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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{2020}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} & 1 &1&1&1&1&1&0&1&1&1&0&0.9&1
\\\hline\textbf{2 Marks Count} & 0 &0&0&0&0&0&1&0&0&0&0&0.1&1
\\\hline\textbf{Total Marks} & 1 &1&1&1&1&1&2&1&1&1&\bf{1}&\bf{1.1}&\bf{2}\\\hline
\end{array}}}$$

Most answered questions in Calculus

2 votes

2 answers

GATE CSE 2024 | Set 1 | Question: 1
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x)=\max \left\{x, x^3\right\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of all points where $f(x)$ is NOT differentiable is $\{-1,1,2\}$ $\{-2,-1,1\}$ $\{0,1\}$ $\{-1,0,1\}$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x)=\max \left\{x, x^3\right\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers....

Arjun

3.5k views

Arjun asked Feb 16

1 votes

2 answers

Memory Based GATE DA 2024 | Question: 3
Evaluate the limit: \[ \lim_{{x \to 0}} \frac{\ln \left(\left(x^2+1\right) \cos x\right)}{x^2} \]

Evaluate the limit:\[\lim_{{x \to 0}} \frac{\ln \left(\left(x^2+1\right) \cos x\right)}{x^2}\]

GO Classes

511 views

GO Classes asked Feb 4

7 votes

2 answers

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 11
Let $f(x)$ be a real-valued function all of whose derivatives exist. Recall that a point $x_0$ in the domain is called an inflection point of $f(x)$ if the second derivative $f^{\prime \prime}(x)$ changes sign at ... only inflection point. $x_0=0$ and $x_0=6$, both are inflection points. The function does not have an inflection point.

Let $f(x)$ be a real-valued function all of whose derivatives exist. Recall that a point $x_0$ in the domain is called an inflection point of $f(x)$ if the second derivat...

GO Classes

861 views

GO Classes asked Jan 28

3 votes

2 answers

GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 37
Which of the following is/are TRUE? There is a differentiable function $f(x)$ with the property that $f(1)=-2$ and $f(5)=14$ and $f^{\prime}(x)\lt 3$ for every real number $x$. There exists a function $f$ ... $a\lt c\lt b$ and $f(c)=0$. If $f$ is differentiable at the number $x$, then it is continuous at $x$.

Which of the following is/are TRUE?There is a differentiable function $f(x)$ with the property that $f(1)=-2$ and $f(5)=14$ and $f^{\prime}(x)\lt 3$ for every real number...

GO Classes

670 views

GO Classes asked Jan 21

1 votes

2 answers

Evaluate the limit without using L' Hospital Rule.

Debargha Mitra Roy

285 views

Debargha Mitra Roy asked Nov 26, 2023

3 votes

2 answers

TIFR CSE 2023 | Part A | Question: 6
For a function $f: \mathbb{R} \rightarrow \mathbb{R},$ consider the following conditions. $\text{(C1)}$ $|f(x)| \leq|x|$ for all $x \in \mathbb{R}$. $\text{(C2)}$ $|f(x)| \leq|x|^{2}$ for all $x \in \mathbb{R}$. $\text{(C3)}$ ... $\text{(C3)}$ only Conditions $\text{(C1)}$ and $\text{(C2)}$ only Conditions $\text{(C2)}$ and $\text{(C3)}$ only

For a function $f: \mathbb{R} \rightarrow \mathbb{R},$ consider the following conditions.$\text{(C1)}$ $|f(x)| \leq|x|$ for all $x \in \mathbb{R}$.$\text{(C2)}$ $|f(x)| \...

admin

467 views

admin asked Mar 14, 2023

8 votes

2 answers

GATE CSE 2023 | Question: 18
Let $\qquad f(x)=x^{3}+15 x^{2}-33 x-36$ be a real-valued function. Which of the following statements is/are $\text{TRUE}?$ $f(x)$ does not have a local maximum. $f(x)$ has a local maximum. $f(x)$ does not have a local minimum. $f(x)$ has a local minimum.

Let $$\qquad f(x)=x^{3}+15 x^{2}-33 x-36$$be a real-valued function.Which of the following statements is/are $\text{TRUE}?$$f(x)$ does not have a local maximum.$f(x)$ has...

admin

5.3k views

admin asked Feb 15, 2023

5 votes

2 answers

GATE CSE 2023 | Question: 21
The value of the definite integral \[ \int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1}\left(4 x^{2} y-z^{3}\right) \mathrm{d} z \mathrm{~d} y \mathrm{~d} x \] is _________. (Rounded off to the nearest integer)

The value of the definite integral \[\int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1}\left(4 x^{2} y-z^{3}\right) \mathrm{d} z \mathrm{~d} y \mathrm{~d} x\]is _________. (Rounde...

admin

8.9k views

admin asked Feb 15, 2023

0 votes

2 answers

GATE CSE 2023 | Memory Based Question: 11
Let $f(x)=x^3+15 x^2-33 x-36$ be a real valued function. Which statement is/are TRUE? $f(x)$ has a local maximum. $f(x)$ does NOT have a local maximum. $f(x)$ has a local minimum. $f(x)$ does NOT have a local minimum.

Let $f(x)=x^3+15 x^2-33 x-36$ be a real valued function. Which statement is/are TRUE?$f(x)$ has a local maximum.$f(x)$ does NOT have a local maximum.$f(x)$ has a local mi...

