PRACTICE PROBLEMS FOR THE FINAL
Problem 1. Suppose you blow up a spherical balloon at a constant
rate of 3 cubic inches per second. How fast is the surface area of the
balloon increasing when the diameter of the balloon is 10 inches?
Problem 2. Suppose that a manufacturer is making a cylindrical can
that will hold a liter of liquid. Suppose that the top of the can is made
of a material that is twice as expensive as the material that composes
the sides of the can and the base of the can. Determine the height and
radius of the can that will minimize the cost of the material used to
make the can.
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Problem 3. Let f (x) = e−x and let g(x) = 3/x.
(a) Determine all local maxima and minima for f and g. Say which of
these points are maxima and which are minima.
(b) Determine the points of inflection of f and g. Determine the intervals for which f and g are concave up and concave down.
(c) Find all vertical and horizontal asymptotes of f and g.
(d) Graph f and g. Label maxima, minima, points of inflection, and
vertical and horizontal asymptotes.
Problem 4. Determine the following limits.
(a)
3x3 + 2x2 − 5
lim 3
.
x→1 x − 3x2 + x + 1
(b)
lim x2 e−x .
x→∞
(c)
lim x log x.
x→0+
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PRACTICE PROBLEMS FOR THE FINAL
Problem 5. Let f (x) = 10x3 − x2 + 4x − 1.
(a) The function f has at least one zero in [0, .5]. Determine the first
digit after the decimal point for one of the zeroes (Hint: Plug in
0,.1,.2, etc and use the Intermediate Value Theorem).
(b) Find the integer n with 0 ≤ n ≤ 5 for which |f (.n)| is smallest.
Use a linear approximation to the function f at .n to approximate
the zero of the function.
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Problem 6. Find the tangent line to ex − ey = 0 at the point (1, 1).
Problem 7. Compute the derivatives of the following functions.
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(a) f (t) = te−t
2x
(b) y =
sin x
(c) g(x) = arctan(x2 + 1)
(d) y = (1 + x)1/x
Problem 8. Below is the graph of f 0 . Where does f have local maxima, local minima, and points of inflection? Where is the graph of f
concave up? Concave down? When is f increasing? Decreasing?
4
3
2
1
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-3
Problem 9. Repeat question 8, except this time assume that the
graph given is the graph of f .
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PRACTICE PROBLEMS FOR THE FINAL
Problem 9. Find all asymptotes of the function
2x + 3
f (x) = √
.
x2 − 1
Problem 10. Let f (x) = ax+b
where a, b, c, and d are constants.
cx+d
(a) Find f −1 (x) in terms of a, b, c, and d.
(b) Compute the derivatives of f and of f −1 .
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in this case, where g = f −1 .
(c) Verify that g 0 (x) = f 0 (g(x))
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