Математический анализ 1 — МИЭФ, 2020 midterm

On the grid, sketch the graph of a function $y=f(x)$ satisfying all the conditions below. Use a solid dot $\bullet$ or an open dot $\circ$ where necessary to indicate whether a point belongs to the graph.

(a) $f$ is defined for every $x$ and $f(-3)=0$.

(b) $\lim_{x\to-\infty} f(x)=0$.

(c) The graph has an asymptote $y=1+x$.

(d) $\lim_{x\to3+}f(x)=5$.

(e) $\lim_{x\to3}f(x)$ exists, but $f$ is not continuous at $x=3$.

(f) $\lim_{x\to0+}f(x)$ exists.

(g) $\lim_{x\to0-}f(x)=+\infty$.

(h) $f$ has exactly three points of discontinuity: one removable, one jump, and one other.

(i) The Intermediate Value Theorem does not apply to $f(x)$ on $[-7,-3]$.

Consider

(a) Find the domain $D$ of $f$.

(b) There are some points $p$ such that $\lim_{x\to p} f(x)$ exists, but $p\notin D$. Find all such points and the corresponding limits.

(c) Find all vertical and slant asymptotes of the graph $y=f(x)$. Justify.

(d) Explain why there is at least one solution of $f(x)=0$ on $(0,1)$.

Consider an infinite sequence $\{a_n\}$ where

(a) Find $L=\lim_{n\to+\infty} a_n$.

(b) Find the smallest index $N$ such that all terms $a_n$ with $n>N$ lie inside $(L-0.01,L+0.01)$.

(c) The sequence $\{b_n\}$ is defined recursively by $b_1=-24$, $b_{n+1}=a_nb_n$. Compute $b_2$ and $b_3$. Is $\{b_n\}$ monotonic?

(d) Is $\{b_n\}$ convergent? If yes, find $\lim_{n\to+\infty} b_n$; otherwise explain why.

At time $t=0$, robots Ace and Bdf start moving from the origin $O$. Every minute each robot makes exactly one step of size 1 either up or right. Their algorithms are:

Here $k\uparrow$ means $k$ steps up and $k\to$ means $k$ steps right. Let $A_n$ and $B_n$ be their positions after $n$ minutes, and let $\alpha_n$ and $\beta_n$ be the slopes of $OA_n$ and $OB_n$.

(a) Sketch the paths of the robots.

(b) Show that $\alpha_n\le\beta_n$ for all $n>0$.

(c) Is $\{\alpha_n\}$ convergent? Justify.

(d) Is $\{\beta_n\}$ convergent? Justify.

Let $\{a_n\}$ and $\{b_n\}$ be two convergent infinite sequences. Which infinite sequences must be convergent?

I. $\{\max(a_n,b_n)\}$.

II. $\{a_n b_{n+2019}-a_{n+2019} b_n\}$.

III. $a_1,b_1,a_2,b_2,\ldots,a_n,b_n,a_{n+1},b_{n+1},\ldots$.

1. I, II and III.
2. I and II only.
3. I and III only.
4. II and III only.

Find $k$ that makes $f$ continuous at $x=-3$:

1. $0.5$.
2. $0.1$.
3. $-0.5$.
4. $-0.1$.

The set of all points $(e^t,t)$, where $t$ is real, is a graph of:

1. $y=e^{1/x}$.
2. $y=\ln x$.
3. $y=\frac1{e^x}$.
4. $y=\frac1{\ln x}$.

Find

1. $\frac{2}{15}$.
2. $\frac15$.
3. $\frac3{20}$.
4. $0$.

Which instruction, applied to the graph $y=\sin x$, produces the shown graph?

1. Shift $\pi$ units right, then compress horizontally by a factor of $\pi$.
2. Compress horizontally by a factor of $\pi$, then reflect in the $x$-axis.
3. Shift $\pi/2$ units right, then compress horizontally by a factor of $\pi$.
4. Compress horizontally by a factor of $\pi$, then shift $1/2$ unit left.

Which choice of $\delta$ is the largest that can be used successfully with arbitrary $\varepsilon$ in an $\varepsilon$–$\delta$ proof of

1. $\delta=\frac{\varepsilon}{4}$.
2. $\delta=\frac{\varepsilon}{2}$.
3. $\delta=\frac{3\varepsilon}{2}$.
4. $\delta=\frac{\varepsilon}{5}$.

Suppose $f$ is defined for all real numbers. Which condition ensures that $f$ has an inverse function?

1. The graph $y=f(x)$ is symmetric with respect to the $y$-axis.
2. $f$ is periodic.
3. $f$ is continuous.
4. $f$ is strictly decreasing.

