# Midterm 2 review page

Midterm solutions

**Review session. **Josh will be holding a review session Sunday, April 1 at 3pm in SC 310. Please send him any specific topics you'd like to have discussed.

**Review material. **Here is a brief outline of the material that you should know for the second midterm.

**Manifolds.**Inverse function theorem, implicit function theorem, manifolds, tangent spaces, manifold recognition, Lagrange multipliers.**The integral.**Partitions, upper/lower sums, upper/lower integral, measure 0 and content 0, integrability criterion theorem, fundamental theorem of calculus, Fubini's theorem.

Some specific things you should be comfortable with:

- Applying the implicit function theorem to test if a subset of R^n is a manifold (either locally or globally).
- Computing tangent spaces.
- Finding maxima/minima using Lagrange multipliers.
- Applying the fundamental theorem of calculus and Fubini's theorem to compute integrals.
- Working with the definition of the integral; showing a function is integrable using partitions.

**Practice midterms. **Practice 1, Practice 2, Practice 3 ; Solutions 1, Solutions 2, Solutions 3

**Frequently asked questions**

** Do I need to know proofs?** Many of the theorems we've discussed (IFT, IFT, integrability criterion, Fubini) have more difficult/involved proofs. I'm not going to ask you to know the proof; it's important that you know what the theorems say (definitely know the statements. why are the hypotheses necessary? what are examples that the theorem does or doesn't apply to? is the theorem if and only if?) and also know how to apply them in both theoretical and computational situations.

** Is the midterm cumulative?** In some sense, yes. While the focus will be on manifolds/integration, we've continued to use tools from earlier in the semester. In particular, you should not forget how to compute derivatives, use the chain rule or derivative magic wands, or work with open/closed/compact sets.