Recent questions tagged calculus

The value of $\sum_{i=0}^{\infty} \sum_{j=1}^{\infty} 2^{-i} 3^{-j}$ is $\_\_\_\_$. (Answer in integer)

For a real number $a$, let $I(a)=\int_{-1}^{1}\left(3 x^{2}-a x+1\right) d x$. Which of the following statements is/are true?The value of $I(a)$ is independent of the val...

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 th...

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...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ 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...

Let \[f(x)=x^3-3x^2+2\]on the interval $(-1,3]$.Which of the following statement(s) is/are TRUE?$f(x)$ has a maximum at $x=0$ only. $f(x)$ has a minimum at $x=2$ only. $f...

Evaluate the following infinite sum:\[\sum_{i=0}^{\infty} \sum_{j=1}^{\infty} \frac{1}{2^i 3^j}.\]

Let a function $f$ be defined as $f:(0,1) \rightarrow\{0,1\}$.For $r \in(0,1), f(r)=1$ if the second digit after decimal is $2,3,6, 7$ and $f(r)=0$ otherwise. Then number...

Let $I(a)=\int_{-1}^1\left(3 x^3-a x+1\right) d x$. Then which of the following is/are correct?The value of $I(a)$ is independent of $a$. The value of $I(a)$ is dependent...

$$f(x)=\begin{cases}C_1 e^{x} - C_2 \ln\!\left(\frac{1}{x}\right), & x>0 \\3, & \text{otherwise}\end{cases}$$Where $C_1, C_2 \in \mathbb{R}$.If $f$ is continuous at $x=0$...

$$f(x)=\left(\frac{|x|}{2}-x\right) \times\left(x-\frac{|x|}{2}\right)$$$f$ has local minimum $f$ has local maximum $f^{\prime}$ is continuous at $x=0$ $f^{\prime}$ is di...