Calculus Multiple Choice and FRQ

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Multiple Choice (1-10)

Find the area or the region bounded by the curves y = x3 and y = x. (5 points)

	0.25
	0.5
	1
	2

Find the area of the region bounded by the graphs of y = -x2 + 3x + 4 and y = 4. (5 points)

	2.7
	4.5
	28.5
	16.5

Find the area of the region bounded by the curves y = x2 and y = cos(x). Give your answer correct to 2 decimal places. (5 points)

_________________________

The region in the first quadrant bounded by the x-axis, the line x = π, and the curve y = cos(cos(x)) is rotated about the x-axis. What is the volume of the generated solid? (5 points)

	1.921
	3.782
	6.040
	8.130

Which of the following integrals will find the volume of the solid that is formed when the region bounded by the graphs of y = e2x, x = 1, and y = 1 is revolved around the line y = -1. (5 points)

The base of a solid in the xy-plane is the circle x2 + y2 = 16. Cross sections of the solid perpendicular to the y-axis are squares. What is the volume, in cubic units, of the solid? (5 points)


	1024π

Find the average value of f of x equals 1 divided by x over the interval [1, 3]. (5 points)

	0.67
	1.10
	0.55
	0.33

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = 5 - x2 on the interval the closed interval from 0 to the square root of 5 . If so, find the x-coordinates of the point(s) guaranteed by the theorem. (5 points)

	No, the Mean Value Theorem for Integrals does not apply
	Yes,
	Yes,
	Yes,

For an object whose velocity in ft/sec is given by v(t) = -t2 + 2, what is its displacement, in feet, on the interval t = 0 to t = 2 secs? (5 points)

	1.333
	2.438
	2.667
	-4

10.

For an object whose velocity in ft/sec is given by v(t) = -3t2 + 6, what is its distance travelled, in feet, on the interval t = 0 to t = 2 secs? (5 points)

	4
	7.314
	3.657
	-6

Multiple Choice (1-30)

Which of the following integrals represents the area of the region bounded in the first quadrant by x = pi over 4 and the functions f(x) = sec2(x) and g(x) = sin(x)? (4 points)

Which integral gives the area of the region in the first quadrant bounded by the x-axis, y = x2, and x + y = 2? (4 points)

Find the area of the region bounded by the graphs of y = 2 - x2 and y = -x. (4 points)

	4.5
	1.5
	-1.833
	None of these

Find the area of the region bounded by the graphs of y = x, y = 4 - 3x, and x = 0. (4 points)

	0.375
	2
	4
	None of these

5.	Find the number a such that the line x = a divides the region bounded by the curves x = y2 − 1 and the y-axis into 2 regions with equal area. Give your answer correct to 3 decimal places.(4 points)

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Use your calculator to find the approximate volume in cubic units of the solid created when the region under the curve y = sin(x) on the interval [0, π] is rotated around the x-axis. (4 points)

	4.93
	1.57
	6.28
	2.00

Find the volume of the solid formed by revolving the region bounded by the graphs of y = x2, x = 4, and y = 1 about the y-axis. (4 points)




	None of these

Which of the following integrals correctly computes the volume formed when the region bounded by the curves x2 + y2 = 25, x = 3, and y = 0 is rotated around the y-axis? (4 points)

The base of a solid in the first quadrant of the xy plane is a right triangle bounded by the coordinate axes and the line x + y = 2. Cross sections of the solid perpendicular to the base are squares. What is the volume, in cubic units, of the solid? (4 points)

10.

The base of a solid in the region bounded by the graphs of y = e-x y = 0, and x = 0, and x = 1. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid? (4 points)

11.

Find the average value of f(x) = sin(x) over the interval [0, π]. (4 points)

	2
	2π

	0

12.

Find the velocity, v(t), for an object moving along the x-axis if the acceleration, a(t), is a(t) = 2t + sin(t) and v(0) = 4. (4 points)

	v(t) = t2 + cos(t) + 3
	v(t) = 2 + cos(t) + 1
	v(t) = t2 - cos(t) + 5
	v(t) = t2 + sin(t) + 4

13.

Find the distance, in feet, a particle travels in its first 2 seconds of travel, if it moves according to the velocity equation v(t)= 6t2 - 18t + 12 (in feet/sec). (4 points)

	4
	5
	6
	-1

14.

For an object whose velocity in ft/sec is given by v(t) = -t2 + 6, what is its displacement, in feet, on the interval t = 0 to t = 3 secs? (4 points)

	9.000
	-9.000
	10.596
	-3.00

15.

An elementary student kicks a ball straight into the air with a velocity of 16 feet/sec. If acceleration due to gravity is -32 ft/sec2, how many seconds after it leaves his foot will it take the ball to reach its highest point? Assume the position at time t = 0 is 0 feet. (4 points)

	1.25
	6
	0.50
	Cannot be determined

16.

