Midterm 1 Review

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Math 2300 – Calculus II – University of Colorado

Fall 2010 – Review for Midterm I

One of the main areas that will be emphasized during this midterm, compared to previous calculus II exams that you may find, is abstract and conceptual understanding. From what we have done in class and on homework/webwork, you have all of the tools to solve these problems. But, they may initially look a bit unfamiliar. Some examples of such problems are given here to help prepare you for the sorts of questions that will be asked on the midterm.

Some Things to Make Sure You Know

  • Substitution of indefinite and definite integrals.

  • Integration by parts of indefinite and definite integrals. It is also always helpful conceptually to know how to derive the formula for integration by parts.

  • Know how to use the table of integrals. More importantly, know how to derive some of the “basic” formulas on and related to that table: \int e^{x}\cos x\, dx and \int\cos^{n}x\, dx are classic examples, and you did one of them as a homework problem.

  • Know how to use the method of partial fractions, including how to apply long division to simplify an integral.

  • Know which trig substitutions to use, and how to use completing the square to rewrite quadratics when needed.

  • Understand how the Left, Right, Mid and Trap approximation methods work and how they relate (when a function is increasing/decreasing, concave up/concave down). Know how Simp relates. Of course, you should be able to calculate any of these and produce the relevant formulae for their calculation (even if you are using a calculator). Know what increase in the factor of subdivisions results in one more decimal of accuracy for each method.

  • Know how to set up the calculations for improper integrals, be able to follow through with the corresponding limit calculations, and be able to explain why this weird limit process is needed.

Examples:

  1. For a continuous function, prove that

    \int f(x)+xf^{{\prime}}(x)\, dx-xf(x)=0
  2. Prove that

    \pi\int _{0}^{\pi}x^{n}e^{{ax}}\, dx=\frac{\pi^{{n+1}}e^{{a\pi}}}{a}-\frac{\pi n}{a}\int _{0}^{\pi}x^{{n-1}}e^{{ax}}\, dx
  3. Give formulae and values for Left(n), Right(n), Mid(n), Trap(n) and Simp(n) for f(x)=e^{x} on [1,3] with n=4. Without explicitly calculating Right(400) or Mid(400), can you give approximations to what they will equal?

© 2011 Jason B. Hill. All Rights Reserved.