Review Worksheet
Math 251, Winter ’15, Gedeon
1. Write the definition of continuity; i.e. what does it mean to say “f (x) is continuous at x = a”?
2. Is the following function continuous at x = 2? Use limits to justify your answer.
x−2
x<2
f (x) =
x2 − 4x + 1
x≥2
3. Sketch the graph of a function that is not continuous at x = 3.
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4. Find the value of c that makes the following function continuous:
2
x +c
x ≤ −1
f (x) =
x3 + 1
x > −1
Will this value of c also make the function differentiable? Explain.
5. What does it mean (in terms of the first and second derivatives) for a function to be increasing
at a decreasing rate?
6. In the fall of 1972 President Nixon announced that the rate of increase of inflation was
decreasing. Suppose I(t) represents the inflation in the year t. Translate Nixon’s statement
into mathematical symbols.
[Hint: Say something about a derivative of I(t).]
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7. Consider the graph of y = f (x), shown below
y
4
2
x
-4
-2
2
4
-2
-4
(a) Compute each of the following limits, if possible. If a limit does not exist, write “DNE”.
i. lim f (x) =
x→−2
ii. lim f (x) =
x→0−
iii. lim f (x) =
x→1
iv. lim f (x) =
x→−1
(b) Is f continuous at x = −1? Use limits to justify your conclusion.
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8. Use the Intermediate Value Theorem to show that there is a root of the given equation in the
specified interval.
√
(a) 3 x = 1 − x on (0, 1).
(b) ex = 3 − 2x on (0, 1).
9. Find an equation of the tangent line to the function f (x) = 7 +
4
1
at x = 1.
x
10. Find the following limit
√
lim
y→7
y+2−3
.
y−7
[Hint: Use L’Hospital’s rule!]
11. A rumor spreads through a small town according to the equation
p(t) =
1
1 + 100e−0.5t
where p(t) is the proportion of the population that knows the rumor at time t. Find lim p(t).
t→∞
Write a sentence that explains the meaning of this limit in the context of the problem.
5
12. Let f be some function so that
f 0 (x) = (x + 2)2 (x − 2)
and
f 00 (x) = (x + 2)(3x − 2).
[The first and second derivatives have been given in factored form for convenience.]
(a) On what interval(s) is the function f increasing?
(b) At what x-value(s) does f have a local extremum? Make sure to state whether each is
a minimum or a maximum.
(c) On what interval(s) is f concave up?
(d) At what x-value(s) does f have a point of inflection?
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(e) Suppose you also know that f (0) = 0. Use this information and the information collected
above to sketch a graph of the function f .
5
4
3
2
1
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
7
2
3
4
13. Use the function f (x) =
√
x + 1 to estimate the value of
√
4.4 using linear approximation.
14. Find the absolute maximum and minimum values of the function f (x) = x4 − 32x + 7 on the
interval [−1, 3].
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15. Compute each of the following derivatives, as indicated.
(a) Find
dy
at the point (2, −3), given xy 2 − 20 = x2 + 2y.
dx
(b) For g(x) = x ln(3x − 2), compute g 0 (1).
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16. If a stone is thrown vertically upward from the surface of Planet X, its height (in meters)
after t seconds is s = 7t − 0.9t2 .
(a) Find and interpret s0 (0). Include units.
(b) Find the acceleration of the stone after 1 second. Include units.
(c) At what value(s) of t is the tangent line to s horizontal? Interpret your answer in terms
of the problem.
10
17. A rectangular storage container with an open top is to have a volume of 10 m3 . The length of
its base is twice the width. Material for the base costs $10 per square meter. Material for the
sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
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18. The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing
when the diameter is 80 cm?
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[Hint: For a sphere, V = πr3 .]
3
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