Which of the following is the general solution of the differential equation dy dx equals the quotient of 8 times x and y ? (4 points)

	y2 = x2 + C
	x2 - y2 = C
	y2 = 4x2 + C
	y2 = 8x2 + C

17.

The slope of the tangent line to a curve at any point (x, y) on the curve is x divided by y . What is the equation of the curve if (2, 1) is a point on the curve? (4 points)

	x + y = 3
	x2 - y2 = 3
	x2 + y2 = 3
	xy = 3

18.

The particular solution of the differential equation dy dt equals 2 times y for which y(0) = 60 is (4 points)

	y = 60e2t
	y = 60 e0.5t
	y = 59 + et
	y = 30et

19.

The temperature of a roast varies according to Newton's Law of Cooling: dT dt equals negative k times the quantity T minus A , where T is the water temperature, A is the room temperature, and k is a positive constant.

If a room temperature roast cools from 68°F to 25°F in 5 hours at freezer temperature of 20°F, how long (to the nearest hour) will it take the roast to cool to 21°F? (4 points)

	1
	9
	12
	24

20.

Find the specific solution of the differential equation dy dx equals the quotient of 4 times y and x squared with condition y(-4) = e. (4 points)




	None of these

21.

Choose the appropriate table for the differential equation dy dx equals the quotient of the quantity x plus 2 and x . (4 points)

x	0	0.5	2
	2	5	undefined

x	0	0.5	2
	undefined	5	2

x	0	0.5	2
	undefined	4	2

Cannot be found without solving the differential equation

22.

A differential equation that is a function of x only (4 points)

	will produce a slope field with parallel tangents along the diagonal
	will produce a slope field that does not have rows or columns of parallel tangents
	will produce a slope field with rows of parallel tangents
	will produce a slope field with columns of parallel tangents

23.

The differential equation dy dx equals the quotient of x and x minus 4 . (4 points)

I.will have a slope field with negative slopes in quadrant I II.will have a slope field with positive slopes in all quadrants III.will produce a slope field with columns of parallel tangents

	I only
	II only
	III only
	None of these

24.

Which of the following differential equations has the slope field below?

Quadrants one and three have negative slopes. Quadrants two and four have positive slopes. As x approaches zero from the positive or negative, the slopes approach horizontal. (4 points)

25.

The differential equation dy dx equals the product of 2 times x and the quantity 4 minus y . (4 points)

I. produces a slope field with horizontal tangents at y = 4 II. produces a slope field with horizontal tangents at x = 0 III. produces a slope field with vertical tangents at x = 0 and y = 4

	I only
	II only
	I and II
	III only

26.

Which of the following values would be obtained using 10 inscribed rectangles of equal width (a lower sum) to estimate the integral from 0 to 1 of x squared, dx ? (4 points)

	0.285
	0.385
	1.380
	2.310

27.

Which definite integral approximation formula is the integral from a to b of f of x, dx is approximately equal to the product of the quantity b minus a over 2 times n and the quantity y sub 0 plus 2 times y sub 1 plus 2 times y sub 2 plus 2 times y sub 3 plus dot dot dot plus 2 times y sub n minus 1 plus y sub n ? (4 points)

	Circumscribed rectangles
	Inscribed rectangles
	Mid-point rule
	Trapezoidal rule

28.

The estimated value of the integral from 1 to 3 of x squared dx , using the trapezoidal rule with 4 trapezoids is (4 points)

	17.50
	11.25
	8.75
	5.63

29.

Given the table below for selected values of f(x), use 6 left rectangles to estimate the value of

the integral from 1 to 10 of f of x dx . (4 points)

x	1	3	4	6	7	9	10
f(x)	4	8	6	10	10	12	16

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30.

Function f(x) is positive, increasing and concave down on the closed interval [a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the left sum, right sum, and trapezoidal rule approximations for the value of the integral from a to b of f of x dx . Which one of the following statements is true? (4 points)

	Trapezoidal rule value < Left sum < Right sum
	Left sum < Trapezoidal rule value < Right sum
	Right sum < Trapezoidal rule value < Left sum
	Cannot be determined without the x-values for the partitions

FRQ QUESTIONS (1-5)

1.	Find the area of the region bounded by the curves , y = 0, and x = 6 obtained by integrating with respect to y. Your work must include the definite integral and the antiderivative.

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2.	Set up, but do not evaluate, the integral which gives the volume when the region bounded by the curves y = Ln(x), y = 1, and x = 1 is revolved around the line y = -1.

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3.	The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x2. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

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4.	Determine if the Mean Value Theorem for Integrals applies to the function f(x) = x3 - 16x on the interval [-1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.

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5.	An object has a constant acceleration of 30 ft/sec2, an initial velocity of -10 ft/sec, and an initial position of 4 ft. Find the position function, s(t), describing the motion of the object.

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