Lesson Plan Day Tomorrow
Tomorrow you'll grab a partner and share your lesson plans with each other. 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 $dV = |\rho^2\sin\phi|d\rho d\theta d\phi$.
Individual Practice
(Cylindrical Coordinates - Disc and Shell method) This sequence of problems has you work with one solid in several different ways. It also has you develop the disc and shell method as by-products of cylindrical coordinates, where the only difference is the order of integration.
1. Draw the solid region in space that is bounded above by $z=9-x^2-y^2$ (so $z=9-r^2$) and below by the $xy$-plane.
2. Set up a triple integral in Cartesian coordinates to compute the volume of this solid.
3. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta drdz$.
4. Compute the two inside integrals from (3.) and simplify to show that $V = \int_{0}^{9} \pi (\sqrt{9-z})^2 dz$. When you're done, you'll have obtained the disc-method $$V = \int dV =\int_a^b \underbrace{\pi (\text{rad of disc at $z$})^2}_{\text{area of disc at height $z$}} \underbrace{dz}_{\text{little height}}.$$
5. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta dzdr$.
6. Compute the two inside integrals and simplify to show that $V = \int_{0}^{3} 2\pi r (9-r^2) dr$. When you're done, you'll have obtained the shell-method $$V = \int dV = \int_a^b \underbrace{(2\pi r)(\text{height of shell at $r$})}_{\text{shell surface area = (circumference)(height)}} \underbrace{dr}_{\text{shell thickness}}.$$
The only difference between the shell and disc methods, from first semester calculus, is the order of integration used.
Textbook practice
If you want more integrals to work with. These all come from Thomas's calculus, the 14th edition. Feel free to tackle these at home as you are reviewing for the exam.
- Double integrals - Swapping order - 15.2: 33-54
- Polar integrals - 15.4: 1-8, 9-22 (swap to polar and then use software to check)
- Triple integrals - 15.5: 21-36
- Cylindrical and Spherical- 15.7: 37-42 (cyl), 55-60 (sph), 65-84 (BEST ones, you have to pick the system, draw the region, set things up - use the Mathematica notebook Integration.nb to check if your bounds are right).
You can also get extra practice problems to work on from the OpenStax calculus text.
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