Practice (31)

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Compute $$\lim_{x\to 4}\frac{3-\sqrt{x+5}}{x-4}$$

Is the $y=\frac{1}{x}$ a continuous function?

Show that $$\lim_{x\to 0}\ \frac{x}{\sin{x}}=1$$

Show the following sequence is convergent:

$$\frac{1}{1^2},\ \frac{1}{2^2},\ \frac{1}{3^2},\ \cdots,\ \frac{1}{n^2},\ \cdots$$

Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.

Show that $$\lim_{x\to 0}\frac{e^x-1}{x}=1$$

Find the value of 


Compute the derivative of $f(x)=x^n$.

Show that $$\frac{d}{dx} e^x = e^x$$

Given $\frac{d}{dx} e^x = e^x$, find the value of $\frac{d}{dx} \ln x$.

Find the derivative of function $y=\sin{x}$.

Find the derivative of $\arcsin{x}$.

Let $f(x)$ be an odd function which is differentiable over $(-\infty, +\infty)$. Show that $f'(x)$ is even.

Compute the limit of the power series below as a rational function in $x$:

$$1\cdot 2 + (2\cdot 3)x + (3\cdot 4)x^2 + (4\cdot 5)x^3 + (5\cdot 6)x^4+\cdots,\qquad (|x| < 1)$$

Compute $$1-\frac{1\times 2}{2}+\frac{2\times 3}{2^2}-\frac{3\times 4}{2^3}+\frac{4\times 5}{2^4}-\cdots$$

Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:

  • it is an even function
  • $f(2)=f(-2)=0$
  • $f(x) > 0$ when $-2 < x < 2$, and
  • the maximum of $f(x)$ is achieved at $x=\pm 1$.

Find the coordinates of the center of mass of the $\frac{1}{4}$ disc defined by

$$\{(x, y) | x\ge 0, y\ge 0, x^2 + y^2 \le 1\}$$

assuming the density is uniform.

Compute $$I=\int \frac{x\cos{x}-\sin{x}}{x^2 + \sin^2{x}} dx$$

Find the maximum and minimal values of the function

$$f(x)=(x^2-4)^8 -128\sqrt{4-x^2}$$

over its domain.

Find all quadratic polynomials $p(x)=ax^2 + bx + c$ such that graphs of $p(x)$ and $p'(x)$ are tangent to each other at point $(2, 1)$.

Evaluate $$I=\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{1}{\tan\theta +\cot\theta}d\theta$$

Determine if the following infinite series is convergent or divergent:

$$\sum_{n=2}^{\infty}\frac{1}{(\ln n)^{\ln \ln n}}$$

Show that $\ln x < \sqrt{x}$ holds for all positive $x$.

Evaluate $$\int_{0}^{\pi}\frac{x\sin{x}}{1+\cos^2 x}dx$$

Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(x)+f(\frac{1}{x})$.