MAT126 Review for Midterm II

5.4 Review basic manipulations of indefinite integrals (from 5.2, Properties 1-4, 5 p.361-362 and Comparison Properties 6-8 p.363) and know the basic anti-differentiation formulas (Table, p.369). Understand both statements of the Fundamental Theorem, p.381: 1-The derivative of a definite integral with respect to its upper endpoint is the value of the integrand at that endpoint ( Examples 2,4,5, Exercises 7,10,11,12,15,17 -remember chain rule!); 2-Evaluation of a definite integral by the difference of the values of an antiderivative at the endpoints (review section 5.3). Understand the principle of the proof of statement 1, as sketched on p.382: the rate of increase of the area is equal to the height of the function.

5.5 Practice as many substitutions as you can. You need to be able to recognize a likely "u", and that comes with practice. (Examples 2,3,4 and the more difficult Examples 5,6,7 -each of those involves a standard "trick"; you need to know these elementary tricks. Exercises 7-34, as many as you can do. For definite integrals, remember that you have to EITHER transform your limits of integration to be the corresponding u-values, OR rewrite your u-antiderivative in terms of the original variable before evaluating. Exercises 39-54, as many as you can do.

5.6 Again, practice is essential: you need to be able to recognize what is the "u" and what is the "dv". I recommend using "Formula 2" (the "u,v" formulation). Once you have chosen "u" and "dv", write down du and v -this requires a preliminary integration!-. Example 1-obvious case; Example 2-less obvious but you should know this "trick" (Exercises 4,6). Examples 3,4 each give an important wrinkle in applying "parts." Applying it several times if necessary (keep very careful track of your signs!! - Exercises 7,8,12); or applying it twice and then solving for the integral (Exercises 13,14). Be familiar with these maneuvers. Exercises 1-24, as many as you can do.

5.9 Remember the relation between left sum Ln, right sum Rn, trapezoidal rule Tn, midpoint rule Mn and Simpson's rule S2n, that is: Tn = (Ln+Rn)/2 and S2n = (2Mn+Tn)/3 (Example 4). This will help you derive the formula for Simpson's rule p.418 which you should know. Understand that Ln underestimates and Rn overestimates if f is increasing, and vice-versa if f is decreasing. Understand that Mn underestimates and Tn overestimates if f is concave-up, and vice-versa if f is concave-down. Exercises 5,7,8,21,23,24. The exercises require a calculator; understand the procedures well enough so that you can carry them out by hand for n = 2, 3 or 4

Use the Chapter Review for further reviewing.

March 25 2008