Differential Calculus Exercise 1 : Sequences and Limits of Sequences Calculate the limiting value of the following sequences for $n \to \infty$. $a_n = \frac{\sqrt{n}}{n}$ $a_n = \frac{5 + n}{2n}$ $a_n = \left(-\frac{1}{4}\right)^n - 1$ $a_n = \frac{2}{n} + 1$ $a_n = \frac{n^3 + 1}{2n^3 + n^2 + n}$ $a_n = 2 + 2^{-n}$ $a_n = \frac{n^2 - 1}{(n + 1)^2} + 5$ Exercise 2 : Limits of Functions Calculate the following limits: ...
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. ...