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Indian Navy AA (Artificer Apprentice) Application of Derivatives — practice questions

Explore 30 free Application of Derivatives questions in Indian Navy AA (Artificer Apprentice) 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 30.

If $f(x) = x^2 - 4x + 3$, the **interval** on which $f$ is decreasing is: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 →For y = sin x, find dy/dx at x = π/3:Read the answer and worked solution →Find minimum value of f(x) = (sin x)² + (cos x)² + 2 (sin x)(cos x):Read the answer and worked solution →By the second derivative test, at a point of local maximum the value of $f''(x)$ is:Read the answer and worked solution →What does the sign of f ′(x) tell you?Read the answer and worked solution →Complete the definition of a critical pointRead the answer and worked solution →Order the first derivative testRead the answer and worked solution →Match each sign change to its verdictRead the answer and worked solution →The second derivative testRead the answer and worked solution →True or false: maxima and minimaRead the answer and worked solution →Order the working rule for absolute maxima and minimaRead the answer and worked solution →Tap every point you must test for an absolute maximumRead the answer and worked solution →Local or absolute?Read the answer and worked solution →Complete the rate of change statementRead the answer and worked solution →The chain rule for related ratesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →True or false: increasing and decreasingRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Match each situation to the right toolRead the answer and worked solution →Tap every statement true of a local maximum at cRead the answer and worked solution →Reading the tests backRead the answer and worked solution →Which test is being described?Read the answer and worked solution →Assertion and reason: rates of changeRead the answer and worked solution →Spot the odd oneRead the answer and worked solution →Complete the second derivative testRead the answer and worked solution →True or false: scope of this chapterRead the answer and worked solution →

Worked practice questions

If $f(x) = x^2 - 4x + 3$, the **interval** on which $f$ is decreasing is:

  1. $x < 2$
  2. $x > 2$
  3. All $x$
  4. $x = 2$
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Correct answer: A. $x < 2$

$f'(x) = 2x - 4 < 0$ ⇔ $x < 2$. So $f$ is decreasing on $(-\infty, 2)$ and increasing on $(2, \infty)$. Vertex/minimum at $x = 2$.

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A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:

  1. $R^2$ (when the rectangle is a square inscribed)
  2. $2R^2$
  3. $R^2/2$
  4. $\pi R^2/2$
Show answer and solution

Correct answer: A. $R^2$ (when the rectangle is a square inscribed)

Half-width = $x$, height $y = \sqrt{R^2 - x^2}$. Area $A = 2xy = 2x\sqrt{R^2 - x^2}$. We have $dA/dx = 2\sqrt{R^2 - x^2} - 2x^2/\sqrt{R^2 - x^2}$. So $dA/dx = 0$. ⇒ $R^2 = 2x^2$ ⇒ $x = R/\sqrt 2$. Then $y = R/\sqrt 2$, area = $2 \cdot (R/\sqrt 2)(R/\sqrt 2) = R^2$.

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For y = sin x, find dy/dx at x = π/3:

  1. cos(π/3) = 1/2
  2. 1/2
  3. 1
  4. √3/2
Show answer and solution

Correct answer: A. cos(π/3) = 1/2

dy/dx = cos x. At π/3: cos(π/3) = 1/2. (Equation of tangent at this point: y - √3/2 = (1/2)(x - π/3).)

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