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

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 Differentiation, from the material on parametric curves and their curvature through partial derivatives, linear / quadratic approximations, and their applications.

Parametric Curves

• Be able to compute the velocity and unit tangent vectors of a curve.
• Be able to compute the normal and binormal vectors to a curve.
• Compute the curvature of a curve
• Qualitatively understand the curvature: where is it high, where is it low, how does it relate to the circle of best fit?

Multivariable Functions

Functions

• Understanding graphs of multivariable functions.
• Understanding level sets of multivariable functions.
• Be able to convert between a level set and a graph for functions \(z=f(x,y)\).

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.
• Be able to compute higher order partial derivatives.

Approximation

• Find 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.
• Find the quadratic approximation to a function (second order Taylor series) at a point

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 using the quadratic approximation.
• Given information about maxes mins and saddles from such a computation, be able to sketch the level sets of a function.
• Be able to draw the level sets of a function using the location of maxes, mins and saddles.

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.

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.