Calculus

Derivatives Exercise 1 Using the definition, prove that if $f(x)=1/x$, then $f’(a)=-1/a^{2}$ for $a\neq 0$. Show that the tangent line to $f$ at $(a, 1/a)$ intersects $f$ only at $(a, 1/a)$. Exercise 2 Using the definition, prove that if $f(x)=1/x^{2}$, then $f’(a)=-2/a^{3}$ for $a\neq 0$. Show that the tangent line at $(a, 1/a^{2})$ intersects $f$ at exactly one other point, on the opposite side of the y-axis. Exercise 3 Prove that if $f(x)=\sqrt{x}$, then $f’(a)=1/(2\sqrt{a})$ for $a>0$. (Hint: Rationalize the difference quotient.) ...

Fundamental Theorem of Calculus Exercise 1 : Derivatives of Integral Functions Find $F’(x)$ for each function: $F(x) = \int_a^{x^3} \sin^3 t dt$ $F(x) = \int_3^x \frac{1}{1 + \sin^6 t + t^2} \int_1^t \sin^3 u du dt$ $F(x) = \int_{15}^x \int_8^y \frac{1}{1 + t^2 + \sin^2 t} dt dy$ $F(x) = \int_x^b \frac{1}{1 + t^2 + \sin^2 t} dt$ $F(x) = \int_a^b \frac{x}{1 + t^2 + \sin^2 t} dt$ $F(x) = \sin \left( \int_0^x \sin \left( \int_0^y \sin^3 t dt \right) dy \right)$ ...

Inverse Functions Exercise 1 Find $ f^{-1} $ for each function $ f $ $ f(x) = x^3 + 1 $ $ f(x) = (x - 1)^3 $ $ f(x) = \begin{cases} x & \text{rational} \ -x & \text{irrational} \end{cases} $ $ f(x) = \begin{cases} -x^2 & x \geq 0 \ 1 - x^3 & x < 0 \end{cases} $ $ f(x) = \begin{cases} x & x \neq a_i \ a_{i+1} & x = a_i \ (i < n) \ a_1 & x = a_n \end{cases} $ $ f(x) = x + \lfloor x \rfloor $ $ f(0.a_1a_2a_3\ldots) = 0.a_2a_1a_3\ldots $ $ f(x) = \frac{x}{1 - x^2}, \ -1 < x < 1 $ Exercise 2 Describe $ f^{-1} $’s graph when $ f $ is: ...

Riemann Sums Exercise 1 : Approximation of Integral Products Let \(f,g\) be continuous on \([a,b]\). For any partition \(P = \{t_0,\ldots,t_n\}\) of \([a,b]\), choose points \(x_i, u_i \in [t_{i-1},t_i]\). Show that sums of the form \[\sum_{i=1}^n f(x_i)g(u_i)(t_i-t_{i-1})\] can be made arbitrarily close to \(\int_a^b fg\) by choosing sufficiently fine partitions \(P\). Exercise 2 : Approximation of Composite Functions Let \(f,g\) be continuous and nonnegative on \([a,b]\). Show that for sufficiently fine partitions \(P\), sums ...

Convexity and Concavity Exercise 1 Show that a function $ f $ is convex on an interval if and only if for all $ x, y $ in the interval and $ 0 < t < 1 $, we have: $$ f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y). $$Exercise 2 Let $ f $ and $ g $ be convex functions with $ f $ increasing. ...

Differentiation Exercise 1 Find $f^{\prime}(x)$ for each $f$: $f(x) = \sin(x + x^2)$ $f(x) = \sin x + \sin x^2$ $f(x) = \sin(\cos x)$ $f(x) = \sin(\sin x)$ $f(x) = \sin\left(\frac{\cos x}{x}\right)$ $f(x) = \frac{\sin(\cos x)}{x}$ $f(x) = \sin(x + \sin x)$ $f(x) = \sin(\cos(\sin x))$ Exercise 2 Find $f^{\prime}(x)$ for each $f$: $f(x) = \sin((x + 1)^2(x + 2))$ $f(x) = \sin^3(x^2 + \sin x)$ $f(x) = \sin^2((x + \sin x)^2)$ $f(x) = \sin\left(\frac{x^3}{\cos x^3}\right)$ $f(x) = \sin(x \sin x) + \sin(\sin x^2)$ $f(x) = (\cos x)^3$ $f(x) = \sin^2 x \sin x^2 \sin^2 x^2$ $f(x) = \sin^3(\sin^2(\sin x))$ $f(x) = (x + \sin^5 x)^6$ $f(x) = \sin(\sin(\sin(\sin(\sin x))))$ $f(x) = \sin((\sin^7 x^7 + 1)^7)$ $f(x) = (((x^2 + x)^3 + x)^4 + x)^5$ $f(x) = \sin(x^2 + \sin(x^2 + \sin x^2))$ $f(x) = \sin(6 \cos(6 \sin(6 \cos 6x)))$ $f(x) = \frac{\sin x^2 \sin^2 x}{1 + \sin x}$ $f(x) = \frac{1}{x - \frac{2}{x + \sin x}}$ $f(x) = \sin \left( \frac{x^3}{\sin \left( \frac{x^3}{\sin x} \right)} \right)$ $f(x) = \sin \left( \frac{x}{x - \sin \left( \frac{x}{x - \sin x} \right)} \right)$ Exercise 3 Find derivatives of: tan, cot, sec, csc. ...

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 $0 < |x-a| < \delta$ ...