Calc Multiple Choice

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Find the particular solution to y ' = sin(x) given the general solution is y = C - cos(x) and the initial condition y of pi over 2 equals 1 . (5 points)

	-cos(x)
	2 - cos(x)
	-1 - cos(x)
	1 - cos(x)

The slope of the tangent to a curve at any point (x, y) on the curve is negative 1 times x divided by y . Find the equation of the curve if the point (2, -2) is on the curve. (5 points)

	x + y = 0
	x2 - y2 = -2
	x2 + y2 = 16
	x2 + y2 = 8

The rate of decay in the mass, M, of a radioactive substance is given by the differential equation dM dt equals negative 1 times k times M , where k is a positive constant. If the initial mass was 200g, then find the expression for the mass, M, at any time t. (5 points)

	M = 200ln(kt)
	M = 2e-kt
	M = 200 ekt
	M = 200 e-kt

The temperature of a cup of hot tea varies according to Newton's Law of Cooling: dT dt equals negative k times the quantity T minus A , where T is the temperature of the tea, A is the room temperature, and k is a positive constant. If the water cools from 100°C to 80°C in 1 minute at a room temperature of 60°C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes. (5 points)

	42
	58
	63
	79

The differential equation dy dx equals the quotient of the quantity y minus 2 and y plus 1 (5 points)

I.produces a slope field with horizontal tangents at y = 2 II.produces a slope field with vertical tangents at y = -1 III.produces a slope field with columns of parallel segments

	I only
	II only
	I and II
	III only

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

slope field with positive slopes in quadrants 1, 2 and 4, negative slopes in quadrant 2, horizontal sopes along the y axis and vertical slopes along the x axis (5 points)

The general solution of the differential equation dy - 0.2x dx = 0 is a family of curves. These curves are all (5 points)

	lines
	hyperbolas
	parabolas
	ellipses

Estimate the value of the integral from negative 1 to 3 of x cubed, dx by using the Trapezoidal Rule with n = 4. (5 points)

	8
	19
	22
	36

The table below gives selected values for the function f(x). With 5 rectangles, using the left side of each rectangle to evaluate the height of each rectangle, estimate the value of the integral from 1 to 2 of f of x, dx . (5 points)

x	1	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	2.0
f(x)	1	0.909	0.833	0.769	0.714	0.667	0.625	0.588	0.556	0.526	0.500

	0.7456
	0.6456
	0.6919
	0.6932

10.

Given f(x) > 0 with f ′(x) < 0, and f ″(x) > 0 for all x in the interval [0, 2] with f(0) = 1 and f(2) = 0.2, the left, right, trapezoidal, and midpoint rule approximations were used to estimate the integral from 0 to 2 of f of x, dx . The estimates were 0.7811, 0.8675, 0.8650, 0.8632 and 0.9540, and the same number of subintervals were used in each case. Match the rule to its estimate. (5 points)

abcde	1.	left endpoint
abcde	2.	right endpoint
abcde	3.	midpoint
abcde	4.	trapezoidal
abcde	5.	actual area

0.7811

0.8650

0.8675

0.9540

0.8632

Which of the following functions grows the fastest as x goes to infinity? (4 points)

	2x
	3x
	ex
	x20

Compare the rates of growth of f(x) = the square root of x and g(x) = Ln(x) as x approaches infinity. (4 points)

	f(x) grows faster than g(x) as x goes to infinity.
	g(x) grows faster than f(x) as x goes to infinity.
	f(x) and g(x) grow at the same rate as x goes to infinity.
	The rate of growth cannot be determined.

What does limit as x goes to infinity of the quotient of f of x and g of x equals 0 show? (4 points)

	g(x) grows faster than f(x) as x goes to infinity.
	f(x) and g(x) grow at the same rate as x goes to infinity.
	f(x) grows faster than g(x) as x goes to infinity.
	L'Hôpital's Rule must be used to determine the true limit value.

Which of the following functions grows at the same rate as the square root of the quantity x raised to the 4th power plus x ? (4 points)

	x
	x2
	x3
	x4

Which of the following functions grows the slowest as x goes to infinity? (4 points)

	5x
	5x
	x5
	They all grow at the same rate.

Let F of x equals the integral from 1 to 3 times x of the natural logarithm of t squared . Use your calculator to find F"(1). (4 points)

	12
	6
	4

2.	Pumping stations deliver gasoline at the rate modeled by the function D, given by with t measure in hours and and R(t) measured in gallons per hour. How much oil will the pumping stations deliver during the 3-hour period from t = 0 to t = 3? Give 3 decimal places. (4 points)

A particle moves along the x-axis with velocity v(t) = sin(2t), with t measured in seconds and v(t) measured in feet per second. Find the total distance travelled by the particle from t = 0 to t = π seconds. (4 points)

	2
	1

	0

Find the range of the function f of x equals the integral from negative 4 to x of the square root of the quantity 16 minus t squared . (4 points)

	[-4, 4]
	[-4, 0]
	[0, 4π]
	[0, 8π]

Use the graph of f(t) = 2t + 4 on the interval [-4, 6] to write the function F(x), where f of x equals the integral from 2 to x of f of t dt . (4 points)

	F(x) = x2 + 6x
	F(x) = x2 + 4x - 12
	F(x) = x2 + 4x - 8
	F(x) = x2 + 8x - 20