M408D First Midterm with Solutions, October 3, 2002

1. A sequence { tex2html_wrap_inline53 } is said to ``grow faster'' than { tex2html_wrap_inline55 } if tex2html_wrap_inline57 . Put the following sequences in order of growth rate, from fastest to slowest. Justify your answers.






Note that tex2html_wrap_inline69 and that tex2html_wrap_inline71 . The sequence tex2html_wrap_inline73 (factorial) grows faster than tex2html_wrap_inline53 (exponential), which grows faster than tex2html_wrap_inline77 (power), which grows faster than tex2html_wrap_inline79 (smaller power), which grows faster than tex2html_wrap_inline55 (log).

2. Evaluate the following limits or improper integrals, or write DNE if the limits do not exist.

a) tex2html_wrap_inline83 . If you let n=1/x, this is tex2html_wrap_inline87 .

b) tex2html_wrap_inline89 . Since tex2html_wrap_inline91 (integrate by parts), we have tex2html_wrap_inline93 , which goes to 1 as tex2html_wrap_inline95 .

c) tex2html_wrap_inline97 . This is a ``0/0'' indeterminate form. Applying L'Hopital's rule once gives tex2html_wrap_inline99 , which equals 1/2.

d) tex2html_wrap_inline101 . This is NOT an indeterminate form. It is 0/(-1)=0.

3. Which of the following series and integrals converge, and which diverge. In each case, give a 1-sentence explanation (e.g., ``converges by ratio test'', or ``diverges by comparison to tex2html_wrap_inline105 '')

a) tex2html_wrap_inline107 diverges by integral test, or by comparison to 1/2k.

b) tex2html_wrap_inline111 converges by ratio test or by root test.

c) tex2html_wrap_inline113 diverges. Terms don't go to zero.

d) tex2html_wrap_inline115 . This is a funny way of writing the alternating series tex2html_wrap_inline117 , which converges.

4. a) Write down the Taylor series for tex2html_wrap_inline119 .


b) Write down the 6-th order Taylor polynomial (i.e., through the tex2html_wrap_inline123 term) for tex2html_wrap_inline125 .

tex2html_wrap_inline127 , so the answer is tex2html_wrap_inline129 .

c) Evaluate tex2html_wrap_inline131 . (That is, the 6-th derivative of f(x), evaluated at x=0.) The coefficient of tex2html_wrap_inline123 , namely 8, is the answer divided by 6!, so the answer must be 8(6!) = 8(720) = 5760.

d) Evaluate tex2html_wrap_inline141 to five decimal places. (No, you don't need a calculator!)

The third-order Taylor polynomial is good enough:
tex2html_wrap_inline143 .

5. a) Find the second-order Taylor polynomial for tex2html_wrap_inline145 around x=4.

tex2html_wrap_inline149tex2html_wrap_inline151 and tex2html_wrap_inline153 . Taking a=4 we have f(a)=2, f'(a) = 1/4 and f''(a)=-1/32, so


[OOPS!  That should be powers of (x-4), not of (x-2).]

b) What is the radius of convergence of the power series tex2html_wrap_inline165 ? Justify your answer.

By the root test, tex2html_wrap_inline167 , so the radius of convergence is 2. (The ratio test would work just as well).

Lorenzo Sadun

Thu Oct 3 10:03:01 CDT 2002