M408C Fall 2001

Prof: Brian Martensen

Calculus Practice Final

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Test:

Answers:

Calculus bonus problems with answers

Disclaimer: These problems are NOT for extra credit. They are designed by me, not the prof, and have no bearing on your mark. Do not spend time on them unless you have finished the homework and understand everything.
Having said that, they are fun problems. The "bonus" is that if you show me a solution that I believe you came up with yourself, I will be impressed.

Sorry for the bad formatting, I'd like to put these in MathML soon.
  1. Prove that sqrt(2) is not rational, that is, it cannot be written a/b for any integers a,b. answer
    hint: try a proof by contradiction.
  2. Prove that the set S = { q in Q : sqrt(2) < q } has no least member. answer
    remark: this is obvious if you take q in R rather than Q. proof: suppose you believe q is the least member. let r = (sqrt(2) + q)/2. then r is in R and sqrt(2) < r, so r is in S, but r < q, a contradiction. note this proof fails when we restrict S to a subset of Q, because r is not rational.
    remark 2: this problem shows that the set T = { q in Q : q < sqrt(2) } has no least upper bound in Q. every rational upper bound of T must be in S, but S has no least member. this implies that any rational sequence approaching sqrt(2) has no limit in Q. more generally, if it were not for irrational numbers, most limits would not exist. in contrast, every subset of R that is bounded above has a least upper bound in R. this number can be used to define the limit.
  3. Without using l'Hopital's rule, find lim (x->0) (sin (sin x))/x. answer
  4. Given that (d/dx)(sin x) = cos x, find (d/dx)(cos x) without using the definition directly (there are two ways). answer
  5. Define F(x) = integral from 1 to x of dt/t. Show (from first principles) that F(xy) = F(x) + F(y). answer
  6. integrate 1/(1-x^2)
    remark: you know how to do this now that we have learned partial fractions.
  7. integrate 1/sqrt(1+x^2)
    remark: I think you know how to do this too. These last two problems were lookaheads.
  8. you learned that (dy/du)(du/dx) = dy/dx - this is the chain rule. is it true that (d2y/du2)(du/dx)2 = d2y/dx2?
  9. A ladder of length 10 meters is leaning against a wall. It starts to slip such that the bottom end moves along the ground at a constant speed of 1 m/s. If the other end stays against the wall, at what speed will it hit the ground? (thanks to Steve McAdam for the problem)