2) has slope
(a) -2(b) (c) (d) 1(e) 2
11)The water level in a cylindrical barrel is falling at a rate of one inch per minute. If the radius of the barrel is ten inches, what is the rate that water is leaving the barrel (in cubic inches per minute) when the volume is cubic inches?
(a) 1(b) (c) (d) (e)
12)An equation of the line tangent to at is
(a) (b) (c)
(d) (e) None of the above
13)Which of the following gives the derivative of the function at the point (2,4)?
(a) (b)
(c) (d)
(e)
14)Let . Determine
(a)
(b)
(c)
(d)
(e)
15)Find the derivative of with respect to x.
(a) (b)
(c) (d)
(e)
16)A curve is generated by the equation . Determine the number of points on this curve whose corresponding tangent lines are horizontal.
(a) 0(b) 1(c) 2(d) 3(e) 4
17)A 13-foot ladder is leaning against a 20-foot vertical wall when it begins to slide down the wall. During this sliding process, the bottom of the ladder is sliding away from the bottom of the wall at a rate of foot per second. Determine the rate at which the top of the ladder is sliding down the vertical wall when the tip of the ladder is exactly 5 feet above the ground.
(a) (b) (c) (d) (e) not enough info
18)Suppose that the function f satisfies . Then the slope of the line tangent to the graph of f at the point x = 2 is
(a) 12(b) (c) 7(d) (e) 24
19)The derivative of is given by which of the following?
(a)
(b)
(c)
(d)
(e)
20)If
(a)(b)(c)
(d)(e)
21)Consider the function . Which of the following is a linear approximation of f at x = 1?
(a)
(b)
(c)
(d)
(e)
22)If the function, represents the position of a particle in meters after t seconds, , then the instantaneous rate of change of the particle at time t = 2 seconds is
(a)(b)(c)
(d)(e)
23)A spherical balloon is being filled with water so that its volume increases at a rate of . How fast is the radius of the balloon increasing when the diameter is 50 cm? (The volume of a sphere is given by )
(a)(b)(c)
(d)(e)Not enough information
24)The derivative of the function may be expressed as a limit by
(a)(b)
(c)(d)Both (B) and (C)
(e)Both (A) and (B)
25)Consider the curve described by the equation Calculate at the point .
(a)(b)(c)(d)(e)
26)Car A is traveling south at 40 mph toward Millville, and Car B is traveling west at 30 mph toward Millville. If both cars began traveling 100 miles outside of Millville at the same time, then at what rate, in mph, is the distance between them decreasing after 90 minutes?
(a) 35.00(b) 47.79(c) 50.00(d) 55.14(e) 68.01
27)The position for a particle moving on the x-axis is given by . At what time, t, on [0,3] is the particle’s instantaneous velocity equal to its average velocity on [0,3]?
(a) 0.535(b) 1.387(c) 1.821(d) 1.869(e) 2.333
28)The cost of producing x units of a certain item is . What is the average rate of change of c with respect to x when the level of production increases from x = 300 to x = 310 units?
(a) 313.6(b) 310.0(c) 214.2(d) 200.0(e) 10
29)Let . Then accurate to three decimal places,
(a) 0.538(b) -0.999(c) 0.009(d) 1.000(e) 0.866
30)Determine the slope of the normal line to the curve at the point (2,1).
(a) 0(b) 2(c) (d) (e)
31)A spherical balloon is being inflated at a rate of 3 cubic inches per second. Determine the rate of change of the radius of the balloon when the balloon’s radius is 5 inches, accurate to three decimal places. The volume of a sphere of radius r is .
(a) 3.000 in/sec(b) 1.667 in/sec(c) 0.010 in/sec
(d) -2.000 in/sec(e) 0.120 in/sec
32)The velocity of a runner in feet per second as a function of time is given in the table below.
t / 0 / 1 / 2 / 3 / 4v(t) / 5 / 10 / 12 / 11 / 9
The approximate acceleration of the runner at time t = 2 seconds is
(a) (b) (c)
(d) (e)
Free Response Questions (all non-calculator)
1985 AB1
Let f be the function given by .
(a)Find the domain of f.
(b)Write an equation for each vertical and each horizontal asymptote for the graph of f.
(c)Find
(d)Write an equation for the line tangent to the graph of f at the point
1992 AB4, BC1
Consider the curve defined by the equation .
(a)Find in terms of y.
(b)Write an equation for each vertical tangent to the curve.
(c)Find in terms of y.
1994 AB5, BC2
A circle is inscribed in a square as shown in the figure below. The circumference of the circle is increasing at a constant rate of 6 inches per second. As the circle expands, the square expands to maintain the condition of tangency.
(a)Find the rate at which the perimeter of the square is increasing. Indicate units of measure.
(b)At the instant when the area of the circle is square inches, find the rate of increase in the area enclosed between the circle and the square. Indicate units of measure.
Derivatives
Test A
Name ______