Recent questions tagged calculus

The value of $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^2+x}\right)$ is equal to$-1$ $-0.5$  $-2$ $0$

Evaluate the following limit:$$\lim _{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{2(2 x-3)}{x^3-3 x^2+2 x}\right]$$$0$$1$$-\frac{1}{2}$$\frac{1}{2}$

Evaluate the integral:$$\int_0^{\frac{\pi}{2}} \frac{\tan ^7 x}{\cot ^7 x+\tan ^7 x} d x$$$\frac{\pi}{4}$ $\frac{\pi}{2}$ $\frac{1}{2}$ $\frac{\pi}{8}$

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by$$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2}, & x<0 \\ \alpha, & x=0 \\ \frac{\beta \sqrt{1-\cos x}}{x}, &...

The maximum value of the function $h(x)=-x^3+2 x^2$ in the interval $[-1,1.5]$ is equal to $\qquad$______ (rounded off to 1 decimal place)

Consider the function given below and pick one or more CORRECT statement(s) from the following choices.$$f(x)=x^3-\frac{15}{2} x^2+18 x+20$$ $f(x)$ has a local minimum at...

Let $x$ be a variable that takes real values, and let $f(x)=x^{3}-3 x$. Which of the following statements is/are true?$f(x)$ has a local maximum at $x=-\sqrt{3}$$f(x)$ ha...

Let $f(x)=\sqrt{x}$. We draw a tangent to the curve $y=f(x)$ at the point on the curve whose $x$ coordinate is equal to $4$. Where does this tangent intersect the $X$-axi...

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

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which is twice differentiable everywhere and suppose that $f^{\prime}(q)=0$ for every rational number $q.$ Find t...