Mock Test 1 — Application of Derivatives

A point moves along the parabola $y^2 = 8x$. At what coordinate position are the rates of change of its abscissa and ordinate identically equal?

The equation of the normal to the curve $y = \sin x$ at the origin $(0, 0)$ is:

The function $f(x) = \frac{x}{\ln x}$ is strictly decreasing over which domain interval?

The maximum value of the function $f(x) = \sin x + \cos x$ on the interval $[0, \pi]$ is:

The coordinate point on the curve $y = x^2 - 4x + 5$ where the tangent line is perfectly horizontal is:

The function $f(x) = x^3 - 3x^2 + 3x + 7$ has how many inflection points?

Evaluate $\lim_{x \to 0} \frac{\tan x - x}{x^3}$ using L'Hôpital's Rule:

The absolute minimum value of the function $f(x) = 2x^3 - 9x^2 + 12x + 1$ on the closed interval $[0, 2]$ is:

If the path of a curve is given by $y = e^x$, the tangent line drawn at $x = 0$ crosses the x-axis at which position?

The curves $y = e^x$ and $y = e^{-x}$ intersect at what angle?

A ladder $5\text{ m}$ long leans against a vertical wall. If the bottom of the ladder is pulled away along the ground at a rate of $2\text{ m/s}$, find the speed (in $\text{m/s}$) at which its top is sliding down the wall when the base is exactly $3\text{ m}$ away from the wall.

Find the total number of critical points for the function $f(x) = |x^2 - 4|$ across the real line.

Using differentials, if the side length of a cube increases by $2\%$, find the percentage error that propagates into the calculated volume.

Assertion (A): The function $f(x) = \tan x - x$ is strictly increasing for all real numbers where it is defined.
Reason (R): The derivative $f'(x) = \sec^2 x - 1 = \tan^2 x \ge 0$, meaning the slope is always non-negative.

Assertion (A): If $f''(c) = 0$, then the point $(c, f(c))$ must be an inflection point of the function.
Reason (R): An inflection point requires both $f''(c) = 0$ (or undefined) and a change in concavity across the point.