List of top Mathematics Questions on Calculus

Show that the function $ f(x) = 7x^2 - 3 $ is an increasing function when $ x>0 $.

If $y=(\tan^{-1} x)^2$, show that $(x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1)\frac{dy}{dx} = 2$.

Let $E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$ are constructed such that their centers and eccentricities are the same as that of $E_1$, and the length of the minor axis of $E_{i+1}$ is the length of the major axis of $E_i$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi} \sum_{i=1}^{\infty} A_i$ is equal to:

Prove that $\int_0^{\pi/2} \log(\cos x) \, dx = -\frac{\pi}{2} \log 2$.

A spherical ball has a variable diameter $\frac{5}{2}(3x + 1)$. The rate of change of its volume w.r.t. $x$, when $x = 1$, is:

Prove that $ \int_{0}^{\pi/2} \sqrt{\frac{1+\cos 4x}{2}} \, dx = 1 $.

The coefficient of $ (y-2) $ in the Taylor's series expansion of $ f(x, y) = x^2 + xy + y^2 $ in powers of $ (x-1) $ and $ (y-2) $ is:

Find: \[ J = \int \frac{\sqrt{x^2 + 1} \left[ \log(x^2 + 1) - 2 \log x \right]}{x^2} \, dx \]

If the equation of the parabola with vertex $\left(\frac{3}{2},\, 3\right)$ and the directrix $x + 2y = 0$ is
$$ ax^2 + by^2 - cxy - 30x - 60y + 225 = 0, $$ then $\alpha + \beta + \gamma$ is equal to:

What is the value of $ \int_0^{\pi/2} \sin x \cos x \, dx$?

If the system of equations \[ x + 2y - 3z = 2, \quad 2x + \lambda y + 5z = 5, \quad 14x + 3y + \mu z = 33 \] has infinitely many solutions, then $ \lambda + \mu $ is equal to:}

Prove that $\int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx = 2$.

At $ t = 2 $, the slope of the vector function $ \vec{f}(t) = 2\hat{i} + 3\hat{j} + 5t^2\hat{k} $ is

If function $ f $ is defined as \[ f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \] then prove that $ f $ is continuous.

The vector function is given by $\vec{f}(t) = t\hat{i} + t^2\hat{j} + 5\hat{k}$, then at point $t = 1$ the slope is

If $\int x \log x \, dx = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c$, then $ f(x) $ is

Prove that $\int_{0}^{\pi/4} \log(1 + \tan x) \, dx = \frac{\pi}{8} \log 2$.

Show that the given function $ f, f(x) = x^3 - 3x^2 + 4x, x \in \mathbb{R} $ is an increasing function in $ \mathbb{R} $.

Find the value of the determinant $ \begin{vmatrix} x+y+2z & x & y \\ z & y+z+2x & y \\ z & x & z+x+2y \end{vmatrix} $.

Evaluate: $ \int \frac{dx}{(x+1)(x+2)} $.

Find the value of the determinant $ \begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{vmatrix} $.

Prove that $\tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x$, where $-\frac{1}{\sqrt{2}} \leq x \leq 1$.

Test whether the function $f: \mathbb{R} \to \mathbb{R}$ defined by \[ f(x) = \begin{cases} x+5, & \text{if } x \le 1 \\ x-5, & \text{if } x > 1 \end{cases} \] is continuous at $x=1$.

If $ I = \int_{0}^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} $, find the value of I.

A car is started to move from a point P at time t = 0 and is stopped at the point Q. The distance x metre covered by the car in t second is given by $x = t^2\left(3-\frac{t}{2}\right)$. Find the time required by the car to reach at the point Q and also find the distance between P and Q.