The homework is due on 3/20. All answers must include proofs or an explanation of the computation process.

  1. The lower base, upper base, and height of the trapezoid are 100, 40, and 30 respectively. Calculate the relative error of the trapezoid area. If the relative error of these measurements is controlled within 1%, find the maximum possible relative error of the area.
  2. Determine the conditions for which the limit $\displaystyle\lim_{(x,y)\to(0,0)}\frac{x^ay^b}{x^2+y^2}$ exists.
  3. In class, we showed that the limit of $\frac{xy^2}{x^2+y^4}$ at $(0,0)$ does not exist. Determine a condition for which the limit of $\frac{x^ay^b}{x^\alpha+y^\beta}$ does not converge due to the reason that the limit along some path $(c_1t^n,c_2t^m)$ is determined by the coefficients $c_1$ and $c_2$. This is an open question, so any condition can be proposed. You only need to provide convincing evidence (give a proof) that your proposed condition is sufficient to the grader.
  4. Given $F(x,y)$, and let’s set $x=x(u,v)$, $y=y(u,v)$, $u=u(s,t)$, and $v=v(s,t)$. Find $\frac{\partial F}{\partial s}$ and $\frac{\partial F}{\partial t}$.