Lesson Plan Day
Grab a partner and share your lesson plans with each other. If you are not ready for this activity, use class time today to work on getting your lesson plan ready. Remember, the goal is to chunk together ideas in a way that makes sense to you. The objectives below are the topics that you will be tested on. As you look for examples to put in your lesson plan, feel free to use the past rapid recalls.
Optimization
- For a function of the form $f(x,y)$ or $f(x,y,z)$, construct (by hand and with software) contour plots, surface plots, and gradient field plots.
- Compute differentials, partial derivatives, and gradients.
- Compute slopes (directional derivatives), tolerances (differentials), and equations of tangent planes.
- Obtain and use the chain rule to analyze a function $f$ along a parametrized path $\vec r(t)$. In particular, calculate slopes and locate maximums and minimums of $f$ along $\vec r$.
- Use Lagrange multipliers to locate and compute extreme values of a function $f$ subject to a constraint $g=c$.
- Apply the second derivative test, using eigenvalues, to locate local maximum and local minimum values of a function $f$ over a region $R$.
Integration
- Set up and compute single, double, and triple integrals to obtain lengths, areas, and volumes. Connect these to the differentials $dx$, $ds$, $dA$, and $dV$.
- Explain how to compute the mass of a wire, planar region, or solid object, if the density is known. Connect this to the differential $dm$.
- Find the average value of a function over a region. Use this to compute the center-of-mass and centroid of a wire, planar region, or solid object.
- Draw regions described by the bounds of an integral, and then use this drawing to swap the order of integration.
- Obtain the cross product and use it to find a vector orthogonal to two given vectors, the area of a parallelogram, and the volume of a parallelepiped.
- Appropriately use polar coordinates $dA = |r| dr d\theta$, cylindrical coordinates $dV = |r|drd\theta dz$, and spherical coordinates $dV = |\rho^2\sin\phi|d\rho d\theta d\phi$.
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