Chapter 3 Review Problems and Study Guide
Calculus AB Bixler
Extrema on an interval.
1.
What points are potential extrema on an interval?
2.
How do you find critical numbers for a function?
3.
Write the steps for finding extrema on an interval?
4.
Find all the critical numbers for f ( x ) 
5.
Find all of the critical numbers for f ( x)  9  x 2
6.
Find all of the extrema in the interval [0, 2π] for y = sin x + cos x
7.
Find the absolute maximum and absolute minimum on the interval [-1, 2]
10
of f ( x )  2
.
x 1
8.
Find the absolute maximum and absolute minimum on the interval [0, 3]
of f(x) = x2 –2x + 1.
x 1
x 1
3
d i
5
Mean Value Theorem (MVT)
9.
What is the mean value theorem and when does it apply?
10.
Find the c guaranteed by the mean value theorem on the interval [2, 3] for
f(x) = 3x – x2
Increasing and Decreasing Functions and the First Derivative Test
x
is decreasing.
x  x2
11.
Find all the open intervals on which f ( x ) 
12.
Find all the open intervals on which f(x) = x3 –3x2 is increasing or decreasing.
13.
Find the x values that give relative extrema for f(x) = 3x5 – 5x3. Justify your
answers using the first derivative test.
14.
Find the relative extrema for y = 2x3 +3x2 –12x. Justify using the first derivative
test.
15.
Use calculus to show that f ( x ) 
2
x
is increasing wherever it is defined.
1 x
Concavity and the second derivative test
16.
State what the second derivative means in terms of concavity, extrema and
points of inflection.
17.
Find the intervals on which the graph of the function f(x) = x4 –4x3 + 2 is concave
upward or downward. Find the points of inflection. Justify everything using
calculus.
18.
Find all of the points of inflection of the graph of f(x) = x4 – x3. Justify analytically.
19.
Let f(x) = x3 – x2 +3. Use the second derivative test to find the relative extrema.
20.
Find all of the relative extrema of the function f(x) = x4 + 4x3. Use the second
derivative test.
Sketching Graphs from extrema, intercepts, concavity, asymptotes,
etc.
21.
Use the techniques learned in this chapter to sketch f(x) = x3 + x2 –6x
22.
Use the techniques learned in this chapter to sketch f(x) = 2x4– 8x2
Optimization Problems
23.
A rancher has 300 feet of fencing to enclose a pasture bordered on one side by a
river. The river side of the pasture needs no fence. Find the dimensions of the
pasture that will maximize the area.
24.
The product of two positive numbers is 588. Minimize the sum of the first and
three times the second.
25.
A manufacturer determines that x employees on a certain production line will
produce y units per month where y = 75x2 – 0.2x4. To obtain a maximum
monthly production, how many employees should be assigned to the production
line?
26.
Two rectangular lots are to be made by fencing in a rectangular lot and putting a
fence across the middle. If each lot contains 1875 square feet, what dimensions
of lots require the minimum amount of fence and how much fence is needed?
27.
An open box is to be constructed from cardboard by cutting out squares of equal
size in the corners and then folding up the sides. If the cardboard is 6 inches by
11 inches, determine the volume of the maximum sized box the can be
constructed.