Calculus Multiple Choice and FRQ

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The graph of f '(x) is continuous and increasing with an x-intercept at x = 4. Which of the following statements must be true? (5 points)

	The graph of f is always concave down.
	The graph of f is always increasing.
	The graph of f has a relative minimum at x = 4.
	The graph of f has an inflection point at x = 4.

Below is the graph of f '(x), the derivative of f(x), and has x-intercepts at x = -3, x = 1, and x = 2 and a relative maximum at x = -1.5 and a relative minimum at x = 1.5. Which of the following statement is true?

Graph of a function that increases from negative infinity to x equals negative 1.5, decreases from x equals negative 1.5 to x equals 1.5, crossing the y axis at y equals 6, and increases from x equals 1.5 to positive infinity with x intercepts at x equals negative 3, 1 and 2 (5 points)

	f is concave up from x = 0 to x = 3.
	f has an inflection point at x = 0.
	f is concave down from x = -1.5 to x = 1.5.
	None of these is true.

The graph of y = f '(x), the derivative of f(x), is shown below. List the intervals where the graph of f is increasing.

Graph consists of 3 line segments from x equals negative 4 to x equals 4. Graph is decreasing from x equals negative 4 to x equals negative 2, increases from x equals negative 2 to x equals 2 and decreases from x equals 2 to x equals 4. There are x intercepts at x equals negative 4, 0 and 4. (5 points)

	(-4, -2) U (2, 4)
	(-2, 2)
	(-4, 0)
	(0, 4)

Which of the following functions grows the fastest as x grows without bound? (5 points)

	f(x) = x10
	g(x) = ln(x10)
	h(x) = 10x
	They all grow at the same rate.

Compare the growth rate of the functions f(x) = x3 + 1 and g(x) = x2 . (5 points)

	f(x) grows faster than g(x).
	g(x) grows faster than f(x).
	f(x) and g(x) grow at the same rate.
	It cannot be determined.

f is a twice differentiable function that is defined for all reals. The value of f "(x) is given for several values of x in the table below.

x	-8	-3	0	3	8
f " (x)	-4	-2	0	4	5

If f " (x) is always increasing, which statement about f(x) must be true? (5 points)

	f(x) is concave downwards for all x.
	f(x) passes through the origin.
	f(x) has a relative minimum at x = 0.
	f(x) has a point of inflection at x = 0.

f is a differentiable function on the interval [0, 1] and g(x) = f(2x). The table below gives values of f '(x). What is the value of g '(0.1)? (5 points)

x	0.1	0.2	0.3	0.4	0.5
f '(x)	1	2	3	-4	5

	1
	2
	4
	Cannot be determined

Use the graph of f(t) = 2t + 1 on the interval [-1, 4] to write the function F(x), where f of x equals the integral from 2 to x of f of t dt . (5 points)

	F(x) = x2 + 3x
	F(x) = 2x + 1
	F(x) = x2 + x - 20
	F(x) = x2 + x - 6

The velocity of a particle moving along the x-axis is v(t) = t2 + 2t + 1, with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance travelled by the particle from t = 0 to t = 2 minutes. (5 points)

_________________________

10.

Find the range of the function f of x equals the integral from 0 to x of the square root of the quantity 4 minus t squared dt . (5 points)

	[0, 4π]
	[0, 2π]
	[-4, 0]
	[0, 4]

FRQ QUESTIONS

The figure below shows the graph of f ', the derivative of the function f, on the closed interval from x = -2 to x = 6. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4.

Graph of a function on the domain negative 2, 6 and range negative 3, 4. Graph increases from negative 2, 3 until 2, 0 and from 4, negative 2.5 to 6, 4. Graph decreases from 2, 0 to 4, negative 2.5. There are x intercepts at 2, 0 and 4, negative 2.5.

Find the x-coordinate of each of the points of inflection of the graph of f. Justify your answer. (10 points)

A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec2 is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10.

What is the position of the car when t = 20? You must show your work answer to 3 decimal places. Include units in your answer.

Linear graph from x equals 0 to x equals 30 with a negative slope. y intercept is y equals 10 and x intercept is x equals 10. (10 points)

3.	Show that f(x) = 2000x4 and g(x) = 200x4 grow at the same rate. (10 points)

A radar gun was used to record the speed of a car (in feet per minute) during selected times in the first 2 minutes of a race. Use a trapezoidal sum with 4 intervals to estimate the distance the car covered during those 2 minutes. Give a 2 decimal place answer and include units. (10 points)

t	0	0.3	1.0	1.6	2
v(t)	0	24.5	27.8	28.3	29.0

Oil flows into a tank according to the rate F of t equals the quotient of t squared plus 1 and the quantity 1 plus t , and at the same time empties out at the rate E of t equals the quotient of the natural log of the quantity t plus 7 and the quantity t plus 2 , with both F(t) and E(t) measured in gallons per minute. How much oil, to the nearest gallon, is in the tank at time t = 12 minutes. You must show your setup but can use your calculator for all evaluations. (10 points)