Calculus Multiple Choice and FRQ

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Find the absolute maximum value of the function

f of x equals x squared plus 2 for x between 0 and 1 inclusive and equals negative x plus 4 for x greater than 1.

. (4 points)

	0
	1
	2
	3

In the answer space below, provide one of the two positive integers whose product is 196 and whose sum is a minimum. (4 points)

_______________________________

A farmer has 2,400 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. Write the function that will produce the largest area if x is the short side of the rectangle. (4 points)

	f(x) = 2400x - x2
	f(x) = x2 - 2400
	f(x) = 2x2 - 2400
	f(x) = 2400x - 2x2

What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 3 - x2? (4 points)

	1
	3
	4
	12

If 1,200 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter without commas. For example, if the answer is 2,000 write 2000. (4 points)

_______________________________

A particle moves with velocity function v(t) = 2t2 - 3t - 3, with v measured in feet per second and t measured in seconds. Find the acceleration of the particle at time t = 2 seconds. (4 points)

	3 divided by 4 feet per second2
	-1 feet per second2
	32 feet per second2
	5 feet per second2

A particle is moving along the x-axis so that its position at t ≥ 0 is given by s(t) = (t)In(5t). Find the acceleration of the particle when the velocity is first zero. (4 points)

	5e
	5e2
	e
	None of these

The driver of a car traveling at 50 ft/sec suddenly applies the brakes. The position of the car is s(t) = 50t - 2t2, t seconds after the driver applies the brakes. How many seconds after the driver applies the brakes does the car come to a stop? (4 points)

	12.5 sec
	10 sec
	15 sec
	60 sec

The position of a particle on the x-axis at time t, t > 0, is s(t) = ln(t) with t measured in seconds and s(t) measured in feet. What is the average velocity of the particle for e ≤ t ≤ 2e? (4 points)

	ln2
	the quotient of 1 and the quantity of 3 times e
	the quotient of the natural logarithm of 2 and e
	the quotient of the natural logarithm of 2 and 2

A particle moves along the x-axis so that at any time t, measured in seconds, its position is given by s(t) = 5cos(t) - sin(3t), measured in feet. What is the acceleration of the particle at time t = π seconds? (4 points)

_______________________________

If y = 2x3 - 4x and

dx, dt equals 4

, find

dy, dt

when x = 1. Give only the numerical answer. For example, if

dy, dt

= 3, type only 3. (4 points)

______________________________

The area A = πr2 of a circular puddle changes with the radius. At what rate does the area change with respect to the radius when r = 5ft? (4 points)

	5π ft2/ft
	5 ft2/ft
	10π ft2/ft
	25π ft2/ft

A rectangular box has a square base with an edge length of x cm and a height of h cm. The volume of the box is given by V = x2h cm3. Find the rate at which the volume of the box is changing when the edge length of the base is 4 cm, the edge length of the base is increasing at a rate of 2 cm/min, the height of the box is 15 cm, and the height is decreasing at a rate of 3 cm/min. (4 points)

	The volume of the box is decreasing at a rate of 192 cm3/min.
	The volume of the box is increasing at a rate of 288 cm3/min.
	The volume of the box is decreasing at a rate of 288 cm3/min.
	The volume of the box is increasing at a rate of 192 cm3/min.

The height of a cylinder with a fixed radius of 4 cm is increasing at the rate of 2 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 14cm. (4 points)

	16π
	28π
	32π
	None of these

A 25-ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 0.18 ft/sec, how fast, in ft/sec, is the top of the ladder sliding down the wall, at the instant when the bottom of the ladder is 20 ft from the wall? Answer with 2 decimal places. Type your answer in the space below. If your answer is a number less than 1, place a leading "0" before the decimal point (ex: 0.35). (4 points)

______________________________

If f(x) is differentiable for the closed interval [-1, 4] such that f(-1) = -3 and f(4) = 12, then there exists a value c, -1< c < 4 such that (4 points)

	f '(c) = 3
	f '(c) = 0
	f(c) = -15
	f (c) = 3

If f(x) = |(x2 - 8)|, how many numbers in the interval 0 ≤ x ≤ 2.5 satisfy the conclusion of the mean value theorem? (4 points)

	Three
	Two
	One
	None

Find the exact value of the following limit:

the limit as x goes to 0 of the quotient of the quantity e raised to the 6 times x power minus 6 times x minus 1 and x squared