Find $a$ such that

1. $\ln2$.
2. $0.5$.
3. $2$.
4. $-\ln2$.

Find $f\circ g\circ h$ if

1. $\frac{e^{2x}}{e^{2x}+1}$.
2. $e^{2x/(x+1)}$.
3. $\left(\frac{e^x}{e^x+1}\right)^2$.
4. $\frac{e^{x^2}}{e^{x^2}+1}$.

Find

1. $\frac\pi6$.
2. $\frac\pi3$.
3. $-\frac\pi6$.
4. $-\frac\pi3$.

Suppose $f$ is odd and the graph $y=f(x)$ has asymptote $y=26x-10$ as $x\to+\infty$. Which is the asymptote as $x\to-\infty$?

1. $y=26x-10$.
2. $y=-26x-10$.
3. $y=26x+10$.
4. $y=-26x+10$.

Which sequences are convergent?

1. $\{c_n\}$ only.
2. $\{a_n\}$ and $\{b_n\}$ only.
3. $\{b_n\}$ and $\{c_n\}$ only.
4. $\{a_n\}$ only.

Let $g$ be continuous on $[0,1]$, with $g(0)=1$ and $g(1)=0$. Which statement is not necessarily true?

1. There exists $h\in[0,1]$ such that $g(h)=e^\pi$.
2. There exists $h\in[0,1]$ such that $g(h)=\frac\pi e$.
3. For all $h\in(0,1)$, $\lim_{x\to h}g(x)=g(h)$.
4. There exists $h\in[0,1]$ such that $g(h)\ge g(x)$ for all $x\in[0,1]$.

If $9^x-8^x$ is equivalent to $cx$ as $x\to0$, then $c=$

1. $2\ln3-3\ln2$.
2. $1$.
3. $3\ln2-2\ln3$.
4. $0$.

A continuous positive function $f$ has domain $x>0$. If asymptotes are $x=0$ and $y=2$, which statement must be true?

1. $\lim_{x\to0+}f(x)=2$ and $\lim_{x\to+\infty}f(x)=0$.
2. $\lim_{x\to0+}f(x)=\infty$ and $\lim_{x\to+\infty}f(x)=2$.
3. $\lim_{x\to2}f(x)=\infty$ and $\lim_{x\to+\infty}f(x)=2$.
4. $\lim_{x\to0+}f(x)=\infty$ and $\lim_{x\to2}f(x)=\infty$.

Compute

1. $e^{10}$.
2. $e^5$.
3. $e^{20}$.
4. $1$.

Graphs of $f$ and $g$ are shown. Which statement is false?

1. $\lim_{x\to1}f(x)g(x+1)$ does not exist.
2. $\lim_{x\to1}f(x)=0$.
3. $\lim_{x\to2}g(x)$ does not exist.
4. $\lim_{x\to1}f(x+1)g(x)$ exists.

Which interval belongs to the domain of

1. $\left(\frac{3\pi}{2}+1,\frac{5\pi}{2}+1\right)$.
2. $\left(\frac{5\pi}{2}-1,\frac{7\pi}{2}-1\right)$.
3. $\left(\frac{7\pi}{2},\frac{9\pi}{2}\right)$.
4. $\left(-\frac\pi2-1,\frac\pi2-1\right)$.

The sequences $x_n,y_n,z_n$ satisfy $x_n<y_n<z_n$. Which statements are false?

I. If $x_n$ and $z_n$ are convergent, then $y_n$ is also convergent.

II. If $\lim z_n=\lim x_n=L\in\mathbb R$, then $\lim y_n=L$.

III. If all three sequences converge, then $\lim x_n<\lim y_n<\lim z_n$.

1. I and II only.
2. I and III only.
3. II only.
4. I only.

Which statement is true about the graph of

on $(-1,1)$?

1. It is monotonic.
2. Its range is all positive real numbers.
3. It is continuous.
4. It is bounded.

Function $f$ is defined for all real $x$. The figure shows part of $y=f(x)$, including the only discontinuity at $x=1$. Which function is continuous on $(-\infty,+\infty)$?

1. $|f(x)-1|$.
2. $|f(x-1)-1|$.
3. $|f(|x-1|)-1|$.
4. $f(x-|x|)$.

Find

1. $\frac12$.
2. $1$.
3. $0$.
4. $\frac14$.

Which graph has exactly one horizontal asymptote and no vertical asymptotes?

1. $y=\frac1{1+x^{2020}}$.
2. $y=\frac1{1+x^{2019}}$.
3. $y=\frac1{2020x-1}$.
4. $y=\frac1{2019x+1}$.

Find

1. $0$.
2. $\frac15$.
3. $\infty$.
4. $\frac12$.