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Syllabus: Limits, Continuity, and Differentiability, Maxima and minima, Mean value theorem, Integration.

$$\small{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count}&1&1&0&1&1&1&0&0.8&1
\\\hline\textbf{2 Marks Count}&0&0&1&0&0&0&0&0.2&1
\\\hline\textbf{Total Marks}&1&1&2&1&1&1&\bf{1}&\bf{1.2}&\bf{2}\\\hline
\end{array}}}$$

Previous GATE Questions in Calculus

8 votes
6 answers
1
Compute $\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}$ $1$ $53/12$ $108/7$ Limit does not exist
asked Feb 7, 2019 in Calculus Arjun 3.1k views
19 votes
6 answers
2
The value of $\int^{\pi/4} _0 x \cos(x^2) dx$ correct to three decimal places (assuming that $\pi = 3.14$) is ____
asked Feb 14, 2018 in Calculus gatecse 7.4k views
2 votes
0 answers
3
Assuming $i=\sqrt{-1}$ and t is real number , $\int_{0}^{\Pi /3} e^{it}dt$ given ans is $\sqrt{3}/2 + i/2 , and i am getting \sqrt{3}/2-i/2$ i solved first put $e^{it}= cost +isint$ then and integrate , but in solution they first integrate $e^{it}$ then put the value of it ... but ans different ... why is this happening .. am i missing something ??
asked Aug 29, 2017 in Calculus sid1221 249 views
19 votes
5 answers
4
The value of $\lim_{x\rightarrow 1} \frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}$ is $0$ is $-1$ is $1$ does not exist
asked Feb 14, 2017 in Calculus Arjun 3.3k views
17 votes
2 answers
5
If $f(x) = R \: \sin ( \frac{\pi x}{2}) + S, f’\left(\frac{1}{2}\right) = \sqrt{2}$ and $\int_0^1 f(x) dx = \frac{2R}{\pi}$, then the constants $R$ and $S$ are $\frac{2}{\pi}$ and $\frac{16}{\pi}$ $\frac{2}{\pi}$ and 0 $\frac{4}{\pi}$ and 0 $\frac{4}{\pi}$ and $\frac{16}{\pi}$
asked Feb 14, 2017 in Calculus Madhav 3.9k views
1 vote
0 answers
6
asked Jan 18, 2017 in Calculus ManojK 255 views
7 votes
1 answer
7
If $f(x_{i}).f(x_{i+1})< 0$ then There must be a root of $f(x)$ between $x_i$ and $x_{i+1}$ There need not be a root of $f(x)$ between $x_{i}$ and $x_{i+1}$. There fourth derivative of $f(x)$ with respect to $x$ vanishes at $x_{i}$. The fourth derivative of $f(x)$ with respect to $x$ vanishes at $x_{i+1}$.
asked Nov 9, 2016 in Calculus makhdoom ghaya 1k views
18 votes
4 answers
8
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 $∀ x ∈ R$
asked Mar 20, 2016 in Calculus Anuraag Nayak 2.5k views
20 votes
4 answers
9
$\lim _{x\rightarrow 4}\frac{\sin(x-4)}{x-4}$=____.
asked Feb 12, 2016 in Calculus Sandeep Singh 3.6k views
24 votes
5 answers
10
Let $f(x)$ be a polynomial and $g(x)=f'(x)$ be its derivative. If the degree of $(f(x)+f(-x))$ is $10$, then the degree of $(g(x) - g(-x))$ is __________.
asked Feb 12, 2016 in Calculus Akash Kanase 5.2k views
26 votes
1 answer
11
If for non-zero $x, \: af(x) + bf(\frac{1}{x}) = \frac{1}{x} - 25$ where a $a \neq b \text{ then } \int_1^2 f(x)dx$ is $\frac{1}{a^2 - b^2} \begin{bmatrix} a(\ln 2 - 25) + \frac{47b}{2} \end{bmatrix}$ ... $\frac{1}{a^2 - b^2} \begin{bmatrix} a(\ln 2 - 25) - \frac{47b}{2} \end{bmatrix}$
asked Feb 16, 2015 in Calculus jothee 3.8k views
23 votes
5 answers
12
The value of $\lim_{x \rightarrow \infty} (1+x^2)^{e^{-x}}$ is $0$ $\frac{1}{2}$ $1$ $\infty$
asked Feb 14, 2015 in Calculus jothee 6.1k views
16 votes
2 answers
13
Compute the value of: $\large \int_{\frac{1}{\pi}}^{\frac{2}{\pi}}\frac{\cos(1/x)}{x^{2}}dx$
asked Feb 13, 2015 in Calculus makhdoom ghaya 3.6k views
35 votes
7 answers
14
Let $f(x)=x^{-\left(\frac{1}{3}\right)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from $-1$ to $1$. Which of the following statements is/are TRUE? $f$ is continuous in $[-1, 1]$ $f$ is not bounded in $[-1, 1]$ $A$ is nonzero and finite II only III only II and III only I, II and III
asked Feb 12, 2015 in Calculus jothee 7.7k views
23 votes
4 answers
15
Consider a function $f(x) = 1- |x| \text{ on } -1 \leq x \leq 1$. The value of $x$ at which the function attains a maximum, and the maximum value of the function are: $0, -1$ $-1, 0$ $0, 1$ $-1, 2$
asked Feb 12, 2015 in Calculus jothee 3k views
16 votes
5 answers
16
$\lim_{x\rightarrow \infty } x^{ \tfrac{1}{x}}$ is $\infty $ 0 1 Not defined
asked Feb 11, 2015 in Calculus makhdoom ghaya 4k views
12 votes
4 answers
17
What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$ $-1$ $0$ $1$ $\pi$
asked Nov 3, 2014 in Calculus Ishrat Jahan 2.8k views
16 votes
3 answers
18
If $f(x)$ is defined as follows, what is the minimum value of $f(x)$ for $x \in (0, 2]$ ? $f(x) = \begin{cases} \frac{25}{8x} \text{ when } x \leq \frac{3}{2} \\ x+ \frac{1}{x} \text { otherwise}\end{cases}$ $2$ $2 \frac{1}{12}$ $2\frac{1}{6}$ $2\frac{1}{2}$
asked Oct 29, 2014 in Calculus Ishrat Jahan 3.6k views
18 votes
2 answers
19
Let $f$ be a function defined by $f(x) = \begin{cases} x^2 &\text{ for }x \leq 1\\ ax^2+bx+c &\text{ for } 1 < x \leq 2 \\ x+d &\text{ for } x>2 \end{cases}$ Find the values for the constants $a$, $b$, $c$ and $d$ so that $f$ is continuous and differentiable everywhere on the real line.
asked Oct 9, 2014 in Calculus Kathleen 2.2k views
15 votes
3 answers
20
The formula used to compute an approximation for the second derivative of a function $f$ at a point $X_0$ is $\dfrac{f(x_0 +h) + f(x_0 – h)}{2}$ $\dfrac{f(x_0 +h) - f(x_0 – h)}{2h}$ $\dfrac{f(x_0 +h) + 2f(x_0) + f(x_0 – h)}{h^2}$ $\dfrac{f(x_0 +h) - 2f(x_0) + f(x_0 – h)}{h^2}$
asked Oct 9, 2014 in Calculus Kathleen 3.1k views
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