MAT 132 Spring 2009 Review for Midterm 2

References are to Stewart, Single Variable Calculus - SBU Edition - 3ed.

6.3 Arclength. Be able to set up the integral giving the length of the curve traced out by the point with coordinates (x(t), y(t)) between t = a and t = b [Formula 1, Example 1 p.463, Exercises 3, 5, 9]. Special case: be able to set up the integral giving the length of the graph of y = f(x) between x = a and x = b [Formula 2 p. 463, Example 2, Exercises 8,9]. Since these integrals are in general difficult or impossible to calculate analytically (using anti-differentiation) be able to apply approximation methods to estimate them [Exercise 11 - use n=4 if doing by hand].

6.4 Average value. Be able to calculate the average value of the function f(x) on the interval [a,b]: divide the integral by (b-a). [Box p. 468, Example 1, Exercises 5, 7].

6.5 Work. Understand that if the force is a constant F, and displaces its application point from a to b, the work done is the product W = F (b-a); and that if the force varies with distance x as F(x) the problem is handled by slicing the interval [a,b] into infinitesimal subintervals of length dx from x to x + dx over which the force can be considered constant. The displacement from x to x + dx involves an infinitesimal amount of work dW = F(x)dx, and these infinitesimal amounts of work are summed by the integral: the total work W is the integral of dW from a to b. Be able to to convert problems of this type into integrals. [Examples 1 and 2 page 472, Exercises 1-6].
    A different application of Calculus is to problems where the distance that points are moved varies with the parameter x. In the simplest examples (e.g. emptying a rectangular tank of water over the top) the force is uniform, but points at one end of the problem (the bottom) have to travel farther than those at the other end (the top; here x is height). The method is to slice the problem perpendicular to the x-direction, and to let dW be work corresponding to lifting out the slice at height x and thickness dx. [Example 3, Exercises 9, 11, 15].
    In more complicated examples, the size of the slice may also vary with x [Example 4, Exercise 17]. Or the force may also vary with x [Exercise 13].
    Be able to apply the "slice and integrate" method to all these problems. Hydrostatic Pressure and Moments are not covered.

6.7 Probability. Be able to check whether a given function f(x) can be a probability density function [Examples 1,2 page 487, 488] and how to interpret the integral from a to b of f(x)dx as a probability [Exercises 1, 3, 5]. Be able to calculate the average value (= the mean) of the random variable described by a probability density function f(x) [Example 3 page 489, Exercise 6c]. Be able to work with probability density functions given by graphs [Exercise 6] and equations [Example 1]; and in particular with exponentially decreasing probability density functions [Example 2, Examples 3 and 4 pages 489, 490, Exercises 7,9].

7.2 Direction fields. Understand how the direction field for the first-order differential equation y' = f(x,y) allows one to "see" the family of solutions [Figures 1-4 page 505]. Be able to sketch the direction field of y' = f(x,y) [Example 1, Exercises 9,10, Exercises 11-14 -choose a 5 by 5 grid of points centered on the initial value]. Given a direction field be able to sketch solutions, and be able to sketch the solution with a given initial value [Figures 7,8 page 506; Example 2; Exercises 1a, 3, 7, 18].
    Euler's Method. Given a first-order differential equation y'=f(x,y), an interval [0,T], an initial value y0 and a number N, be able to apply Euler's method with N steps to find approximate values y(T/N), y(2T/N), ... , y(T) for the solution satisfying y(0) = y0 [Example 3 page 509, Example 4 page 510, Exercises 21-24].

7.3 Separable equations. Understand what a separable equation is and be able to solve it by integration, applying the "separate, integrate, solve" method [Examples 1a, 2 page 514]. Understand how the "+C" from one of your integrals turns into the undetermined constant in the general solution, and how an initial value determines what that constant must be [Example 1b, Exercises 9-14].
    Mixing problems. Be able to apply the "rate in - rate out" method to convert a mixing problem to a separable differential equation, and then be able to solve the equation [Example 6, problems 35, 37].