Recent questions and answers in Calculus

1
$\lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}$ equals $1$ $-1$ $\infty$ $-\infty$
2
The function $y=|2 - 3x|$​ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{3}{2}$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{2}{3}$ is continuous $∀ x ∈ R$ except $x=3$ and differentiable $∀ x ∈ R$
3
The value of the integral $\displaystyle{}\int_{-1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $– \frac{1}{2}$ $1$
4
What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$ $-1$ $0$ $1$ $\pi$
5
The map $f(x) = a_0 \cos \mid x \mid +a_1 \sin \mid x \mid +a_2 \mid x \mid ^3$ is differentiable at $x=0$ if and only if $a_1=0$ and $a_2=0$ $a_0=0$ and $a_1=0$ $a_1=0$ $a_0, a_1, a_2$ can take any real value
6
A function $f(x)$ is continuous in the interval $[0,2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true? There exists a $y$ in the interval $(0,1)$ such that $f(y) = f(y+1)$ For every $y$ ... maximum value of the function in the interval $(0,2)$ is $1$ There exists a $y$ in the interval $(0,1)$ such that $f(y)$ = $-f(2-y)$
7
Solve min $x^{2}+y^{2}$ subject to \begin {align*} x + y &\geq 10,\\ 2x + 3y &\geq 20,\\ x &\geq 4,\\ y &\geq 4. \end{align*} $32$ $50$ $52$ $100$ None of the above
8
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is $\alpha$ $[\alpha]$ $1$ $\dfrac{[\alpha] + [\alpha +1]}{2}$
9
The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$
10
A solution for the differential equation $x’(t) + 2x(t) = \delta(t)$ with initial condition $x(\overline{0}) = 0$ $e^{-2t}u(t)$ $e^{2t}u(t)$ $e^{-t}u(t)$ $e^{t}u(t)$
11
The function $f(x)=x^{5}-5x^{4}+5x^{3}-1$ has one minima and two maxima two minima and one maxima two minima and two maxima one minima and one maxima
12
$\underset{x \rightarrow 0}{\lim} \dfrac{x^{3}+x^{2}-5x-2}{2x^{3}-7x^{2}+4x+4}=?$ $-0.5$ $(0.5)$ $\infty$ None of the above
13
$\displaystyle \int_{0}^{\dfrac{\pi}{2}} \sin^{7}\theta \cos ^{4} \theta d\theta=?$ $16/1155$ $16/385$ $16\pi/385$ $8\pi/385$
14
$\displaystyle \lim_{x \rightarrow a}\frac{1}{x^{2}-a^{2}} \displaystyle \int_{a}^{x}\sin (t^{2})dt=$? $2a \sin (a^{2})$ $2a$ $\sin (a^{2})$ None of the above
15
$\displaystyle \lim_{x \rightarrow 0}\frac{1}{x^{6}} \displaystyle \int_{0}^{x^{2}}\frac{t^{2}dt}{t^{6}+1}=$? $1/4$ $1/3$ $1/2$ $1$
16
A ladder $13$ feet long rests against the side of a house. The bottom of the ladder slides away from the house at a rate of $0.5$ ft/s. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is $5$ feet from the house? $\dfrac{5}{24} \text{ ft/s} \\$ $\dfrac{5}{12} \text{ ft/s} \\$ $-\dfrac{5}{24} \text {ft/s} \\$ $-\dfrac{5}{12} \text{ ft/s}$
17
What is the maximum value of the function $f(x) = 2x^{2} – 2x + 6$ in the interval $[0,2]?$ $6$ $10$ $12$ $5,5$
18
The value of the Integral $I = \displaystyle{}\int_{0}^{\pi/2} x^{2}\sin x dx$ is $(x+2)/2$ $2/(\pi-2)$ $\pi – 2$ $\pi + 2$
19
The function $f\left ( x \right )=\dfrac{x^{2}-1}{x-1}$ at $x=1$ is : Continuous and differentiable Continuous but not differentiable Differentiable but not continuous Neither continuous nor differentiable
