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

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

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Hot questions in Calculus

7 votes
3 answers
641
TIFR CSE 2010 | Part A | Question: 3
The function $f (x) = 2.5 \log_e \left( 2 + \exp \left( x^2 - 4x + 5 \right)\right)$ attains a minimum at $x = $? $0$ $1$ $2$ $3$ $4$
The function $f (x) = 2.5 \log_e \left( 2 + \exp \left( x^2 - 4x + 5 \right)\right)$ attains a minimum at $x = $?$0$$1$$2$$3$$4$
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3 votes
2 answers
642
TIFR2010-Maths-B-8
The function $f(x)$ defined by $f(x)= \begin{cases} 0 & \text{if x is rational } \\ x & \text{if } x\text{ is irrational } \end{cases}$ is not continuous at any point is continuous at every point is continuous at every rational number is continuous at $x=0$
The function $f(x)$ defined by $$f(x)= \begin{cases} 0 & \text{if x is rational } \\ x & \text{if } x\text{ is irrational } \end{cases}$$is not continuous at any po...
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1 votes
1 answer
643
Limit of Logarithematic and Exponentials
Find the value of the following limit $\lim_{x \to 0} x^{\sin x}$ ... $0 \cdot \infty$ form now? How do I proceed further?
Find the value of the following limit $$\lim_{x \to 0} x^{\sin x}$$My attempt:$$\begin{align*}\text{Let: }\\y &= \lim_{x \to 0} \Bigl [ x^{\sin x} \Bigr ]\\[1em]\text{The...
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2 votes
1 answer
644
TIFR-2014-Maths-A-3
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $\displaystyle \lim_{x \to +\infty} f'(x)=1$, then $f$ is bounded $f$ is increasing $f$ is unbounded $f'$ is bounded
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $\displaystyle \lim_{x \to +\infty} f'(x)=1$, then$f$ is bounded $f$ is increasing $f$ is unboun...
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1 votes
1 answer
645
Which book is best for calculus?
Between Higher engineering mathematics and Gilbert Strang which one is more appropriate for GATE?
Between Higher engineering mathematics and Gilbert Strang which one is more appropriate for GATE?
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8 votes
2 answers
646
TIFR CSE 2013 | Part A | Question: 16
The minimum of the function $f(x) = x \log_{e}(x)$ over the interval $[\frac{1}{2}, \infty )$ is $0$ $-e$ $\frac{-\log_{e}(2)}{2}$ $\frac{-1}{e}$ None of the above
The minimum of the function $f(x) = x \log_{e}(x)$ over the interval $[\frac{1}{2}, \infty )$ is$0$$-e$$\frac{-\log_{e}(2)}{2}$$\frac{-1}{e}$None of the above
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6 votes
2 answers
647
TIFR CSE 2014 | Part A | Question: 16
Let $x_{0}=1$ and $x_{n+1}= \frac{3+2x_{n}}{3+x_{n}}, n\geq 0$. $x_{\infty}=\displaystyle \lim_{n\rightarrow \infty}x_{n}$ is $\left(\sqrt{5}-1\right) / 2$ $\left(\sqrt{5}+1\right) / 2$ $\left(\sqrt{13}-1\right) / 2$ $\left(-\sqrt{13}-1\right) / 2$ None of the above
Let $x_{0}=1$ and$x_{n+1}= \frac{3+2x_{n}}{3+x_{n}}, n\geq 0$.$x_{\infty}=\displaystyle \lim_{n\rightarrow \infty}x_{n}$ is$\left(\sqrt{5}-1\right) / 2$$\left(\sqrt{5}+1\...
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2 votes
3 answers
648
limit
limx->0 (cot x)1/logx ? ans: 0(m getting 1/e)
limx->0 (cot x)1/logx ?ans: 0(m getting 1/e)
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1 votes
2 answers
651
f(x) continuous / differentiable ?
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2 votes
2 answers
652
limits
Find the value of: $\lim_{\theta \to \pi/2} \left ( 1 - 5 \cot\theta \right )^{\tan\theta}$ $e^{5}$ $e^{-5}$ $e^{1/5}$ $e^{-1/5}$
Find the value of: $$\lim_{\theta \to \pi/2} \left ( 1 - 5 \cot\theta \right )^{\tan\theta}$$$e^{5}$ $e^{-5}$ $e^{1/5}$ $e^{-1/5}$
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3 votes
2 answers
653
Limits Indeterminate forms
$\lim_{x \rightarrow 0^{+}} x^{\frac{1}{\ln x}}$
$\lim_{x \rightarrow 0^{+}} x^{\frac{1}{\ln x}}$
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0 votes
4 answers
654
Limits
What is $\lim_{ x \to 0} (1-x)^{\frac{1}{x}}$ ? Also please explain the result?
What is $\lim_{ x \to 0} (1-x)^{\frac{1}{x}}$ ? Also please explain the result?
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0 votes
1 answer
655
funtion
suppose a set A= { x/x ∈ N and X< 9 } and set B ={ x/x+5 = 8 and X∈ N } 1) how many one to one funtions are possible from set B to A 2) how many on to funtions are possible from set A to B
suppose a set A= { x/x ∈ N and X< 9 } and set B ={ x/x+5 = 8 and X∈ N }1) how many one to one funtions are possible from set B to A2) how many on to funtions are po...
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0 votes
2 answers
656
Is the function differentiable at x=1
The question is f(x) = | x-1 | + | x+1 | is differentiable at x=1 or not . Now , when x<1 , the first part becomes : -(x-1) , i.e 1-x and why should we not change the sign of second part ,i.e (x+1) ?
The question is f(x) = | x-1 | + | x+1 | is differentiable at x=1 or not . Now , when x<1 , the first part becomes : -(x-1) , i.e 1-x and why should we not change the sig...
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2 votes
1 answer
657
limit
Value of $\lim_{x \to 0} \frac{x^2 \sin \left(\frac{1}{x}\right)} {\sin x}$ is
Value of $\lim_{x \to 0} \frac{x^2 \sin \left(\frac{1}{x}\right)} {\sin x}$ is
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6 votes
1 answer
658
TIFR CSE 2011 | Part A | Question: 17
What is $\lim_{x \to 0} \frac{2^x-1}{x}$ $0$ $\log_2(e)$ $\log_e(2)$ $1$ None of the above
What is $$\lim_{x \to 0} \frac{2^x-1}{x}$$$0$$\log_2(e)$$\log_e(2)$$1$None of the above
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0 votes
1 answer
660
maxima-minima query
consider the following function f(x) = x3 /3+ x2 /2 -6x+1000 find the intervals on which f is increasing. Is my solution right approach for this kind of numerical?? Caption
consider the following function f(x) = x3 /3+ x2 /2 -6x+1000find the intervals on which f is increasing.Is my solution right approach for this kind of numerical?? Caption...
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