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HPCET Application of Derivatives — practice questions

Explore 69 free Application of Derivatives questions in HPCET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Application of Derivatives questions and exercises with answers

Showing questions 1–30 of 69.

The **slope of the tangent** to the curve $y = f(x)$ at point $(a, f(a))$ is:Read the answer and worked solution →A function $f$ is **strictly increasing** on an interval if, on that interval:Read the answer and worked solution →At a **local maximum** of a differentiable function $f$:Read the answer and worked solution →The slope of the tangent to $y = x^3 - 3x + 1$ at $x = 2$ is:Read the answer and worked solution →Water is poured at $36\,\text{m}^3/\text{s}$ into a cylindrical vessel of base radius 3 m. The rate at which water level rises is:Read the answer and worked solution →If $f(x) = x^2 - 4x + 3$, the **interval** on which $f$ is decreasing is:Read the answer and worked solution →Find the approximate value of $\sqrt{25.5}$ using differentials (given $\sqrt{25} = 5$):Read the answer and worked solution →$f(x) = 2x^3 - 3x^2 - 12x + 5$. The **local maximum value** occurs at:Read the answer and worked solution →The function $f(x) = x^3 - 3x + 1$ has critical points at:Read the answer and worked solution →Find the **maximum value** of $f(x) = \sin x + \cos x$ for $x \in [0, 2\pi]$.Read the answer and worked solution →A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:Read the answer and worked solution →Find the **shortest distance** from origin to the curve $xy = 4$ (in the first quadrant):Read the answer and worked solution →If $y = a \ln x + b x^2 + x$ has extreme values at $x = 1$ and $x = 2$, then $(a, b)$ is:Read the answer and worked solution →The function $f(x) = \dfrac{x}{1 + x^2}$ attains its maximum value on $[0, \infty)$ at:Read the answer and worked solution →If $f(x) = x^x$ for $x > 0$, then $f$ has a minimum at $x = $:Read the answer and worked solution →The volume of a sphere is increasing at $8\pi$ cm³/s. The rate of change of surface area when the radius is 2 cm is:Read the answer and worked solution →If the tangent to the curve $y = e^{2x}$ at point $(a, e^{2a})$ passes through origin, then $a$ equals:Read the answer and worked solution →A wire of length $L$ is bent to form a rectangle. The **maximum area** enclosed is:Read the answer and worked solution →Slope of tangent to y = f(x) at point (a, f(a)) is:Read the answer and worked solution →Slope of normal line to y = f(x) at x = a (if f'(a) ≠ 0):Read the answer and worked solution →At a stationary point of f, derivative f' equals:Read the answer and worked solution →Function f is INCREASING on interval (a, b) if:Read the answer and worked solution →Local maximum of f at x = c (if f'(c) = 0):Read the answer and worked solution →Mean Value Theorem says that for continuous f on [a, b] differentiable on (a, b), there exists c ∈ (a, b) with:Read the answer and worked solution →For y = x³ - 3x² + 2, find critical points:Read the answer and worked solution →Point of inflection of curve y = f(x) is where:Read the answer and worked solution →For y = sin x, find dy/dx at x = π/3:Read the answer and worked solution →Approximation by differentials: Δy ≈ f'(x) Δx. Using this, sqrt(25.5) is approximately:Read the answer and worked solution →Rate of change of area of a circle (A = πr²) with respect to radius:Read the answer and worked solution →If two functions f and g satisfy f'(x) = g'(x) for all x on an interval, then:Read the answer and worked solution →

Worked practice questions

The **slope of the tangent** to the curve $y = f(x)$ at point $(a, f(a))$ is:

  1. $f(a)$
  2. $f'(a)$
  3. $1/f'(a)$
  4. $f''(a)$
Show answer and solution

Correct answer: B. $f'(a)$

Tangent slope at $x = a$ equals $f'(a)$, the derivative evaluated at that point. Normal slope = $-1/f'(a)$ (perpendicular).

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A function $f$ is **strictly increasing** on an interval if, on that interval:

  1. $f'(x) = 0$
  2. $f'(x) > 0$
  3. $f'(x) < 0$
  4. $f'(x) = \infty$
Show answer and solution

Correct answer: B. $f'(x) > 0$

$f'(x) > 0$ throughout an open interval ⇒ strictly increasing there. $f'(x) < 0$ ⇒ strictly decreasing. $f'(x) = 0$ at isolated points OK (e.g., $x^3$ is strictly increasing even though $f'(0) = 0$).

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At a **local maximum** of a differentiable function $f$:

  1. $f'(x) > 0$
  2. $f'(x) = 0$ and $f''(x) < 0$
  3. $f'(x) = 0$ and $f''(x) > 0$
  4. $f''(x) = 0$
Show answer and solution

Correct answer: B. $f'(x) = 0$ and $f''(x) < 0$

**At local max:** $f'(x) = 0$ (critical/stationary point) AND $f''(x) < 0$ (curve concave down). **At local min:** $f'(x) = 0$ and $f''(x) > 0$. $f''(x) = 0$ is inconclusive — use higher-derivative test.

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