List of top Mathematics Questions on Calculus

Show that the function \( f(x) = x^3 + 10x + 7 \), \( x \in \mathbb{R} \), is strictly increasing.
Evaluate: \( \int_0^{\pi/2} (1 - \cos 4x) \, dx \).
Find the area of the region bounded by the curve \( y^2 = 4x \), the X-axis, and the lines \( x = 1 \), \( x = 4 \) for \( y \geq 0 \).
Write \( \int \cot x \, dx \).
\( \int 3 \cos^3 x \, dx = \).
If y is a function of x and \( \log (x + y) = 2xy \), then the value of \( y'(0) = \).
The equation of the tangent at any point of the curve \[ x = a \cos 2t, \, y = 2 \sqrt{2a} \sin t \, \text{with} \, m \, \text{as its slope is} \]
Between any two real roots of the equation \( e^x \sin x = 1 \), the equation \( e^x \cos x = -1 \) has:

If \[ \int x \sin x \sec^3 x \, dx = \frac{1}{2} \left[ f(x) \sec^2 x + g(x) \left( \frac{\tan x}{x} \right) \right] + c, \] \(\text{then which of the following is true?}\)
 

Which of the following numbers is the coefficient of \( x^{100} \) in the expansion of \[ \log_e \left( \frac{1 + x}{1 + x^2} \right), \text{where} |x| < 1? \]
Number of points at which \( f(x) \) is not differentiable, where \[ f(x) = |\cos x| + 3 \text{in} [-\pi, \pi] \]

If \( f(x) = \lim_{x \to 0} \frac{6^x - 3^x - 2^x + 1}{\log_e 9 (1 - \cos x)} \) \(\text{ is a real number, then }\) \( \lim_{x \to 0} f(x) = \)

The value of \[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{x \sin x}{\cos^2 x} \, dx \text{ is:} \]

Let \[ f(x) = \frac{x^2 - 1}{|x| - 1}. \] \(\text{Then the value of}\) \[ \lim_{x \to 1} f(x) \text{ is:} \]

If \[ \int f(x) \, dx = g(x), \text{ then } \int x^5 f(x^3) \, dx \] is
If \[ \lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \lim_{x \to k} \frac{x^3 - k^2}{x^2 - k^2}, \text{ then find } k \]

The maximum value of \( f(x) = (x - 1)^2 (x + 1)^3 \) is equal to \[ \frac{2^p 3^q}{3125} \,\, \text{then the ordered pair of} (p, q) \text{ will be} \]

Suppose that the temperature at a point (x, y) on a metal plate is \( T(x, y) = 4x^2 - 4xy + y^2 \). An ant, walking on the plate, traverses a circle of radius 5 centered at the origin. What is the highest temperature encountered by the ant?

If \( x^m y^n = (x+y)^{m+n} \), then \( \frac{dy}{dx} \) is
 

A particle is at rest at the origin. It moves along the x -axis with an acceleration \( x - x^2 \), where x is the distance of the particle at time t. The particle next comes to rest after it has covered a distance

The function \( f(x) = \begin{cases} (1+2x)^{1/x}, & x \neq 0 \\ e^2, & x=0 \end{cases} \) is
 

If \( a < b \), then \( \int_a^b (|x-a| + |x-b|) dx \) is equal to
Which of the following is NOT true?
\( f(x) = x + |x| \) is continuous for

The value of \( \int \frac{(x^2-1)dx}{x^3\sqrt{2x^4 - 2x^2 + 1}} \) is