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# Advanced Calculus I -- Fall 2016 -- final exam

Name:_________________________ $\qquad$ 8 problems, 80 points total

- Write the problems in the given order, each problem on a separate page.

$\bullet$ **1.** (a) State the definition of the Riemann sum. (b) Use part (a) to explain the expression below:
$$0^2\cdot .25+.25^2\cdot .25+.5^2\cdot .25+.75^2\cdot .25.$$

$\bullet$ **2.** From the definition, prove that $f(x)=x^3$ is uniformly continuous on $[0,1]$.

$\bullet$ **3.** Prove:
$$ \lim _{x\to 0}x\sin\frac{1}{x}=0.$$

$\bullet$ **4.** (a) State theorems that relate differentiability, continuity, and integrability to each other. (b) Provide examples that relate differentiability, continuity, and integrability to each other.

$\bullet$ **5.** Is it possible to have a function $f$, a point $a$, and a sequence $\{x_n\}$ convergent to $a$ such that

- $\lim_{x\to a} f(x)$ does not exist but
- $\lim_{n\to \infty} f(x_n)$ does?

$\bullet$ **6.** From the definition, prove that l.u.b. is unique.

$\bullet$ **7.** From the definition, prove that if $f$ is differentiable at $a$ then so is $-f$.

$\bullet$ **8.** State the Bolzano-Weierstrass Theorem and provide an example that shows that the condition of the theorem cannot be removed.