Worksheet 3 - Download as PDF

							ESP – Math 179                      Worksheet 3                            Spring 2010




1. Assume that −x4 ≤ τ (x) ≤ x2 . What is limx→0 τ (x)? Is there enough information given
to evaluate limx→−1 τ (x)?

2. Use the Squeeze Theorem from Section 2.6 to evaluate the limit.
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 (a) limx→0 x cos x
             √      π
 (b) limx→0+ xecos( x )

3. Use the Squeeze Theorem to show that if limx→c |g(x)| = 0, then limx→c g(x) = 0.

4. Use Theorem 2 in Section 2.6 to evaluate the limit.
             tan x
 (a) limx→0    x
             √
 (b)   limx→0 x+2 sin x
                  x
               ex sin x
 (c) limx→0       x


5. Evaluate the limit.
               6 sin x
 (a) limx→0      6x
                  x
 (b) limx→ π
           4   sin 11x

               1−cos x
 (c) limx→0     sin x


6. Use the Intermediate Value Theorem from Section 2.7 to show that

                                         ex + ln x = 0

has a solution in the interval (0, 1).

7. Let µ(x) = 2x − x3 .

 (a) Show that µ(x) has a zero in the interval [1, 1.5].
 (b) Show that µ(x) has a zero in the interval [1.25, 1.5].
 (c) Determine whether [1.25, 1.375] or [1.375, 1.5] contains a zero.


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8. In Chapter 2, you learned several different ways to tell if a limit exists and (if it does)
to evaluate limits. Use the methods of Chapter 2 to evaluate the following limits:
               x3 −x
 (a) limx→1    x−1

               e3x −ex
 (b) limx→0     ex −1

               x3 −ax2 +ax−1
 (c) limx→1         x−1

                [x]
 (d) limx→0−     x
             sin 5x
 (e) limx→0  sin 2x
             √
 (f)   limx→3 x+1−2
                x−3



9. Define the derivative of f (x) at x = a (denoted by f (x)) by the following limit (if it
exists):

                                           f (a + h) − f (a)
                               f (a) = lim                   .
                                       h→0         h
For the following functions, compute f (x) and f (0).

 (a) f (x) = x2 − x + 4

 (b) f (x) = x3 + 2x − 1
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 (c) f (x) =   x

 (d) f (x) = sin x (Hint: Look up the sum and difference formulas for sin x for help)

 (e) f (x) = ex


10. We interpret the derivative of f (x) at x = a as being the instantaneous slope of the
graph at the point x = a. Let f (x) = |x|. Can f (0) exist? Why or why not? (Hint: Graph
the function, and think about the instantaneous slope of the graph to the left of 0 and to
the right of 0.)




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