MAT 132 Spring 2009 --- Review for Final --- Updated 5/6/09

The final examination is comprehensive, so you should use the review sheets from Midterm 1 and Midterm 2 as well as this one. Also the review items at the end of Chapters 6, 7, 7 may be useful

References are to Stewart, Calculus - Concepts and Contexts, 3nd Ed.

7.4 Exponential growth and decay. y' = ky is an important special case of a separable equation. The general solution is y=Cekt, called "exponential growth" if k is positive, "exponential decay" if k is negative. Understand that C = y(0) [Exercises 1,2]. Be able to calculate k from C and one other data point [Exercise 3]. Be able to calculate C and k from two data points [Exercise 4]. Special case: the half-life. Be able to calculate k from the half-life, and vice-versa [Example 3, Exercises 8, 9, 10].
    Compound interest. Understand that $1 invested for one year at annual interest t (in dollars) gives $1+t if compounded annually, $(1 + t/2)2 if compounded twice a year $(1 + t/n)n if compounded n times a year, and $et if compounded continuously [Example 5 page 531, Exercises 18a, 19a].

7.5 The Logistic Equation. Understand the concept of carrying capacity K and why dP/dt = kP(1-P/K) is a natural way of encoding K into the growth equation. Recognize the shape of the slope fields that come from this kind of equation (k always positive in this context!). [Example 1]. Understand the concept of equilibrium solution [Problem 1d]. Be able to write down the equation from data giving k and K. Be able to use Euler's method (small number of steps) to estimate P(t) given k, K, P(0) and t. [Problem 3a].

7.6 Predator-Prey Systems. Understand the meaning of the constants k, a, r, b in Equation 1 p. 545 [Example 1, Exercise 1]. Understand why, when the constants are positive, the Phase portrait of the system has the shape given in Figures 1, 2 p. 546; explanation p. 547 [Problem 3]. Be able to locate the equilibrium solutions [Problem 9].

Complex Numbers - Appendix I. Understand the identification of the complex number x + iy with the point (x,y) in the plane. Be able to add and multiply complex numbers. [Exercises 1, 2, 3, 4]. Be able to write the complex number x + iy in "polar" form r ei theta. [Box, p.A72, Example 4 p.A73, Exercises 25, 26]. Understand the geometric explanation of multiplication (multiply the r's and add the theta's) [Box, p.A73]

Second Order Differential Equations (we only study an important special case: linear homogeneous equations with constant coefficients). Review the Notes. Be able to substitute y = elambda x for the unknown function y and to solve for lambda. This method gives two solutions (remember how to get them in the special case where the lambda equation has a double root), say y1 and y2. Be able to write the general solution as c1y1 + c2y2 [Exercises 1, 2 in Notes] and to use initial conditions to calculate c1 and c2 [Exercises 6, 9 in Notes]. The fundamental physical example is a horizontal spring fixed at one end and with a mass of 1 gram at the other. Then, taking y = 0 as the rest position, the position of the mass is given by the solution of y'' + py' + ky = 0 where k is the spring constant and p is the damping constant [Exercises 11-15]

8.1 Know what a sequence is, and be able to check convergence/divergence in simple cases: for rational functions of n as in Example 3 [Exercises 10, 11]; using Theorem 2 p.559 and l'Hôpital's rule [Example 4, Exercises 15, 19, 21].

8.2 Understand that the sum of an infinite series only makes sense as the limit of the sequence of partial sums (Definition 2 p.568). Fundamental example: the geometric series a + ar + ar2 + ar3 + ... (Example 1) converges to a/(1-r) if |r| < 1 and diverges otherwise. Be able to sum this series correctly [Example 2, Exercises 13, 14, 15, 16]. Also, be able to use this technique to express a repeating decimal as a fraction [Example 4, Exercises 31-34]. Second fundamental example: the harmonic series 1 + 1/2 + 1/3 + ..., which diverges (Example 7). This is fundamental because it shows that the terms can go to zero and the series may still diverge (Note 2 p. 572).

