$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Study Guide II

Write a study guide for yourself in preparation for Midterm I.
You can write the guide however helps you most, but here are some possible suggestions:

• Write down the important definitions for each topic you want to review. Explain them in your own words!
• Either come up with an example problem (or find one on WebAssign) for each topic you want to review, and write out its solution by hand! You’ll be handwriting solutions on the exam, so this is a good way to get yourself in that mindset and away from online work.
• For each topic you want to review, write to yourself how confident you are with it (maybe on like a 1-10 scale, if that’s helpful to you): this way when you’re studying you can look back and easily remember which things you were struggling with.
• If there’s multiple ways to do something, make a note of that for yourself!

The second midterm will focus mostly on the chapter Multivariate Differentiation, and the beginnings of Multivariate Integration (Only the first section, double integrals). Here’s a short cliffnotes of what we have covered (essentially a table of contents of the notes above.)

Differentiation

Functions

• Understanding graphs of multivariable functions.
• Understanding level sets of multivariable functions.

Partial Derivatives

• Know the definition and conceptual meaning of partial derivatives.
• Be able to use partial derivatives to tell when a function is increasing / decreasing / concave up /concave down in a coordinate direction.
• Computing implicit partial derivatives, and using them to find slopes.
• Be able to compute higher order partial derivatives.
• Being able to express in words what a partial differential equation says. Understanding qualitative behavior of partial differential equations.

Linearization

• Find implicit and parametric tangent planes to graphs \(z=f(x,y)\).
• Find normal vectors to graphs of functions \(z=f(x,y)\).
• Use differentials to estimate the value of a function near a point where you can compute it.
• Use differentials for error analysis: quantification of error in measurements.

The Gradient

• Know the definition of the gradient, how to compute it.
• Be able to use the gradient to compute directional derivatives.
• Know the geometry of the gradient: specifically, the meanings of its direction and magnitude.
• Know the relationship between gradients of a function and its level sets.

Extrema

• Be able to find the critical points of a function
• Be able to classify the critical points of a function as maxes mins or saddles.
• Given information about maxes mins and saddles from such a computation, be able to sketch the level sets of a function.

Optimization

• Understand how to deal with constraints via substitution.
• Know the concepts behind the technique of Lagrange multipliers
• Be able to use the technique of Lagrange multipliers to find the maxima and minima of a function along a constraint.
• Be able to combine Lagrange multipliers with the first and second derivative test of the previous section to solve constrained optimization problems with inequalities.

Integration

Double Integrals

• Be able to compute the double integral of a function over a rectangle.
• Be able to compute an iterated integral with variable bounds.
• Be able to change the order of integration on a double integral.