Calculus Practice Set 2

Let $y = y(x)$ be determined by $x^5 - y = \tan(x-y)$, find $y(0)$ and $y'(0)$.

Let function $y=y(x)$ be determined by parametric equations:

Find $\frac{d^2y}{dx^2}\bigg|_{x=\sqrt{2}}$.

Find the limit:

Find the limit:

Find the limit:

Calculate the integral:

Calculate the integral:

Prove: When $0 \leq x < +\infty$, $\arctan 3x \leq \ln(1+4x)$. When is equality achieved (i.e., when $x=0$)?

Find $\sum_{n=0}^{\infty}\frac{(-1)^{2n}}{(2n+1)(2n-2)}x^{2n+2}$, determine its radius of convergence, interval of convergence, and sum function.

Let constant $a > 0$, $f(x) = \frac{1}{3}ax^3 - x$. Determine the maximum and minimum values of $f(x)$ on interval $\left[0, \frac{1}{a}\right]$.

Find the curve $y^2 = x+2$ and the line $l: y = x$ such that the region bounded by the curve and the line equals $x=2$, find the solid of revolution's volume when rotated around the line.

Prove: if the continuous function $f(x)$ satisfies the improper integral's Cauchy convergence criterion, show:

(1) $\lim_{x \to \infty}f(x) = \lim_{x \to \infty}g(x) = 0$;

(2) $f(x), g(x)$ are small order positive infinitesimals of $(x_0)^n$ as $n$ increases, and $g'(x) \neq 0$;

(3) $\lim_{x \to \infty}\frac{f(x)}{g(x)} = A(\infty\infty)$, then $\lim_{x \to \infty}\frac{f(x)}{g(x)} = \lim_{x \to \infty}\frac{f(x)}{g'(x)} = A$.

Please verify that given condition (3) does not hold, then $\lim_{x \to \infty}\frac{f(x)}{g(x)}$ exists, but cannot use L'Hôpital's rule.

Let $f(x) = -\cos\pi x + (2x-3)^3 + \frac{1}{2}(x-1)$. Verify by calculating that $f(x) = 0$ has a unique zero.