Continuous Functions Exercise 1 For which functions $ f $ does there exist a continuous function $ F: \mathbb{R} \to \mathbb{R} $ such that $ F(x) = f(x) $ for all $ x $ in the domain of $ f $? $ f(x) = \frac{x^2 - 4}{x - 2} $ $ f(x) = \frac{|x|}{x} $ $ f(x) = 0 $ for irrational $ x $ $ f(x) = \frac{1}{q} $ for $ x = \frac{p}{q} $ in lowest terms ...
Functions Exercise 1 Let $f(x) = \frac{1}{1 + x}$. Find: $f(f(x))$ and determine its domain $f\left(\frac{1}{x}\right)$ $f(cx)$ $f(x + y)$ $f(x) + f(y)$ For which numbers $c$ does there exist $x$ such that $f(cx) = f(x)$? For which numbers $c$ does $f(cx) = f(x)$ hold for two different values of $x$? Exercise 2 Let $g(x) = x^2$ and $h(x) = \begin{cases} 0 & \text{if } x \text{ rational} \ 1 & \text{if } x \text{ irrational} \end{cases}$. Find: ...
Integrals Exercise 1 : Proofs Prove $\int_{0}^{b} x^{3} dx = \frac{b^4}{4}$ using equal partitions and $\sum_{i=1}^{n} i^3$. Prove $\int_{0}^{b} x^{4} dx = \frac{b^5}{5}$ analogously. Show $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^p}{n^{p+1}} = \frac{1}{p+1}$. Prove $\int_{0}^{b} x^p dx = \frac{b^{p+1}}{p+1}$. Exercise 2 For $0 < a < b$, find $\int_{a}^{b} x^p dx$ using partitions with fixed ratios $r = t_i/t_{i-1}$: Show $t_i = a \cdot c^{i/n}$ where $c = b/a$. For $f(x) = x^p$, derive: $$ U(f,P) = (b^{p+1} - a^{p+1}) \frac{c^{p/n}}{1 + c^{1/n} + \cdots + c^{p/n}} $$ and find $L(f,P)$. Conclude $\int_{a}^{b} x^p dx = \frac{b^{p+1} - a^{p+1}}{p+1}$. Exercise 3 Evaluate by symmetry: ...
Limits and Continuity Exercise 1 Find the following limits: $\lim_{x \to 1} \frac{x^2 - 1}{x^3 - 1}$ $\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$ $\lim_{x \to 3} \frac{x^3 - 8}{x^2 - 4}$ (Note the changed limit point) $\lim_{x \to y} \frac{x^n - y^n}{x - y}$ $\lim_{y \to x} \frac{x^n - y^n}{x - y}$ $\lim_{h \to 0} \frac{\sqrt{a + h} - \sqrt{a}}{h}$ Exercise 2 Find the following limits: $\lim_{x \to 1} \frac{1 - \sqrt{x}}{1 - x}$ $\lim_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x}$ $\lim_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x^2}$ Exercise 3 In each of the following cases, determine the limit $l$ for the given $a$, and prove that it is the limit by showing how to find a $\delta$ such that $| f(x) - l | < \varepsilon$ where $\forall x$ satisfying $0 < |x-a| < \delta$ ...