. (4 points)

__________________________

For which pair of functions f(x) and g(x) below will the

limit as x goes to infinity of the product of f of x and g of x does not equal 0

? (4 points)

	f(x) = 10x + e-x; g(x) = 1 divided by the quantity 5 times x
	f(x) = x2; g(x) = e-4x
	f(x) =(Lnx)3; g(x) = 1 divided by x
	f(x) = square root of x ; g(x) = e-x

Evaluate

the limit as x goes to negative infinity of the product of x raised to the 4th power and e raised to the x power

. (4 points)

	-1
	0
	1
	e

Which of the following is the best linear approximation for f(x) = sin(x) near x = π seconds? (4 points)

	y = -x + π - 1
	y = -1
	y = -x + π
	y = -x - π

If f(-1) = -3 and

f prime of x equals the quotient of 4 times x squared and the quantity x cubed plus 3

, which of the following is the best approximation for f(-1.1) using local linearization? (4 points)

	-7.2
	2.8
	-1.2
	-3.2

The local linear approximation of a function f will always be less than or equal to the function's value if, for all x in an interval containing the point of tangency, (4 points)

	f '(x) < 0
	f '(x) > 0
	f "(x) < 0
	f "(x) > 0

If f is continuous at x = c, then f is differentiable at x = c. (4 points)

	True
	False

If f is a function such that

the limit as x approaches a of the quotient of the quantity f of x minus f of a and the quantity x minus a equals 5

, then which of the following statements must be true? (4 points)

	f(a) = 5
	The slope of the tangent line to the function at x = a is 5.
	The slope of the secant line through the function at x = a is 5.
	The linear approximation for f(x) at x = a is y = 5

Given f '(x) = (x + 1)(6 + 3x), find the x-coordinate for the relative minimum on the graph of f(x). (5 points)

	0
	-1
	-2
	None of these

Use the graph of f '(x) below to find the x values of the relative maximum on the graph of f(x):

graph is increasing from x equals negative 2 to x equals 0.5 and again from x equals 2 to x equals 2.2, and is decreasing from x x equals 0.5 to x equals 1.5

(5 points)

	0.5
	2
	1
	1.5

Given the position function s(t), s(t) = t3 - 5t, where s is measured in meters and t is in seconds, find the velocity and acceleration of the particle at t = 3 seconds. (5 points)

	12 m/sec; 22 m/sec2
	22 m/sec; 18 m/sec2
	22 m/sec; 6 m/sec2
	14 m/sec; 3 m/sec2

Given the position function,

s of t equals negative t cubed divided by 3 plus 13 times t squared divided by 2 minus 30 times t

, between t = 0 and t = 9, where s is given in feet and t is measured in seconds, find the interval in seconds where the particle is moving to the right. (5 points)

	3 < t < 9
	5 < t < 9
	The particle never moves to the right.
	The particle always moves to the right.

Given the relationship 2x2 + y3 =10, with y > 0 and

dy, dt

= 3 units/min., find the value of

dx, dt

at the instant x = 1 unit. (5 points)

	-9 units/min
	-1 units/min
	negative 13 halves units/min
	negative 1 ninth units/min

The base of a triangle is decreasing at the rate of 1 ft/sec, while the height is increasing at the rate of 2 ft/sec. At what rate is the area of the triangle changing when the base is 10 ft and the height is 70 ft? (5 points)

	-25 ft2/sec
	-45 ft2/sec
	-50 ft2/sec
	45 ft2/sec

Find

the limit as x goes to 1 of the quotient of the natural logarithm of x squared and the quantity x squared minus 1

. (5 points)

	0 divided by 0
	1
	0
	1 divided by 2

Evaluate

the limit as goes to infinity of the quotient of the 24th power of 1 plus x and x minus 1 over x

. (5 points)

	0
	24
	23
	∞

Which of the following statements is/are true? (5 points)

I. If f '(x) exists and is nonzero for all x, then f(1) ≠ f(0). II. If f is differentiable for all x and f (-1) = f(1), then there is a number c, such that |c| < 1 and f '(c) = 0. III. If f '(c) = 0, then f has a local maximum or minimum at x = c.

	I only
	II only
	I and III only
	I and II only

10.

The local linear approximation of a function f will always be greater than or equal to the function's value if, for all x in an interval containing the point of tangency, (5 points)

	f ' < 0
	f ' > 0
	f " < 0
	f " > 0