Use the graph of f(x) below to estimate the value of f ‘(-2):
Given the graph of f ‘(x), the derivative of f(x), which of the following statements are true about the graph of f(x)? (4 points)
I. The graph of f has a critical point at x = 0.
II. The graph of f has a critical point at x = 1.
III. The graph of f has a point of inflection at x = 1.
I and III only3.
Find the critical point of the function f(x) = x2 – 2x + 7. (4 points)
For what values of x is the graph of concave upward? (4 points)
No values of x
x < 6
x > -6
x > 65.
At what value(s) of x does f(x) = x4 – 18×2 have a critical point where the graph changes from decreasing to increasing?(4 points)0 and -3 only
0 and 3 only
-3, and 3 only
The maximum value of f(x) = 2×3 – 18×2 + 48x – 1 on the interval [0, 3] is (4 points)
In which interval is the function f(x) = – 4×2 + 12x + 1 increasing? (4 points)
(-∞, 2) only
(-∞, 2) U (6, ∞)
(6, ∞) only
(2, 6) only3.
What are the critical values on the graph of y = x4 + 6×2 – 5? (4 points)
x = -1 and 1
x = -3, 0, and 3
x = 0
There are no critical values for this function.4.
Find the x-coordinate of the point(s) of inflection on the graph of y = x3 – 12×2 – 5. (4 points)
x = -2
x = 4
x = 2
x = 125.
Which one of the following statements is true? (4 points)If f ′′(x) > 0 on the interval (a, b), then f is concave up on the interval (a, b).
If f ”(c) = 0, then x = c is an inflection point on the graph of f(x).
If f is continuous on a closed interval [a, b], then f attains an absolute maximum value and f attains an absolute minimum inside the interval [a, b].
None of these are true.
Find the absolute maximum value of the function . (4 points)
None of these2.
In the answer space below, provide one of the two positive integers whose product is 196 and whose sum is a minimum. (4 points) 3.
A farmer has 1,800 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) = -2×2 + 1800x
f(x) = -x2 + 1800x
f(x) = x2 – 1800
f(x) = 2×2 – 18004.
What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 12 – x2? (4 points)
If 600 cm 2 of material is available to make a box with a square base and a closed 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) = – t2 + 5t – 3, with v measured in feet per second and t measured in seconds. Find the acceleration of the particle at time t = 3 seconds. (4 points)
feet per second2
-1 feet per second2
-3 feet per second2
6 feet per second22.
A particle is moving along the x-axis so that its position at t ≥ 0 is given by s(t) = (t)In(2t). Find the acceleration of the particle when the velocity is first zero. (4 points)
None of these3.
The driver of a car traveling at 60 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)
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)
A particle moves along the x-axis so that at any time t, measured in seconds, its position is given by s(t) = 4sin(t) – cos(2t), measured in feet. What is the acceleration of the particle at time t = 0 seconds? (4 points)
If y = 2x 3 – 4x and , find when x = 1. Give only the numerical answer. For example, if = 3, type only 3. (4 points)
The area A = πr2 of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 3ft? (4 points)
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 6 cm is increasing at the rate of 3 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 20cm. (4 points)
None of these
A 10-ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 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 6 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)