Application of Derivatives

The rate of change of the area of an equilateral triangle with respect to its side length $a$ when $a = 4$ cm is:

If the cost function is $C(x) = 2x^2 + 5$, the marginal cost when $x = 10$ is:

The rate of change of the volume of a sphere with respect to its radius $r$ when $r = 6$ cm is:

A particle moves such that its displacement $s$ at time $t$ is given by $s = t^3 + 2t^2 + t$. Its velocity at $t = 1$ is:

A point moves on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$. If $\frac{dx}{dt} = 2$ at the point $(0, 3)$, then $\frac{dy}{dt}$ is:

A particle moves along the curve $y = x^3 - 3x^2 + 5$. If the $x$-coordinate is increasing at $2$ units/s, find the rate of change of the $y$-coordinate when $x = 1$.

A cylinder has a constant radius of $2$ cm. If the height increases at a rate of $3$ cm/s, what is the rate of change of its volume?

The mass of a spherical raindrop is $M = \frac{4}{3}\pi r^3 \rho$, where $\rho$ is the density. If the radius $r$ increases at $0.1 \text{ cm/s}$ and the density $\rho$ decreases at $0.2 \text{ g/cm}^3/\text{s}$, find the rate of change of mass when $r = 1 \text{ cm}$ and $\rho = 5 \text{ g/cm}^3$.

Two roads intersect at an angle of $60^{\circ}$. Car A is $10 \text{ km}$ from the intersection and moving towards it at $40 \text{ km/h}$. Car B is $20 \text{ km}$ from the intersection and moving away from it at $50 \text{ km/h}$. Find the rate of change of the distance between them.

Two cars start from the same point. Car A travels East at $40 \text{ km/h}$ and Car B travels North at $30 \text{ km/h}$. After $2$ hours, the distance between them is increasing at:

The function $f(x) = x^2 - 4x + 6$ is strictly increasing in which of the following intervals?

At what value of $x$ does the function $f(x) = x^2 - x + 1$ stop decreasing and start increasing?

Find the interval where $f(x) = 7 - 4x - x^2$ is strictly increasing.

Is the function $f(x) = 3x + 17$ strictly increasing on $\mathbb{R}$?

Consider the function $f(x) = \sin(x) + \cos(x)$ on the interval $[0, 2\pi]$. On which of these sub-intervals is the function strictly decreasing?

Find the values of $x$ for which $f(x) = \frac{x}{x^2+1}$ is strictly decreasing.

Determine the interval where $f(x) = x(x-2)^2$ is increasing.

A function $f(x) = x + \cos(x)$ is defined on $[0, 2\pi]$. Determine the nature of this function over the given interval.

An entrepreneur estimates that the revenue $R$ from selling $x$ units is $R(x) = 400x - \frac{x^3}{3}$. Determine the interval of $x$ where the revenue is strictly increasing.

A particle moves along a line such that its position is $s(t) = 2t^3 - 9t^2 + 12t + 5$. Find the interval of time $t$ where the particle's velocity is strictly decreasing.

The slope of the tangent to the curve $y = \log(\cos x)$ at $x = 0$ is:

Find the slope of the tangent to the curve $x = \theta + \sin \theta, y = 1 - \cos \theta$ at $\theta = \frac{\pi}{2}$.

Find the slope of the tangent to the curve $y = \sin^{-1} x$ at $x = 0$.

The slope of the tangent to the line $y = 7x - 3$ at any point on it is:

The sum of the intercepts on the coordinate axes made by any tangent to the curve $\sqrt{x} + \sqrt{y} = 3$ is:

The value of $k$ for which the line $y = x + k$ is a tangent to the parabola $y^2 = 8(x + 2)$ is:

The equation of the normal to the curve $y = \tan x$ at the origin is:

The points on the curve $y = 2x^3 - 15x^2 + 36x - 21$ where the tangent is parallel to the x-axis are:

The equation of the tangent to the curve $y = x^2 - x + 1$ at the point where it crosses the y-axis is:

If the tangent to the curve $y = x^3 - 3x + 2$ at a point is parallel to the chord joining $(1, 0)$ and $(2, 4)$, then the x-coordinate of the point is:

The maximum value of $f(x) = \cos^2 x$ is:

The function $f(x) = x^3 - 27x + 5$ has a local maximum at:

If the first derivative $f'(x)$ changes sign from negative to positive at $x = c$, then $c$ is a point of:

The minimum value of $f(x) = 2x + \frac{2}{x}$ for $x > 0$ is:

An open box with a square base is to be made out of a given quantity of cardboard of area 48 sq. units. What is the maximum volume of the box?

The maximum value of the function $f(x) = \cos x + \sin^2 x$ is:

What is the minimum value of the function $f(x) = x^x$ for $x > 0$?

The maximum volume of a box with a square base and open top, made from a given quantity of material of area $S$, is:

The minimum value of the function $f(x) = e^x + e^{-x}$ is:

The local maximum value of $f(x) = 2x^3 - 3x^2 - 12x + 4$ is:

Find the absolute minimum of $f(x) = x^2$ on the interval $[-2, 3]$.

Find the absolute minimum value of $f(x) = \ln(x)$ on the interval $[1, e]$.

Identify the absolute minimum value of $f(x) = 1/x$ on the closed interval $[1, 5]$.

Find the absolute minimum value of $f(x) = x^2 - 4x + 6$ on $[0, 5]$.

A particle moves along a path such that its position at time $t$ is given by $s(t) = \sin(t) + \cos(t)$ for $t \in [0, \pi]$. Determine the absolute minimum value of the displacement function $s(t)$ in this interval.

Find the absolute maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x + 1$ on the closed interval $[1, 5]$. The graph of a similar cubic function is shown below for reference.

Determine the absolute maximum value of $f(x) = 2\cos(x) + x$ on $[0, \pi]$.

A power line is to be connected from a power plant $P$ to a factory $F$ across a river. The river is $3$ km wide. The factory is located $8$ km downstream on the opposite bank. The cost of laying the cable under water is Rs $500,000$ per km and on land is Rs $300,000$ per km. If $x$ is the distance from point $A$ (directly opposite to $P$) where the cable leaves the river, find the minimum total cost $C(x)$ for $x \in [0, 8]$.

A rectangle is inscribed in a semi-circle of radius $R = 4$ units such that two of its vertices lie on the diameter and the other two lie on the semi-circular arc. Using the closed interval $[0, 4]$ for the distance $x$ of a vertex from the center, find the absolute maximum area of the rectangle $A(x)$.

For a given curved path $y = x^2$ on the interval $x \in [-1, 2]$, find the point on the curve that is at the absolute maximum distance from the point $(0, 5)$.