Математический анализ 1 — МИЭФ, 2023 midterm

Let $f(x)=e^{\sqrt{2x^2-1}}$.

(a) Is $f$ even, odd, or neither? Justify.

(b) Find the domain of $f$. Justify.

(c) Find the range of $f$. Justify.

(d) Find all points where $f$ is continuous. Justify.

(e) Find $\lim_{h\to0}\frac{\ln(f(1+h))-\ln(f(1))}{h}$.

Let $f(x)=\frac{2x-2}{x^2+x-2}$.

(a) For what $x$ is $f$ discontinuous?

(b) At each discontinuity $a$, find $\lim_{x\to a}f(x)$ or justify non-existence.

(c) Find all slant or vertical asymptotes. Justify.

(d) A rational function $g(x)=\frac{b}{c+x}$ satisfies $f(x)=g(x)$ for all $x\in D[f]$. Find $b,c$.

Consider $a_n=\frac{2^n}{n!}$.

(a) Classify $\{a_n\}$ as increasing, nondecreasing, nonincreasing, decreasing, or neither. Justify.

(b) Is $\{a_n\}$ bounded? Justify.

(c) What conclusion follows from (a) and (b)? Name all theorems used.

(d) If $x_n\to L$, find $R=\lim_{n\to\infty}\frac{x_{n+1}}{x_n}$.

(e) Calculate $A=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ for $a_n=2^n/n!$. Are $A$ and $R$ consistent?

Let $f$ be bounded and continuous on $[0,\infty)$ and $f(x)\ne0$ for all $x\ge0$. Which statements must be true?

(a) There exists $c\in[0,\infty)$ such that $f(x)\le f(c)$ for all $x\ge0$.

(b) $f$ does not change sign on $[0,\infty)$.

(c) $\lim_{x\to+\infty}f(x)=L$ for some finite $L$.

If a statement is always true, explain why and mention relevant theorems. If false, provide a counterexample.

The graph of $y=f(x)$ is shown below. Which of the alternatives a–e represents the graph of

$$y=f(|x|)?$$

Нужно проверить: "Coordinates are approximate because the source graph provides only tick marks, not an explicit formula."

1. a
2. b
3. c
4. d
5. e

Let

$$f(x)=2x, \qquad g(x)=\frac{4}{x-1}.$$

The complete set of solutions of

$$f(g(x))=g(f(x))$$

is:

1. $x_1=\frac13$.
2. $x_1=2$.
3. $x_1=3$.
4. $x_1=-1,\ x_2=2$.
5. $x_1=\frac13,\ x_2=2$.

Let

$$f(x)=x^5-1.$$

Then the inverse function $f^{-1}$ is:

1. $$f^{-1}(x)=\frac{1}{\sqrt[5]{x}+1}.$$
2. $$f^{-1}(x)=\frac{1}{\sqrt[5]{x+1}}.$$
3. $$f^{-1}(x)=\sqrt[5]{x-1}.$$
4. $$f^{-1}(x)=\sqrt[5]{x}-1.$$
5. $$f^{-1}(x)=\sqrt[5]{x+1}.$$

Find the principal period of

$$f(x)=3-2\cos^2\left(\frac{\pi x}{3}\right).$$

1. $1$.
2. $2$.
3. $3$.
4. $5$.
5. $6$.

Let

$$f(x)=\cos(\arctan x).$$

What is the range of $f$?

1. $(0,1)$.
2. $(0,1]$.
3. $[0,1]$.
4. $(-1,1)$.
5. $[-1,1]$.

Let

$$f(x)=\left|\sin x-\frac12\right|.$$

The maximum value attained by $f$ is:

1. at most $\frac12$;
2. $1$;
3. $\frac32$;
4. $\frac34$;
5. greater than $\frac32$.

Match the five graphs with the five sequences.

Нужно проверить: "The plotted values are qualitative; exact numerical coordinates are not supplied."

