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Indian Navy Pilot Limits and Continuity — practice questions

Explore 52 free Limits and Continuity questions in Indian Navy Pilot Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Limits and Continuity questions and exercises with answers

Showing questions 1–30 of 52.

Find d/dx (x² sin x) by product rule:Read the answer and worked solution →d/dx (xˣ) where x > 0: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 →Match each kind of matrix to its conditionRead the answer and worked solution →Sort each matrix by its shapeRead the answer and worked solution →Tap every matrix that must be squareRead the answer and worked solution →

Worked practice questions

Find d/dx (x² sin x) by product rule:

  1. 2x cos x
  2. 2x sin x
  3. x² cos x + 2x sin x
  4. x² + sin x
Show answer and solution

Correct answer: C. x² cos x + 2x sin x

**Product rule:** (fg)' = f'g + fg'. With f = x², g = sin x: derivative = 2x × sin x + x² × cos x = 2x sin x + x² cos x.

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d/dx (xˣ) where x > 0:

  1. xˣ × (1 + ln x)
  2. xˣ × x
  3. x × xˣ⁻¹
  4. xˣ
Show answer and solution

Correct answer: A. xˣ × (1 + ln x)

Let y = xˣ. Take ln: ln y = x ln x. **Differentiate:** (1/y) y' = ln x + 1. So y' = xˣ × (ln x + 1) = xˣ × (1 + ln x).

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What does the sign of f ′(x) tell you?

Read exercise and reference answers
Items
  1. Item
    f ′(x) > 0 throughout
    Category
    Increasing
  2. Item
    x₁ < x₂ gives f(x₁) < f(x₂)
    Category
    Increasing
  3. Item
    f ′(x) < 0 throughout
    Category
    Decreasing
  4. Item
    x₁ < x₂ gives f(x₁) > f(x₂)
    Category
    Decreasing
  5. Item
    f(x) = c for all x
    Category
    Constant
  6. Item
    f ′(x) = 0 throughout the interval
    Category
    Constant
Categories
  1. Item
    Increasing
  2. Item
    Decreasing
  3. Item
    Constant
Your syllabus gives f ′(x) ≥ 0 on an interval as the test for increasing and f ′(x) ≤ 0 for decreasing. Everything else in the chapter builds on reading that sign. Source: syllabus Maths Part I, Ch 6 Application of Derivatives.

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