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Math 2413 Review 3
Sections 3.1–3.7, 3.9
You may use a non-graphing scientific calculator on Test 2.
1. Below is the graph of the function f on the interval  4,4 . Its vertical asymptote is x  2
and lim f  x    .
x 2
b) Find the absolute maximum of f on  4,4 .
a) Find the critical numbers of f .
c) Find the absolute minimum of f on  4,4 . d) Where does f have local maxima?
f) Find the maximum of f on  4, 2 .
e) Where does f have local minima?
g) Find the maximum of f on  2,3 .
i) Find the minimum of f on  3,0  .
2. For f  x   x3  12 x on  0,4 ,
a) Find the critical numbers.
c) Find the absolute minimum.
h) Find the maximum of f on  0,2 .
j) Find the minimum of f on  4,  52  .
b) Find the absolute maximum.
3. The graph of the derivative of f is given.
a) Where is f increasing?
b) Where is f decreasing?
c) Where does f have local maxima?
d) Where does f have local minima?
e) Which is larger, f  2  or f  3 ?
f) Which is larger, f  1 or f  3 ?
4. The graph of the function f on the interval  4,4 is given. The graph consists of linear,
quadratic, and cubic pieces.
a) Identify the intervals where f is increasing. b) Identify the intervals where f is decreasing.
c) Identify the intervals where f is constant. d) Identify the intervals where f is concave up.
e) Identify the intervals where f is concave down. f) Find the inflection points.
g) Find the absolute maximum of the function.
h) Find the absolute minimum of the
function.
5. Make a sketch illustrating the Mean Value Theorem, and summarize the theorem.
6. Determine all the values of c guaranteed by the Mean Value Theorem for the function
3
f  x    x  1 on the interval  1,1 .
7. Use the Intermediate Value Theorem to show that the function f  x   sin x  x  1 has at
 
least one zero on the interval 0,  . Use Rolle’s Theorem to show that it has at most one
 2
zero on this interval.
8. Use the 2nd Derivative Test to determine the local extrema of f  x   12 x  cos x on  0,2  .
9. Find the differential dy of the function y  f ( x)  3 sin(2 x) .
10. For the function y  f ( x)  x5  2 x , evaluate and compare dy and y for
x  2.5 x  dx  0.1
11. a) Find the linearization of the function f  x   3 x at x  8 .
b) Use your linearization to estimate the value of 3 9 .
c) Suppose the side of a cube is measure to be 5 inches, with a maximum possible error of
0.20 inch. What is the maximum propagated error when calculating the volume of the cube?
12. A box is to be made out of a piece of cardboard that measures 10 by 16 inches. Squares of
equal size will be cut out of each corner, and then the ends will be folded up to form an open
rectangular box. What size square should be cut from each corner to obtain the maximum
volume?
13. A fence is to be built to enclose a rectangular area of 800 square feet. The fence along three
sides is to be made of material that costs $6 per foot. The material for the fourth side costs
$18 per foot. Find the dimensions of the rectangle that will allow the most economical fence
to be built.
14. Suppose that the altitude (in feet) of a jet airplane t seconds after takeoff is given by the
function h(t )  6t 3  6t 2  48t  7 . Find the maximum altitude attained by the jet.
15. Find the length of the shortest line segment that is
cut off by the first quadrant and passes through the
point

l1

3,3 .
3
3
3

;0   
Note: l    l1  l2 

cos sin 
2

3
l2
3

3
16. A page of a book is to contain 27 square inches of print. If the margins at the top, bottom,
and left side are 2 inches and the margin at the right side is 1 inch, what size page would use
the least paper?
17. Find the point(s) on the parabola y  18 x 2 closest to the point  0,6  .
18. a) Use the Intermediate Value Theorem to show that the equation x3  x  1  0 has at least
one solution in the interval  0,1 .
b) Use Rolle’s theorem to show that it has at most one solution in the interval  0,1 .
19. Let f  x   4 x3  12 x 2  1 . Find the critical values. Use the second derivative test, if
possible, to determine any local extrema.
x 4 x3
20. Let f  x     2 x 2  4 x  1 . Find all relative maxima and relative minima.
4 3
21. Find the absolute maximum and absolute minimum of f  x    x 2  4  over the interval
2
[1,3] .
22. Find the absolute maximum and absolute minimum of f ( x)  6  3 x 
interval [5, 1] .
12
over the
x2
For Problems #21–25, determine each of the following items and sketch the graph.
a) Domain
b) Critical numbers
c) x-intercept(s)
d) y-intercept(s)
d) Horizontal asymptote(s)
f) Vertical asymptote(s)
g) Slant asymptote(s)
h) Intervals on which f is increasing
i) Intervals on which f is increasing
j) Intervals on which f is concave up k) Intervals on which f is concave down
l) Local extrema
m) Inflection points
n) Sketch the graph of f.
23. f  x   2 x 4  5 x3
24. f ( x)  x3  18 x 2  81x
25. f  x  
6 x  12
 x  1
2
26. f  x   x x  1 Note: the derivatives are f   x  
3
x3
27. f ( x) 
1  x2
4  x  34 
3  x  1
2
3
and f   x  
4  x  32 
9  x  1
5
3
.