Calculus Multiple Choice and FRQ

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Which of the following definite integrals could be used to calculate the total area bounded by the graph of y = sin(x), the x-axis, x = 0, and x = π (4 points)

	the integral from 0 to pi of the sine of x, dx
	the integral from 0 to pi of the negative sine of x, dx
	2 times the integral from 0 to pi of the sine of x, dx
	one half times the integral from 0 to pi of the sine of x, dx

Suppose

the integral from 2 to 8 of g of x, dx equals 13

, and

the integral from 6 to 8 of g of x, dx equals negative 3

, find the value of

2 plus the integral from 2 to 6 of g of x, dx

. (4 points)

	16
	18
	8
	32

Evaluate the integral

the integral from 0 to 2 of the absolute value of x, dx

. (4 points)

	-2
	0
	2
	4

Use your graphing calculator to evaluate to three decimal places the value of

the integral from negative 1 to 1 of the product 2 and the square root of 1 minus x squared over 2, dx

. (4 points)

	3.771
	3.636
	1.571
	1.111

the integral from 3 to 5 of 1 divided by the quantity x plus 1, dx is equal to the integral from 4 to 6 of 1 divided by u, du

(4 points)

	True
	False

Find the average value of f(x)=e2x over the interval [2, 4]. (4 points)

	1463.18
	731.59
	1517.78
	23.60

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

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

	55 over 3
	16 over 3
	16
	12

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

	7.67
	-3.00
	-0.33
	3.00

A girls throws a tennis ball straight into the air with a velocity of 64 feet/sec. If acceleration due to gravity is -32 ft/sec2, how many seconds after it leaves the girl's hand will it take the ball to reach its highest point? Assume the position at time t = 0 is 0 feet. (4 points)

	4 secs
	2 secs
	1 sec
	Cannot be determined

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

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 (3, 1) is a point on the curve? (4 points)

	x2 + y2 = 8
	x + y = 8
	x2 - y2 = 8
	xy = 8

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

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

Find the specific solution of the differential equation

dy dx equals the quotient of 2 times y and x squared

with condition y(-2) = e. (4 points)

	y equals negative 1 minus 2 divided by x
	y equals e raised to the negative 2 over x power
	y equals negative 1 times e raised to the 1 over x power
	None of these

Choose the appropriate table for the differential equation

dy dx equals the quotient of the x and quantity x minus 2

. (4 points)

0.5

dy over dx

negative one third

undefined

0.5

dy over dx

undefined

-1

undefined

0.5

dy over dx

-3

Cannot be found without solving the differential equation

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

The differential equation

dy dx equals the quotient of x and y squared

. (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

Which of the following differential equations is consistent with the following slope field?

In quadrant one, all slopes are positive. Greater values of y have slopes approaching horizontal and y less than 1 have slopes approaching vertical. For quadrant two slopes are negative with slopes approaching horizontal for greater values of y. Quadrant three also has all negative slopes. Larger negative values of y have slopes approaching horizontal. All slopes are positive in quadrant four.

(4 points)

	dy over dx equals x divided by y squared
	dy over dx equals x divided by y
	dy over dx equals x squared divided by y
	dy over dx equals x squared divided by y squared

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

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

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 n and the quantity y sub 0 plus y sub 1 plus y sub 2 plus y sub 3 plus dot dot dot plus y sub n minus 1

? (4 points)

	Left rectangles
	Right rectangles
	Average value
	Trapezoidal rule

The estimated value of

the integral from 0 to 2 of x squared dx

, using the trapezoidal rule with 4 trapezoids is (4 points)

	2.75
	5.50
	1.88
	3.75

Given the table below for selected values of f(x), use 6 trapezoids 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

________________________

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