Math Insight

Overview of: Differentiation exam

The differentiation module exam for Math 1241 is based on part 3 from the Math 1241 thread, with the knowledge from part 1 also assumed.

Material for the exam

  1. Introduction to the derivative (See section from the Math 1241 thread)

    1. The slope of the secant line as average rate of the change, and the slope of the tangent line as instantaneous rate of change.
    2. The tangent line as the limit of a secant line with the two points approaching each other.
    3. The derivative is the slope of the tangent line, which we also call the slope of the function itself.
    4. The limit as the idea of getting as close as possible with actually going all the way.

  2. Derivatives of polynomials (See section from the Math 1241 thread)

    Be able to calculate the derivative of polynomials or the derivative of powers with other exponents.

  3. Derivatives of exponential and logarithmic functions (See section from the Math 1241 thread)

    Be able to calculate the derivative of the exponential function and the logarithm.

  4. Differentiation rules (See section from the Math 1241 thread)

    1. Be able to calculate the derivative of the product and quotient of two functions.
    2. Be able to calculate the derivative of simple compositions using the chain rule, especially those of the form $f(x)=e^{p(x)}$ or $f(x)=\ln(p(x))$, where $p(x)$ is a polynomial.
    3. Be able to combine product and chain rule to differential functions of the form $f(x)=p(x)e^{q(x)}$, where $p(x)$ and $q(x)$ are polynomials.

  5. Partial derivatives (See section from the Math 1241 thread)

    The partial derivative is just like the ordinary derivative, where you hold all variables but one constant.

    Be able to describe what a partial derivative means in the context of an example.

  6. Jacobian matrix or matrix of partial derivatives (See section from the Math 1241 thread)

    A vector valued function can be decomposed into its component functions.

    Each component function has partial derivatives.

    The Jacobian matrix, or matrix of partial derivatives, is a way of organizing all of these partial derivatives into an array. Each row corresponds to each of the component functions. Each column corresponds to differentiation with respect to a different variable.

Study aids

  1. Sample problems

    All questions that may appear in the differentiation exam are available so that you can practice them, as outlined below. In both these problems and on the actual exam, the set of problems as well as values of numbers, variables, parameters, and other quantities are selected randomly. You will want to generate multiple versions of the problems to see the larger array of problems. Given the random nature, we cannot guarantee that you will actually see all the problems that will appear on the exam. But the more problems you work on, the greater the chance you will work on problems that will show up on the test.

    1. Free form questions

      Review questions: Elementary derivative problems and Review questions: Elementary partial derivative problems contain the short answer problems that could appear on the exam.

    2. Computer-scored problems

      Review questions: Derivative intuition practice, matching, Review questions: Basic derivative practice, Review questions: Partial derivative practice, and Review questions: Jacobian practice, 2 x 2 matrices contain problems where the computer will the score your answer for you, so you can see how well you did. These computer generated scores do not count toward your grade. When these questions appear on the exam, they will be hand-graded. However, for these types of problems, only your final answer counts, and it doesn't matter if you show your work.

    3. Quiz problems

      and contain additional computer-graded problems that could show up on the exam. Even after the due date of quizzes, you can still generate more attempts and check out how well you can answer those questions. Again, when these questions appear on your exam, only your final answer counts, and it doesn't matter if you show your work.

  2. Worksheets

    The worksheets from the Math 1241 thread are also good review.

Exam rules

  1. You are allowed 50 minutes to take the exam.
  2. You are allowed to have one-half of one letter sized (8.5in x 11 in) sheet of paper of notes. Double-sided is OK. No restrictions to what can be on the sheet of paper.
  3. You are allowed to have a calculator in the exam. However, graphing calculators are NOT allowed on this exam.
  4. No textbook or electronic equipment (other than calculator) allowed.

Points and due date summary

Total points: 200
Due: Oct. 18, 2013, 11:59 p.m.

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Math 1241, Fall 2013