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Calculus 3, Spring 2013

Final Exam

The final exam is Tues, May 14, 2:00 – 3:50 pm in our classroom. There will be a calculator and a non-calculator part, so please bring a calculator. I have posted one more homework assignment with the solutions available (no credit, just for practice) involving optimization and double and triple integrals.

Good things to study include the tests, your solutions to problems we did from the notes, as well as the online homework problems.

Here's an overview of the objectives we studied from each chapter

Vectors

  1. Define, draw, and explain what a vector is in 2 and 3 dimensions.
  2. Add, subtract, multiply (scalar, dot product, cross product) vectors. Be able to illustrate each operation geometrically.
  3. Use vector products to find angles, length, area, projections, and work.
  4. Use vectors to give equations of lines and planes, and be able to draw lines and planes in 3D.

Curves (Parametric Equations)

In this chapter we generalize the idea of a functions $y=f(x)$ to a parametric curve $\vec r(t)=(x,y)$.

  1. Model motion in the plane using parametric equations, and find velocity, speed, and acceleration.
  2. Find derivatives and tangent lines for parametric equations.

Functions

This chapter generalized our understanding of functions.

  1. Describe uses for, and construct graphs of, space curves and parametric surfaces. Find derivatives of space curves, and use this to find velocity, acceleration, and find equations of tangent lines.
  2. Describe uses for, and construct graphs of, functions of several variables. For functions of the form $z = f(x,y)$, this includes both 3D surface plots and 2D contour plots. For functions of the form $w=f(x,y,z)$, construct plots of level surfaces.
  3. Describe uses for, and construct graphs of, vector fields and transformations.
  4. If you are given a description of a vector field, curve, or surface (instead of a function or parametrization), explain how to obtain a function for the vector field, or a parametrization for the curve or surface.

Polar Coordinates

This chapter explored one particular plane transformation more in depth.

  1. Be able to convert between rectangular and polar coordinates in 2D.
  2. Graph polar functions in the plane. Find intersections of polar equations, and illustrate that not every intersection can be obtained algebraically (you may have to graph the curves).
  3. Find derivatives and tangent lines in polar coordinates.

Derivatives

In this chapter, we generalized the Calculus 1 concept of limit and derivative to multiple variables. We use a matrix for the multivariable generalization of the derivative, even though the textbook does everything without mentioning the matrix underlying the calculations. Using a matrix (the total derivative) both simplifies things and is more powerful and general.

  1. Find limits and be able to explain when a function does not have a limit by considering different approaches.
  2. Compute partial derivatives. Explain how to obtain the total derivative from the partial derivatives (using a matrix).
  3. Find equations of tangent lines and tangent planes to surfaces. We’ll do this three ways.
  4. Find derivatives of composite functions, using the chain rule (matrix multiplication).

Motion

We skipped this chapter.

Line integrals

In this chapter, we generalize the idea of integrals. We also set up the framework for average value type of problems. When we integrate over vector fields, we get the concepts of work, flow, and flux. Finally, we generalize the fundamental theorem of calculus to work with line integrals.

  1. Describe how to integrate a function along a curve. Use line integrals to find the area of a sheet of metal with height $z = f(x,y)$ above a curve $\vec r(t) = (x, y)$ and the average value of a function along a curve.
  2. Find the following geometric properties of a curve: centroid, mass, center of mass
  3. Given a vector field, compute the work (which is the same as the flow or circulation) along and the flux across piecewise smooth curves.
  4. Determine if a field is a gradient field (hence conservative), and use the fundamental theorem of line integrals to simplify work calculations.

Optimization

In this chapter, we deal with functions like $f\colon \mathbb{R}^n\to\mathbb{R}$. We generalize the idea of $f_x$ and $f_y$ to the idea of a directional derivative. We discuss the importance of the gradient vector. We also generalize the second derivative test from calculus 1 and also use the gradient vector to get a way to optimize within some constraints.

Note that the book does not use eigenvalues and the matrix derivative in the second derivative test. I will expect you to be able to do the test using the matrix derivative and eigenvalues, like we discussed in class.

  1. Explain the properties of the gradient, its relation to level curves and level surfaces, and how it can be used to find directional derivatives.
  2. Find equations of tangent planes using the gradient and level surfaces. Use the derivative (tangent planes) to approximate functions, and use this in real world application problems.
  3. Explain the second derivative test in terms of eigenvalues. Use the second derivative test to optimize functions of several variables.
  4. Use Lagrange multipliers to optimize a function subject to constraints.

Multiple integrals

In chapters 10 and 12, we generalize integration to double and triple integrals. We also generalize the calculus 1 method of substitution to other transformations like polar, cylindrical, and spherical. For these problems, the focus is setting up an integration problem. I will expect you to be able to evaluate simple double or triple integrals involving trig functions, polynomials, and $u$-substitution. Note that we skipped chapter 11.

  1. Explain how to setup and compute a double and triple integrals. Be able to set up multiple versions of the same problem by slicing in different ways.
  2. For planar or 3d regions, find area or volume, mass, centroids, center of mass
  3. Explain how to change coordinate systems in integration, with emphasis on polar (for double integrals), cylindrical, and spherical (for triple integrals) coordinates.
  4. Explain what the Jacobian is, and show how to use it.

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