Calculus

Evaluate $\lim_{x \to 0} (1 + x)^2$.

Find the limit $\lim_{x \to 1} \ln(x)$.

Evaluate $\lim_{x \to \infty} \frac{2x + 5}{3x - 1}$.

Calculate $\lim_{x \to 2} \frac{x^2 - 4}{x + 2}$.

What is $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$?

Calculate $\lim_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}$.

Determine $\lim_{x \to 0} \frac{e^{3x} - 1}{2x}$.

A theoretical model for energy dissipation $E$ uses the limit $E = \lim_{x \to 0} \frac{8^x - 2^x}{x}$. Calculate the exact value of this energy constant.

The tension in a cable is described by the limit $\lim_{x \to 1} \frac{x + x^2 + x^3 - 3}{x - 1}$. Find the magnitude of this tension at the critical point $x=1$.

A student is analyzing the behavior of the function $f(x) = \frac{\log(x+h) - \log(x)}{h}$ as $h \to 0$. This represents the derivative of the common logarithm. What is the limit of this expression?

Find the rate of change of the area of a circle with respect to its diameter $D$. (Area $A = \frac{\pi D^2}{4}$)

The power $P$ in a circuit with constant voltage $V = 10$ is $P = 10I$, where $I$ is the current. Find the rate of change of power with respect to current.

Find the rate of change of the area of a circle with respect to its circumference when the radius is $5\text{ units}$.

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is $4\text{ units}$.

The diagonal of a square is increasing at a rate of $5\text{ cm/s}$. Find the rate of increase of the area of the square when the side length is $10\text{ cm}$.

A point moves along the curve $y = 2x - x^2$. If the x-coordinate increases at $3\text{ units/s}$, find the rate of change of the y-coordinate at $x = 2$.

In a conical vessel where the height is twice the radius, water is poured at a rate of $3\pi\text{ cm}^3\text{/s}$. Find the rate at which the water level is rising when the height of the water is $2\text{ cm}$.

The volume of a sphere is increasing at $3 \text{ cm}^3\text{/s}$. The rate of increase of its surface area when the radius is $2 \text{ cm}$ is:

The distance $s$ moved by a particle in time $t$ is $s = t^3 - 6t^2 + 12t + 8$. The velocity of the particle when the acceleration is zero is:

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is $\tan^{-1}(0.5)$. Water is poured into it at a constant rate of $5 \text{ m}^3\text{/h}$. The rate at which the level of the water is rising at the instant when the depth of water in the tank is $4 \text{ m}$ is:

The graph of the function $f(x) = x^2 - 4x$ is shown below. Find the value of the derivative $f'(x)$ at the point where the curve intersects the positive $x$-axis.

Find the slope of the tangent to $f(x) = x^2 - 4$ at the x-intercept $x = 2$.

Differentiate $y = \sqrt{x}$ (expressed as $x^{1/2}$).

What is the derivative of $f(x) = \tan(x)$?

Calculate the derivative of $f(x) = \sin(x)\cos(x)$.

The graph of $f(x) = x^3 - 3x$ has two stationary points. What is the value of the derivative $f'(x)$ at $x=1$?

Find $\frac{dy}{dx}$ if $y = \frac{x^2 - 1}{x^2 + 1}$.

A rectangle is inscribed in a semicircle of radius $R=2$ with its base on the diameter. If the area is $A(x) = 2x\sqrt{4-x^2}$, find the value of $x$ (half the width) that maximizes the area by finding where $A'(x)=0$.

Find the derivative of $f(x) = \frac{\sin x + \cos x}{\sin x - \cos x}$ with respect to $x$ at $x = 0$.

The pressure $P$ and volume $V$ of a gas satisfy $PV = c$. If the volume is increasing at a rate proportional to the volume itself, $\frac{dV}{dt} = kV$, find the rate of change of pressure $\frac{dP}{dt}$.