Choose the correct matching:

1. $$v_n=\frac{2a}{n+1},\quad w_n=\frac{an}{n+1},\quad x_n=\frac{a}{2^{n-1}},\quad y_n=\frac{(a-2)^n}{n!},\quad z_n=\frac{a-2}{\sqrt[n]{n}}.$$
2. $$v_n=\frac{a}{2^{n-1}},\quad w_n=\frac{(a-2)^n}{n!},\quad x_n=\frac{2a}{n+1},\quad y_n=\frac{an}{n+1},\quad z_n=\frac{a-2}{\sqrt[n]{n}}.$$
3. $$v_n=\frac{2a}{n+1},\quad w_n=\frac{an}{n+1},\quad x_n=\frac{a-2}{\sqrt[n]{n}},\quad y_n=\frac{(a-2)^n}{n!},\quad z_n=\frac{a}{2^{n-1}}.$$
4. $$v_n=\frac{(a-2)^n}{n!},\quad w_n=\frac{a-2}{\sqrt[n]{n}},\quad x_n=\frac{a}{2^{n-1}},\quad y_n=\frac{2a}{n+1},\quad z_n=\frac{an}{n+1}.$$
5. $$v_n=\frac{2a}{n+1},\quad w_n=\frac{an}{n+1},\quad x_n=\frac{(a-2)^n}{n!},\quad y_n=\frac{a}{2^{n-1}},\quad z_n=\frac{a-2}{\sqrt[n]{n}}.$$

If

$$a_1=2, \qquad a_n=a_{n-1}+\frac13$$

for every integer $n>1$, then $a_{101}$ equals:

1. $35-\frac23$.
2. $35-\frac13$.
3. $35$.
4. $35+\frac13$.
5. $35+\frac23$.

Which of the following sequences converge?

$$\text{I. }a_n=\frac{5n}{2n-1}, \qquad \text{II. }c_n=\frac{e^n}{1+e^n}, \qquad \text{III. }b_n=\frac{e^n}{n}.$$

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

Which statements are true?

I. All bounded sequences are convergent.

II. If a sequence is unbounded, then it is infinitely large.

III. An unbounded sequence may have two limits.

1. I only.
2. II only.
3. II and III only.
4. I and III only.
5. None of the statements is true.

Find

$$\lim_{n\to\infty} \left( \frac{n^4+2n^2+4}{n^4-2n^2+4} \right)^{n^2}.$$

1. $1$.
2. $e$.
3. $e^2$.
4. $e^4$.
5. $+\infty$.

Let sequences $\{x_n\}$, $\{y_n\}$, and $\{z_n\}$ be such that both difference sequences

$$\{x_n-y_n\} \qquad\text{and}\qquad \{y_n-z_n\}$$

are infinitesimally small.

Which of the following sequences must be convergent?

$$\text{I. }\{x_n-z_n\}, \qquad \text{II. }\{x_n+z_n\}, \qquad \text{III. }\left\{\frac{x_nz_n}{y_n^2}\right\}.$$

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

The graph of a function $f$ is shown below.

For which values of $c$ does

$$\lim_{x\to c}f(x)=1?$$

Нужно проверить: "The exact height of the horizontal ray for x>3 is not labelled; only that it is below 1 matters."

1. $0$ only.
2. $0$ and $3$ only.
3. $-2$ and $0$ only.
4. $-2$ and $3$ only.
5. $-2$, $0$, and $3$.

Find

$$\lim_{x\to0}\frac{e^{2x}-1}{\tan x}.$$

1. The limit does not exist.
2. $-1$.
3. $0$.
4. $1$.
5. $2$.

Among the following choices of $\delta$, which is the largest that can be used successfully for arbitrary $\varepsilon>0$ in an $\varepsilon$–$\delta$ proof of

$$\lim_{x\to2}(1-3x)=-5?$$

1. $\delta=3\varepsilon$.
2. $\delta=\varepsilon$.
3. $\delta=\frac12\varepsilon$.
4. $\delta=\frac14\varepsilon$.
5. $\delta=\frac15\varepsilon$.

Let

$$f(x)=\sin\left(\frac{x+1}{x^2}\right).$$

Which statements are true?

I. The graph of $f$ has a horizontal asymptote $y=0$.

II. The graph of $f$ has a horizontal asymptote $y=1$.

III. The graph of $f$ has a vertical asymptote at $x=0$.

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

Find the asymptote of

$$y=\frac{x^3}{\sqrt{x^4+x^2+1}}$$

as $x\to-\infty$.

1. $y=1$.
2. $y=x$.
3. $y=-x$.
4. $y=x+1$.
5. $y=-x+1$.

Let

$$f(x)=\frac{x^3}{x^2-4}+|x|.$$

The graph of $y=f(x)$ has:

1. two vertical asymptotes, one horizontal asymptote, and one slant asymptote with nonzero slope;
2. two vertical asymptotes and two slant asymptotes with slopes of different signs;
3. one vertical asymptote and two slant asymptotes with positive slopes;
4. one vertical asymptote and three slant asymptotes with positive slopes;
5. three different asymptotes.