GO Classes

4.0k views

GO Classes asked Feb 5, 2023

0 votes

2 answers

limits
the value of $lim_{x-> 0}(1-sinx.cosx)^{cosec2x}$ is: A $e^{2}$ B $e^{-2}$ C$e^{1/2}$ D $e^{-1/2}$

the value of $lim_{x- 0}(1-sinx.cosx)^{cosec2x}$ is:A $e^{2}$B $e^{-2}$C$e^{1/2}$D $e^{-1/2}$

someshawasthi

393 views

someshawasthi asked Jan 16, 2023

0 votes

2 answers

TIFR CSE 2022 | Part A | Question: 6
Let $f$ be a polynomial of degree $n \geq 3$ all of whose roots are non-positive real numbers. Suppose that $f(1)=1$. What is the maximum possible value of $f^{\prime}(1)?$ $1$ $n$ $n+1$ $\frac{n(n+1)}{2}$ $f^{\prime}(1)$ can be arbitrarily large given only the constraints in the question

Let $f$ be a polynomial of degree $n \geq 3$ all of whose roots are non-positive real numbers. Suppose that $f(1)=1$. What is the maximum possible value of $f^{\prime}(1)...

admin

715 views

admin asked Sep 1, 2022

1 votes

2 answers

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 1
Evaluate the limit $ \lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x} $ $1$ $\frac{1}{2}$ $2$ $0$

Evaluate the limit$$\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x}$$$1$$\frac{1}{2}$$2$$0$

GO Classes

697 views

GO Classes asked Aug 28, 2022

8 votes

2 answers

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 5
Which of the following functions satisfy the conditions of Rolle's Theorem on the interval $[-1,1]?$ $ \begin{aligned} &f(x)=1-x^{2 / 3}\\ &g(x)=x^{3}-2 x^{2}-x+2\\ &h(x)=\cos \left(\frac{\pi}{4}(x+1)\right) \end{aligned} $ Rolle's Theorem applies to: both $f$ and $g$ both $g$ and $h$ $g$ only $h$ only

Which of the following functions satisfy the conditions of Rolle's Theorem on the interval $[-1,1]?$ $$\begin{aligned}&f(x)=1-x^{2 / 3}\\&g(x)=x^{3}-2 x^{2}-x+2\\&h(x)=\c...

GO Classes

1.0k views

GO Classes asked Aug 28, 2022

7 votes

2 answers

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 8
Choose the CORRECT statement - The function $f(x)=\exp \left(-x^{2}\right)-1$ has the root $x=0$. If a function $f$ is differentiable on $[-1,1]$, then there is a point $x$ in that interval where $f^{\prime}(x)=0$. If $1$ is a ... at $1 .$ If $f^{\prime \prime}(0)0$ then there is a point in $(0,1)$, where $f$ has an inflection point.

Choose the CORRECT statement -The function $f(x)=\exp \left(-x^{2}\right)-1$ has the root $x=0$.If a function $f$ is differentiable on $[-1,1]$, then there is a point $x$...

GO Classes

966 views

GO Classes asked Aug 28, 2022

2 votes

2 answers

GO Classes Test Series 2023 | Calculus | Test | Question: 14
Which of the following limit is/are correct? $\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=1$ $\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=e$ $\displaystyle{}\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}=e^{2}$ $\displaystyle{}\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}=e$

Which of the following limit is/are correct?$\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=1$$\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=e$$\displa...

GO Classes

494 views

GO Classes asked Aug 28, 2022

3 votes

2 answers

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 17
If $f(x)=e^{x} g(x), g(0)=2$ and $g^{\prime}(0)=1$, then $f^{\prime}(0)$ is $1$ $3$ $2$ $0$

If $f(x)=e^{x} g(x), g(0)=2$ and $g^{\prime}(0)=1$, then $f^{\prime}(0)$ is$1$$3$$2$$0$

GO Classes

392 views

GO Classes asked Aug 28, 2022

1 votes

2 answers

Applied Mathematics Calculus and Limit Question
Evaluate the question of the following limits. $\lim_{x\rightarrow 1} \frac{x}{(x-1)^{2}}$

Evaluate the question of the following limits. $\lim_{x\rightarrow 1} \frac{x}{(x-1)^{2}}$

kidussss

871 views

kidussss asked Jul 8, 2022

0 votes

2 answers

Applied Mathematics Calculus and Limit Question
Evaluate the question of the following limits. $\lim_{x\rightarrow \infty} \frac{2x^{3}+3x-5}{5x^{3}+1}$

Evaluate the question of the following limits. $\lim_{x\rightarrow \infty} \frac{2x^{3}+3x-5}{5x^{3}+1}$

kidussss

469 views

kidussss asked Jul 8, 2022

0 votes

2 answers

Calculus self doubt
Calculus looks difficult for me. Any resources to learn calculus for GATE

Calculus looks difficult for me. Any resources to learn calculus for GATE

kabilan45

534 views

kabilan45 asked Nov 25, 2021

4 votes

2 answers

100

GATE CSE 2021 Set 1 | Question: 20
Consider the following expression.$\displaystyle \lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}$The value of the above expression (rounded to 2 decimal places) is ___________.

Consider the following expression.$$\displaystyle \lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}$$The value of the above expression (rounded to 2 decimal places) is ____...

Arjun

6.4k views

Arjun asked Feb 18, 2021

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