Problems for you

Old exam problem sets, and the compendium, are found in the left margin link "Problems for seminars etc.". Please start doing a little bit of each of the following topics, and then on Tuesday we will discuss briefly what the first seminar should focus on: 

• Let f(x) = g(||x||). What is the gradient of f, in matrix form?
• General differentiation/integration drill for trig. functions: 
  • Differentiate f(t) = tan t = sin t / cos t, and write the answer in two ways: one which has only "cos" (no sin nor tan) and one which has only "tan" (no sin nor cos).
  • From the previous point, you should be able to show that the derivative of arctan t is 1/(1+t^2).
  • Here is a one which you should not spend too much time on, but which shows the method of substitution in a way that might be a bit less familiar: 
    Find an antiderivative of 1/(1-x^2)^(1/2) (sorry, Messages does not support math: Take (1 minus x squared), and then raise to the power of -1/2). Use the substitution x = sin t (yes, it is x = sin t, not u = sin x). 
• The Leibniz rule: Compendium 4-01, 4-02 and if you have not had enough already: 4-04. 4-10 is likely to be given for the seminar too.
• Double integrals: 4-05 and 4-08.