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Recent questions and answers in Calculus

16 votes
6 answers
1
$\lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}$ equals $1$ $-1$ $\infty$ $-\infty$
answered 5 days ago in Calculus Surya_Dev Chaturvedi 4.4k views
18 votes
4 answers
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 $∀ x ∈ R$
answered Dec 4, 2020 in Calculus StoneHeart 2.6k views
0 votes
1 answer
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}$ $1$
answered Nov 14, 2020 in Calculus raja11sep 204 views
12 votes
4 answers
4
What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$ $-1$ $0$ $1$ $\pi$
answered Oct 27, 2020 in Calculus jainanmol123 2.9k views
0 votes
1 answer
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
answered Oct 15, 2020 in Calculus sparta 114 views
53 votes
7 answers
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 $f(y)$ = $-f(2-y)$
answered Oct 9, 2020 in Calculus prajjwalsingh_11 9.9k views
9 votes
6 answers
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
answered Oct 2, 2020 in Calculus ankitgupta.1729 836 views
3 votes
1 answer
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 $\alpha$ $[\alpha]$ $1$ $\dfrac{[\alpha] + [\alpha +1]}{2}$
answered Sep 9, 2020 in Calculus neeraj_bhatt 144 views
0 votes
0 answers
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 $1$ $0$ $+\infty$ $1/2$
asked Aug 30, 2020 in Calculus soujanyareddy13 183 views
0 votes
0 answers
10
A solution for the differential equation $x’(t) + 2x(t) = \delta(t)$ with initial condition $x(\overline{0}) = 0$ $e^{-2t}u(t)$ $e^{2t}u(t)$ $e^{-t}u(t)$ $e^{t}u(t)$
asked Aug 28, 2020 in Calculus Lakshman Patel RJIT 68 views
2 votes
2 answers
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
asked Apr 2, 2020 in Calculus Lakshman Patel RJIT 120 views
0 votes
1 answer
12
0 votes
1 answer
13
0 votes
0 answers
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
asked Apr 2, 2020 in Calculus Lakshman Patel RJIT 45 views
0 votes
1 answer
15
$\displaystyle \lim_{x \rightarrow 0}\frac{1}{x^{6}} \displaystyle \int_{0}^{x^{2}}\frac{t^{2}dt}{t^{6}+1}=$? $1/4$ $1/3$ $1/2$ $1$
asked Apr 2, 2020 in Calculus Lakshman Patel RJIT 68 views
0 votes
1 answer
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? $\dfrac{5}{24} \text{ ft/s} \\$ $\dfrac{5}{12} \text{ ft/s} \\$ $-\dfrac{5}{24} \text {ft/s} \\$ $-\dfrac{5}{12} \text{ ft/s}$
asked Apr 2, 2020 in Calculus Lakshman Patel RJIT 72 views
0 votes
1 answer
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
asked Mar 31, 2020 in Calculus Lakshman Patel RJIT 235 views
0 votes
1 answer
20
0 votes
1 answer
22
0 votes
1 answer
23
0 votes
1 answer
24
0 votes
0 answers
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.
asked Mar 31, 2020 in Calculus Lakshman Patel RJIT 69 views
0 votes
1 answer
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 $\dfrac{3\pi}{2}$
asked Mar 31, 2020 in Calculus Lakshman Patel RJIT 126 views
0 votes
1 answer
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
asked Feb 11, 2020 in Calculus Lakshman Patel RJIT 90 views
0 votes
1 answer
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
asked Feb 10, 2020 in Calculus Lakshman Patel RJIT 178 views
2 votes
1 answer
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 $0$
asked Sep 23, 2019 in Calculus Arjun 278 views
4 votes
4 answers
31
$\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$
asked Sep 23, 2019 in Calculus Arjun 427 views
3 votes
2 answers
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 $\infty$ $0$ $\log_e 2$ $1$
asked Sep 23, 2019 in Calculus Arjun 304 views
2 votes
2 answers
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 $-15$ $22$ $11$ $0$
asked Sep 23, 2019 in Calculus Arjun 201 views
2 votes
3 answers
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
asked Sep 23, 2019 in Calculus Arjun 147 views
2 votes
1 answer
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
asked Sep 23, 2019 in Calculus Arjun 220 views
2 votes
1 answer
36
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
asked Sep 23, 2019 in Calculus Arjun 170 views
2 votes
2 answers
37
$\underset{x \to 2}{\lim} \dfrac{1}{1+e^{\frac{1}{x-2}}}$ is $0$ $1/2$ $1$ non-existent
asked Sep 23, 2019 in Calculus Arjun 163 views
3 votes
1 answer
38
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 $36$ $\infty$ $25$ $21$
asked Sep 23, 2019 in Calculus Arjun 139 views
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