8.3 Be able to apply the integral test, and determine convergence or divergence of a series from covergence or divergence of the corresponding improper integral (discussion on pp. 577, 578) [Exercises 6-8].
   Be able to apply the comparison test: suppose 2 series, the sum of {an} and and the sum of {bn} are related by the inequality 0 < an < bn for every n; then if the smaller series diverges the larger one must diverge, and if the larger one converges the smaller one must converge. [Example 3, Exercises 9, 10, 16, 17].
   Be able to apply the integral test to the "tail" (the sum from the (n+1)-st term on) of a series to estimate the error involved in approximating the infinite sum by the sum of the first n terms [Example 6, Example 7, Exercises 28, 29].

8.4 Understand that an alternating series (terms are alternately positive and negative) will converge if the terms are decreasing in absolute value and if their limit is 0 (Box, page 587) [Example 1, Exercises 3, 5, 7] and the "Alternating Series Estimation Theorem" (p. 588) [Example 4, Exercises 12, 13, 17].
   Understand what absolute convergence means, and that it implies convergence [Theorem 1 p.590, Example 7, Exercises 19, 21, 23].
   VERY IMPORTANT FOR POWER SERIES: Be able to apply the ratio test to a series (Box, p. 591) [Examples 8,9, Exercises 31, 33, 35].

8.5 Power series. Understand what a power series is ("an infinite polynomial") and that in general the convergence of c0 + c1x + c2x2 + ... will depend on x. The Ratio test is very useful here [Examples 1, 2 ]. Understand the difference between the radius of convergence and the interval of convergence (p. 596) [Examples 4, 5, Exercises 5, 7, 9, 13.]

8.6 A power series in x defines a function f(x). [Example 1: 1/(1+x2) = 1 - x2 + x4 - ... ; note that the domain of definition of the function (here it is the whole line) may be different from the interval of convergence of the series (here it is (-1, 1)! Exercises 5, 7, 9].
   Be able to calculate new power series by term-by-term differentiation or integration of old ones (Theorem 2 p. 601) [Examples 5, 6, 7, Exercises 11, 12, 13].

8.7 Taylor series. If a function f(x) has derivatives of every order at 0, then the power series

f(0) + f'(0)x + f''(0)x2/2 + ... + f(n)(0)xn/n! + ...

is called the Taylor series for f(x) about 0; also called the Maclaurin series for f(x). More generally if f(x) has derivatives of every order at x=a, then the power series

f(a) + f'(a)(x-a) + f''(a)(x-a)2/2 + ... + f(n)(a)(x-a)n/n! + ...

is called the Taylor series for f(x) about a [Examples 1, 3, Exercises 3, 4, 6]. Be able to apply Taylor's Inequality (p. 608) -you do not need to memorize it- to show that ex, sin x, cos x are equal to the sum of their Maclaurin series [Example 2, Exercise 11, 13, 15]. Be able to use Taylor series to integrate functions that can't be handled by ordinary methods, like e-x2 [Example 8, Exercises 35, 36].

8.8 The Binomial Series. Understand that the Binomial Theorem
(a+b)k = ak + C(k,1)ak-1b + C(k,2)ak-2b2 + ... + C(k,k-2)a2bk-2 + C(k,k-1)abk-1 + bk
where C(k,n) is the binomial coefficient C(k,n) = [k(k-1)(k-2)...(k-n+1)]/n!, generalizes to cases where k is not a positive integer. This applies in particular to (1+x)k and gives the binomial series [Box, page 618]. When k is a positive integer this series terminates, as above. Otherwise it is a power series with radius of convergence equal to one. [Examples 1, 2, Exercises 1, 3, 7, 9].

8.9 Applications of Taylor Polynomials. Understand how to estimate how good an approximation you get using the n-th Taylor polynomial Tn(x) for a function f instead of f(x) itself. See numbers 2. and 3. on p.622; Examples 1, 2, Exercises 11, 15, 16, 17 -part (c) optional.