GATE CS Calculus: Limits, Continuity, Differentiability questions

Practise Calculus: Limits, Continuity, Differentiability questions for GATE CS. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

20 on this page

SubjectTopicYearType

Consider the function

f: \mathbb{R} \rightarrow \mathbb{R}

defined as follows:

f(x)=\left\{\begin{array}{lr} c_{1} e^{x}-c_{2} \log _{\mathrm{e}}\left(\frac{1}{x}\right), & \text{ if } x>0 \\ 3 & \text{ otherwise } \end{array}\right.

c_{1}, c_{2} \in \mathbb{R}

f

is continuous at

x=0

c_{1}+c_{2}=

f: \mathbb{R} \rightarrow \mathbb{R}

be defined as follows:

f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right)

Which of the following statements is/are true?

Consider a function

f:(0,1) \rightarrow\{0,1\}

defined as follows. For a real number

r \in(0,1), f(r)=1

if the second digit after the decimal point in

r

is one of the four digits

2,3,6

7

f(r)

0

. The number of points in

(0,1)

f

is discontinuous is

\_\_\_\_\_\_

. (answer in integer)

Consider the given function

f(x)

f(x)=\left\{\begin{array}{ll} a x+b & \text{ for } x<1 \\ x^{3}+x^{2}+1 & \text{ for } x \geq 1 \end{array}\right.

If the function is differentiable everywhere, the value of

b

must be ________. (rounded off to one decimal place)

f: \mathbb{R} \rightarrow \mathbb{R}

be a function such that

f(x)=\max \left\{x, x^3\right\}, x \in \mathbb{R}

\mathbb{R}

is the set of all real numbers. The set of all points where

f(x)

is NOT differentiable is

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 following limit is ________________.

\lim_{x \rightarrow 0^{+}} \frac{\sqrt{x}}{1-e^{2\sqrt{x}}}

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 ___________.

f: \mathbb{R} \rightarrow \mathbb{R}

is a continuous function on the interval

[-3, 3]

and a differentiable function in the interval

(-3,3)

such that for every

x

in the interval,

f’(x) \leq 2

f(-3)=7

f(3)

is at most __________

\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}

\displaystyle \lim_{x\rightarrow 1} \frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}

f(x) = R \: \sin ( \frac{\pi x}{2}) + S, f’\left(\frac{1}{2}\right) = \sqrt{2}

\int_0^1 f(x) dx = \frac{2R}{\pi}

, then the constants

R

S

\lim _{x\rightarrow 4}\frac{\sin(x-4)}{x-4}=\_\_\_\_\_\_\_\_\_\_\_\_

f(x)

be a polynomial and

g(x)=f'(x)

be its derivative. If the degree of

(f(x)+f(-x))

10

, then the degree of

(g(x) - g(-x))

\displaystyle \lim_{x\rightarrow \infty } x^{ \tfrac{1}{x}}

f(x)=x^{-\left(\frac{1}{3}\right)}

A

denote the area of region bounded by

f(x)

and the X-axis, when

x

-1

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

\displaystyle \lim_{x \rightarrow \infty} (1+x^2)^{e^{-x}}

f(x)

is continuous in the interval

[0,2]

. It is known that

f(0) = f(2) = -1

f(1) = 1

. Which one of the following statements must be true?

f(x) =x \sin x

satisfies the following equation:

f''(x) + f(x) +t \cos x = 0

t

Let the function

f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) & \\ \sin(\frac{\pi}{3}) & \cos(\frac{\pi}{3}) & \tan(\frac{\pi}{3}) \end{vmatrix}

\theta \in \left[ \frac{\pi}{6},\frac{\pi}{3} \right]

f'(\theta)