20
The greatest and the least value of $f(x)=x^4-8x^3+22x^2-24x+1$ in $[0,2]$ are $0,8$ $0,-8$ $1,8$ $1,-8$
21
The value of improper integral $\displaystyle\int_{0}^{1} x\ln x =?$ $1/4$ $0$ $-1/4$ $1$
22
Maxima and minimum of the function $f(x)=2x^3-15x^2+36x+10$ occur; respectively at $x=3$ and $x=2$ $x=1$ and $x=3$ $x=2$ and $x=3$ $x=3$ and $x=4$
23
What is the derivative w.r.t $x$ of the function given by $\large \Phi(x)= \displaystyle \int_{0}^{x^2}\sqrt t\:dt$, $2x^2$ $\sqrt x$ $0$ $1$
24
$\underset{x\to 0}{\lim} \dfrac{(1-\cos x)}{2}$ is equal to $0$ $1$ $1/3$ $1/2$
25
The minimum value of $\mid x^2-5x+2\mid$ is $-5$ $0$ $-1$ $-2$
26
Differential equation, $\dfrac{d^2x}{dt^2}+10\dfrac{dx}{dt}+25x=0$ will have a solution of the form $(C_1+C_2t)e^{-5t}$ $C_1e^{-2t}$ $C_1e^{-5t}+C_2e^{5t}$ $C_1e^{-5t}+C_2e^{2t}$ where $C_1$ and $C_2$ are constants.
27
Consider the function $f(x)=\sin(x)$ in the interval $\bigg [​\dfrac{ \pi}{4},\dfrac{7\pi}{4}\bigg ]$. The number and location(s) of the minima of this function are: One, at $\dfrac{\pi}{2} \\$ One, at $\dfrac{3\pi}{2} \\$ Two, at $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2} \\$ Two, at $\dfrac{\pi}{4}$ and $\dfrac{3\pi}{2}$
28
What is the area of the largest rectangle that can be inscribed in a circle of radius $R$? $R^{2}/2$ $\pi \times R^{2}/2$ $R^{2}$ $2R^{2}$ None of the above
29
Consider a function $f:[0,1]\rightarrow [0,1]$ which is twice differentiable in $(0,1).$ Suppose it has exactly one global maximum and exactly one global minimum inside $(0,1)$. What can you say about the behaviour of the first derivative $f'$ and and second derivative $f''$ ... $f'$ is zero at at least two points, $f''$ is zero at at least two points
30
Let $a_n=\bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1 – \frac{1}{\sqrt{n+1}} \bigg), \: n \geq 1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
31
$\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$
32
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
33
If $f(x)$ is a real valued function such that $2f(x)+3f(-x)=15-4x$, for every $x \in \mathbb{R}$, then $f(2)$ is $-15$ $22$ $11$ $0$
34
If $f(x) = \dfrac{\sqrt{3} \sin x}{2+\cos x}$, then the range of $f(x)$ is the interval $[-1 , \sqrt{3}{/2}]$ the interval $[-\sqrt{3}{/2}, 1]$ the interval $[-1, 1]$ none of these
35
The integral $\int _0^{\frac{\pi}{2}} \frac{\sin^{50} x}{\sin^{50}x +\cos^{50}x} dx$ equals $\frac{3 \pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{4}$ none of these
Let the function $f(x)$ be defined as $f(x)=\mid x-1 \mid + \mid x-2 \:\mid$. Then which of the following statements is true? $f(x)$ is differentiable at $x=1$ $f(x)$ is differentiable at $x=2$ $f(x)$ is differentiable at $x=1$ but not at $x=2$ none of the above
$\underset{x \to 2}{\lim} \dfrac{1}{1+e^{\frac{1}{x-2}}}$ is $0$ $1/2$ $1$ non-existent
It is given that $e^a+e^b=10$ where $a$ and $b$ are real. Then the maximum value of $(e^a+e^b+e^{a+b}+1)$ is $36$ $\infty$ $25$ $21$