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:

The function $f(x) = \pi x + e$ is:

The function $f(x) = 10 - (x-2)^2$ for $x > 2$ is:

The function $f(x) = (x+1)^3$ is:

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 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: