Quizzes For Exam 1
Quiz 1 (due 16 Jan by 11:59PM) [soln posted 11AM 29 Jan]: Find description of the hyperplane in $E^4$ that contains the points $e_1-e_2$, $2e_2$, $e_3+e_4$, and $e_1-3e_4$.
Quiz 2 (due 5 Feb) [soln posted 1:30PM on 9 Feb]: Find the limit at $x_0=(0,0)$ it exists (if not, explain why!): $\displaystyle\lim_{x \rightarrow (0,0)} \dfrac{x^2y^2}{x^2y^2+(x-y)^2}$. (hint: check along the line determined by vector $v=e_1$ but also along the line determined by the vector $v=e_1+e_2$.)
For Exam 2
Quiz 3 (due 21 Feb): Calculate for $f, g \in \mathscr{B}([0,1])$, $f(x)=x^4$, $g(x)=x^2-1$: $\lVert f-g\rVert$.
Quiz 4 (due 7 Feb): Find the $q=3$ Taylor approximation of the function $f(x)=\cos(x)$ using $a=0$ and $b=x$ (see 5 March notes)..
For Exam 3
Quiz 5 (due 27 March) [soln]: For functions of one variable, it is impossible for a continuous function to have two local maxima and no local minimum. Try to explain why that could be (hint: draw a picture). After you explain that, show that the function $f(x,y)=-(x^2-1)^2-(x^2y-x-1)^2$ has only two critical points, but has a local maximum at both.
Quiz 6 (due 11 April) [soln]: In the 7 April class we proved that $\overline{V}(A\cup B)\leq \overline{V}(A) + \overline{V}(B)$ as part of Lemma 5. Prove the rest of Lmma 5, i.e. prove $\underline{V}(A \cup B) \geq \underline{V}(A)+\underline{V}(B)$.
Quiz 7 [soln]: Prove that $\underline{\displaystyle\int} f+g \mathrm{d}V \geq \underline{\displaystyle\int} f \mathrm{d}V + \underline{\displaystyle\int} g \mathrm{d}V$.