Practice (15)

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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.

Consider the ellipse $x^2+\frac{y^2}{4}=1$. What is the area of the smallest diamond shape with two vertices on the $x$-axis and two vertices on the $y$-axis that contains this ellipse?

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


$$I= \iiint \limits_S \frac{dx dy dz}{(1+x+y+z)^2}$$

where $S=\{x\ge 0, y\ge 0, z\ge 0, x+y+z\le 1\}$.

Which one of the numbers below is larger?

$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$

Show that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by

$$f(x,y)=x^4+6x^2y^2 + y^4 -\frac{9}{4}x-\frac{7}{4}$$

achieves its minimal value, and determine all the points $(x, y)\in\mathbb{R}^2$ at which it is achieved.

For $n=1, 2,\dots$, let $x_n=\displaystyle\sum_{k=n+1}^{9n}\frac{k}{9n^2 + k^2}$. Find the value of $\displaystyle\lim_{n\to\infty}x_n$.

Let $s\in\mathbb{R}$. Prove that

$$\sum_{n\ge 1}(n^{\frac{1}{n^s}}-1)$$

converges if and only if $s > 1$.

Determine whether the following series converge? $$\sum_{n=1}^{\infty}\left(1-\cos{\frac{\pi}{n}}\right)$$

Note that the point $(2, 1)$ is always on the curve $x^4 + ky^4 = 16+k$ regardless of the value of $k$. If for a particular non-zero value of $k$, $y'(2)=y''(2)$ along this curve. Find this $k$.