Mathematics 132 – Calculus for Physical and Life Sciences 2 Review Sheet for Midterm 3 April 20, 2005

General Information The third midterm will be held the evening of Wednesday, April 27, from

6:00–7:30. Material from Section 8.6 through Section 10.4 will be covered. We will review for the exam in class on Tuesday and Wednesday next week.

Topics to be Covered Here is a brief list of topics:

• Probability density and cumulative distribution functions; mean and median of a density.

• Numerical series: partial sums and convergence; finite and infinite geometric series. Convergence tests: integral, comparison, and ratio tests, alternating series.

• Power series, radius and interval of convergence; Taylor polynomials and series, series manipulations, the error estimate for Taylor polynomials.

Some Sample Problems

The following problems are typical of test-type questions. Note that a single problem may call for techniques spanning the coverage of the test. Do not attempt to memorize how to do “all the types of problems”; instead, strive to understand what each idea is used for, and select your approaches accordingly.

1. Let x be the size of an email message in kilobytes (KB). An ISP finds that the fraction of emails between x and x + Δx KB in size is about p(x) = cxe^{−0.01x}Δx.

(a) Find the value of c

(b) Calculate the fraction of email messages that are at most 100 KB in size, the fraction that are at least 50 KB in size, and the median size of an email message.

2. In a certain population, the height x of an individual in inches is normally distributed, with a mean of 68 and a standard deviation of 4. In other words, heights are modeled by the density function p(x) = \frac{1}{4\sqrt{2π}} e^{−(x−68)^{2}/32}.

(a) Write down a definite integral whose value gives the fraction of the population whose height is between 68 and 72 inches, and perform the change of variables z = (x − 68)/4 on your integral.

(b) Write out the Taylor series for

erf(x) = ∫_{0}^{x} e^{−z^{2}/2} dz,

giving both the first four non-zero terms and the summation form.

(c) Suppose you intend to estimate the integral in part (a) using the series in part (b). How many decimal places would you obtain by using the first four non-zero terms? Calculate this estimate.

(d) How many terms are needed to guarantee five decimals of accuracy?

3. Determine (with justification!) whether or not the following series converge:

∑_{n=0}^{∞} \frac{n + 2^{n}}{3^{n}} ,  ∑_{k=1}^{∞} \frac{(−1)^{k}}{\sqrt{k}} ,  ∑_{k=0}^{∞} \frac{k}{e^{k}} ,  ∑_{n=0}^{∞} \frac{n^{2} + 4n + 1}{3n^{3} + 2n^{2} + 10000},  ∑_{n=1}^{∞} \frac{1}{n^{1.01}} .

4. Let f(x) = \sqrt{1 + x} = (1 + x)^{1/2}. Find the degree-4 Taylor polynomial of f centered at a = 0. Try to find a factorial expression for the general term of the Taylor series.

5. Consider the geometric series f(x) = ∑_{k=0}^{∞} x^{k} = \frac{1}{1 − x}.

(a) Use series manipulations to find the Taylor series of xf′(x).

(b) Use series manipulations to find the Taylor series of − ln(1 − x).

(c) Find the radius of the series in part (b), and investigate convergence at the endpoints.

(d) Use parts (a) and (b) to evaluate the series ∑_{k=1}^{∞} \frac{1}{k · 2^{k}} and ∑_{k=1}^{∞} \frac{k}{2^{k}}.

6. For each of the given power series, find the interval of convergence.

f(x) = ∑_{n=1}^{∞} \frac{(2x)^{n}}{\sqrt{n}} , g(x) = ∑_{n=1}^{∞}(−1)^{n−1} \frac{(x − 5)^{n}}{n · 3^{n}} .

(In particular, give the radius, and investigate convergence at the endpoints.)

7. Consider the problem of estimating ∫_{0}^{π} \frac{sin t}{t} dt.

(a) Discuss the relative merits of the numerical methods from Sections 7.5–7.6, Taylor series, and any other methods you can think of.

(b) Use series operations to find the Taylor series of

Si(x) = ∫_{0}^{x} \frac{sin t}{t} dt.

Give five non-zero terms and the summation form.

(c) How many terms are needed in part (b) to guarantee six decimals of accuracy when x = π? (Suggestion: Set up an estimate for the remainder, then use trial-and-error with a calculator.)