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Indian Navy Pilot Application of Derivatives — practice questions

Explore 32 free Application of Derivatives questions in Indian Navy Pilot 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 32.

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 →Slope of tangent to y = f(x) at point (a, f(a)) is:Read the answer and worked solution →Point of inflection of curve y = f(x) is where: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 →Linear approximation of f(x) near x = a:Read the answer and worked solution →If $y = f(x)$, then $\dfrac{dy}{dx}$ represents the: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 →

Worked practice questions

If the tangent to the curve $y = e^{2x}$ at point $(a, e^{2a})$ passes through origin, then $a$ equals:

  1. 0
  2. $1/2$
  3. 1
  4. $\ln 2$
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Correct answer: B. $1/2$

Slope at $(a, e^{2a})$: $y' = 2e^{2a}$. **Tangent:** $y - e^{2a} = 2e^{2a}(x - a)$. Passes through (0, 0): $-e^{2a} = 2e^{2a}(-a)$ ⇒ $1 = 2a$ ⇒ $a = 1/2$.

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Slope of tangent to y = f(x) at point (a, f(a)) is:

  1. f(a)
  2. 1/f'(a)
  3. f''(a)
  4. f'(a) (derivative at the point)
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Correct answer: D. f'(a) (derivative at the point)

Derivative f'(a) is the slope of the tangent line at x = a. **Equation of tangent:** y - f(a) = f'(a)(x - a).

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Point of inflection of curve y = f(x) is where:

  1. Always at origin
  2. f'' = 0 and changes sign (concavity changes)
  3. f = 0
  4. f' = 0
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Correct answer: B. f'' = 0 and changes sign (concavity changes)

**Inflection point:** where f''(x) changes sign (concave up ↔ concave down). **Necessary condition:** f''(x) = 0 (or undefined). **Sufficient:** sign change of f''.

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