Calculus

Differentiate $f(x) = \frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2}$ with respect to $x$.

Find $\frac{dy}{dx}$ for $y = 11x^3 + 1$.

What is the derivative of $f(x) = 2x^4 - 4x^3 + 6x^2$?

If $y = \frac{1}{2}x^2 - 5$, find $\frac{dy}{dx}$.

For which range of values is the function $f(x) = 2x^3 - 3x^2 - 12x + 5$ increasing?

Find the value of $y$ at the stationary point of the curve $y = x^2 - 4x + 7$.

Find the equation of the tangent to the curve $y = x^2 - x$ at the point $(2, 2)$.

The tangent to $y = x^2$ at $x = 3$ and the tangent to $y = x^2$ at $x = -1$ intersect at point $Q$. Find the $y$-coordinate of $Q$.

The gradient of $y = x^3 + px + q$ at $(2, 1)$ is $7$. Find $q$.

A curve has equation $y = kx^2 - x^4$. Find the value of $k > 0$ such that the maximum value of $y$ is $4$.

What is the gradient of the tangent to $y = 4x - \frac{1}{2}x^2$ at $x = 2$?

Find the $x$-coordinate on the curve $y = x^3 - 3x^2$ where the tangent is horizontal (excluding $x=0$).

What is the gradient of the tangent to $y = x^2 - x$ at the origin?

Find the gradient of the normal to $y = x^4$ at $x = 1$.

A curve has the equation $y = \frac{k}{x}$. The normal to the curve at $x = 2$ is $y = 2x - 3$. Find $k$.

Find the equation of the normal to the curve $y = 4 - e^{2x}$ at the point where it crosses the $y$-axis.

Find the equation of the normal to the curve $y = 2x^3 - 3x^2$ at the point where $x = 1$.

The normal to the curve $y = 2\sqrt{x}$ at $x = 1$ intersects the $y$-axis at which point?

Find the equation of the tangent to $y = 1 + \ln(x-1)$ at the point where $x = 2$.

The normal to the curve $y = x^2 + 2x$ at a certain point has the equation $y = -0.25x + c$. Find the value of $c$.

Find the $x$-coordinate of the stationary point for the function $f(x) = x^2 - 30x$.

Find the $x$-coordinate of the stationary point for $f(x) = 12x - x^2$.

Find the $x$-coordinate of the stationary point for $f(x) = \frac{1}{2}x^2 - 4x$.

Find the $x$-coordinate of the stationary point for the function $f(x) = 3x^2 - 18x$.

Find the $x$-coordinate of the stationary point of $y = ax^2 + 6x - 4$ given that it is a maximum point at $x = 3$. Find $a$.

If $y = kx^2 - 4x + 5$ has a stationary point at $x = 2$, find the value of the constant $k$.

Find the coordinates of the local minimum point of the curve $y = x^3 - 27x$.

Find the coordinates of the local maximum of $y = \frac{x^2}{x^3 + 16}$ for $x > 0$.

The displacement of a particle is given by $s(t) = t^2 e^{-t}$. At what time $t > 0$ is the displacement maximum?

Find the minimum value of $y = x^2 - 8\ln x$ for $x > 0$.

Find the integral: $\int (x^2/4) \, dx$.

Find $\int_{2}^{4} 3x^2 \, dx$.

Evaluate $\int (2x + 3) \, dx$.

Calculate the area under $y = 6x$ from $x=0$ to $x=2$.

Evaluate the definite integral $\int_1^3 (x - 2)^3 \, dx$.

Integrate $\int \frac{(x+1)^2}{x^4} \, dx$.

Find the area under the curve $y = x^2 + 1$ from $x = -1$ to $x = 1$.

Find the area bounded by $y = x^2 - 2x$ and $y = x$.

The area under $y = kx^2$ from $x=0$ to $x=3$ is 18. Find $k$.

Find the area of the region bounded by the curve $y = e^x$, $x=0$, $x=1$ and the $x$-axis.

The displacement of a particle is $s = t^2 + 4t$. Find the average velocity of the particle between $t = 0$ and $t = 2$ seconds.

An object decelerates at a constant rate $a = -2$ m/s². If its initial velocity is $10$ m/s, how long does it take to stop?

A particle's velocity is $v = 4t + 3$. If the initial displacement is $0$, what is the displacement at $t = 2$ seconds?

The displacement of a particle is $s = 5t^3$. Find the velocity at $t = 1$ second.

A particle's displacement is $s = \frac{1}{3}t^3 - 4t$. Find the total distance traveled by the particle in the first 3 seconds.

The velocity of a particle is $v = 6t^2 - 2t^3$. Find the maximum acceleration reached by the particle for $t \ge 0$.

The acceleration of a particle is $a = \cos(2t)$. Find the change in velocity from $t = 0$ to $t = \pi/4$.

The displacement $s$ of a particle is given by $s = \sqrt{t^2 + 1}$. Find the acceleration $a$ in terms of $s$.

A particle's acceleration is $a(t) = 2t - 5$. If $v(2) = 0$, find the initial velocity $v(0)$.

A particle moves with acceleration $a(t) = \frac{-1}{(t+1)^2}$. If $v(0) = 1$, find the velocity as $t \to \infty$.