Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 31
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41Worksheet # 24: Review for Exam III
1. (a) Find the linearization, L(x), to f (x) =
1
2x at x =
4.
(b) Use the result of (a) to approximate
11.
(c) Find the absolute error in the approximation of
11 by using your calculator.
2. (a) Describe in words and diagrams how to use the first and second derivative tests to identify and
classify extrema of a function f (x).
(b) Use the first derivative test to identify and classify the extrema of the function
3
2
f (x) = 2x
+ 3x
72x
47.
2
3. Find the absolute minimum of the function f (t) = t+ 1
t
on the interval [ 1, 1]. Be sure to specify
the value of t where the minimum is attained.
4. For each of the following functions (i) Find the intervals on which f is increasing or decreasing. (ii) Find
the local maximum and minimum values of f . (iii) Find the intervals of concavity and the inflection
points.
4
2
2
(a) f (x) = x
2x
+ 3
(b) f (x) = e
+ e
4
3
2
5. For what values of c does the polynomial p(x) = x
+cx
+x
have two inflection points? One inflection
point? No inflection points?
6. (a) State the Mean Value Theorem. Use complete sentences.
(b) Does there exist a function f such that f (0) =
1, f (2) = 4, and f (x)
2 for all x?
7. Evaluate the following limits;
8
3
2012
7x
+ 3x
1
2012
x
(a) lim
(c)
lim
3
8
2
2013
21x
13x
+ x
2013
x
2012
5x
2012
x
(d)
lim
(b) lim
2
2013
9x
+ 1
2013
x
8. (a) State L’Hopital’s Rule for limits in indeterminate form of type 0/0. Use complete sentences, and
include all necessary assumptions.
x + 2
e
x
1
(b) Evaluate lim
(d) Evaluate lim
2
x
2
9x
+ 1
0
2
e
3
(c) Evaluate lim
x
ln(x)
(e) Evaluate lim
x + 2
2
0
2
9. A poster is to have an area of 180 cm
with 1 cm margins at the bottom and sides and 2 cm margins
at the top. What dimensions will give the largest printed area. Be sure to explain how you know you
have found the largest area.
(a) Draw a picture and write the constraint equation.
(b) Write the function you are asked to maximize or minimize and determine its domain.
(c) Find the maximum or minimum of the function that you found in part (b).
10. Find a positive number such that the sum of the number and twice its reciprocal is small as possible.
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