Find

$$\lim_{x\to+\infty} \frac{\ln\left(1+\sqrt{x}+\sqrt[3]{x}\right)} {\ln\left(1+\sqrt[3]{x}+\sqrt[4]{x}\right)}.$$

1. $1$.
2. $0$.
3. $\frac23$.
4. $\frac34$.
5. $\frac32$.

Let $f$ be continuous at $x=2$, and

$$f(x)= \begin{cases} \dfrac{\sqrt{2x+5}-\sqrt{x+7}}{x-2}, & x\ne2,\\[8pt] k, & x=2. \end{cases}$$

Then:

1. $k=0$.
2. $k=\frac16$.
3. $k=\frac13$.
4. $k=1$.
5. $k=\frac75$.

Let $f$ be a continuous function defined by

$$f(x)=\sqrt{\tan^2x-1}.$$

Which interval could be the domain of $f$?

1. $\left(\frac{3\pi}{4},\pi\right)$.
2. $\left(\frac{\pi}{4},\frac{\pi}{2}\right)$.
3. $\left(\frac{\pi}{4},\frac{3\pi}{4}\right)$.
4. $\left(-\frac{\pi}{4},0\right)$.
5. $\left(-\frac{3\pi}{4},-\frac{\pi}{4}\right)$.

Let $f$ be continuous on the closed interval $[-3,6]$. If

$$f(-3)=-1, \qquad f(6)=3,$$

then the Intermediate Value Theorem guarantees that:

1. $f(0)=0$;
2. $f(x)\leq f(c)$ for some $c\in[-3,6]$;
3. $-1\leq f(x)\leq3$ for every $x$ between $-3$ and $6$;
4. $f(c)=0$ for at least one $c$ between $-1$ and $3$;
5. $f(c)=1$ for at least one $c$ between $-3$ and $6$.

According to the Intermediate Value Theorem, which statements are true?

I. The equation

$$x^{20}+x^{23}-2022=0$$

has at least one root in $[-10,10]$.

II. The function

$$f(x)=\log_2x$$

takes the value $11$ in $[2022,2202]$.

III. The function

$$f(x)=\frac{1}{x^2-6x+8}$$

takes the value $0$ in $[1,3]$.

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

Let $a,b,c,d\in\mathbb R$. The polynomial equation

$$x^7+ax^5+bx^3+cx+d=0$$

must have:

1. only one root;
2. at least one root;
3. an even number of roots;
4. no negative roots;
5. no positive roots.

At how many points do the graphs

$$y=x^{12} \qquad\text{and}\qquad y=2^x$$

intersect?

1. None.
2. One.
3. Two.
4. Three.
5. Four.

Let $k$ be the number of real solutions of

$$e^x+x-2=0$$

in $[0,1]$, and let $n$ be the number of real solutions outside $[0,1]$.

Which statement is true?

1. $k=0$ and $n=1$.
2. $k=1$ and $n=0$.
3. $k=n=1$.
4. $k>1$.
5. $n>1$.

Let

$$f(x)= \begin{cases} \dfrac{\sqrt{x^2}-x}{x}, & x\ne0,\\[8pt] 0, & x=0. \end{cases}$$

Which statement is true?

1. $f$ is continuous for every $x$.
2. $f$ has a removable discontinuity at $x=0$.
3. $f$ has a jump discontinuity at $x=0$, and $$\lim_{x\to0^+}f(x)=f(0).$$
4. $f$ has a jump discontinuity at $x=0$, and $$\lim_{x\to0^-}f(x)=f(0).$$
5. $f$ has a Type II discontinuity at $x=0$.

Consider

$$f(x)= \begin{cases} 2^{1/x}, & x<0,\\ 1, & 0\leq x\leq1,\\ \ln(x^2+x-2), & x>1. \end{cases}$$

The function $f$ has:

1. only one jump discontinuity;
2. two jump discontinuities;
3. one Type II discontinuity only;
4. one jump discontinuity and one Type II discontinuity;
5. one jump discontinuity and one removable discontinuity.

Consider

$$f(x)=\arctan\left( \frac1x+\frac1{x-1}+\frac1{x-2} \right).$$

Let $D$ be the number of Type II discontinuities of $f$, and let $A$ be the number of distinct asymptotes of $f$.

Find $D+A$.

1. $1$.
2. $3$.
3. $6$.
4. $2$.
5. $8$.

A function $f$ has a Type II discontinuity at $x=a$.

Which of the following statements may be true?

I. The left-hand limit

$$\lim_{x\to a^-}f(x)$$

exists.

II. $f(a)$ exists.

III. $f$ is bounded in some neighbourhood of $a$.

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