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AP Intermediate 1st Year Limits and Derivatives — practice questions — page 2

Explore 50 free Limits and Derivatives questions in AP Intermediate 1st Year Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Limits and Derivatives questions and exercises with answers

Showing questions 31–50 of 50.

If f(x) = 3x² + 5x + 7, then f′(x) is:Read the answer and worked solution →d/dx [(x² + 1)(x + 1)] using product rule equals:Read the answer and worked solution →lim(x→0) (tan x) / x equals:Read the answer and worked solution →lim(x→0) (e^x − 1) / x equals:Read the answer and worked solution →The derivative of f(x) = √x with respect to x is:Read the answer and worked solution →lim(x→2) (x³ − 8) / (x² − 4) equals:Read the answer and worked solution →If f(x) = 1/x, then f′(x) is:Read the answer and worked solution →lim(x→∞) (3x² + x + 1) / (5x² + 2) equals:Read the answer and worked solution →Key in the limitRead the answer and worked solution →Which one does NOT belong?Read the answer and worked solution →Limits and derivatives form the introduction to:Read the answer and worked solution →Approaching a point from either side gives limits numbering:Read the answer and worked solution →The value expected from points to the right of a is the:Read the answer and worked solution →The value expected from points to the left of a is the:Read the answer and worked solution →If the two one sided limits coincide, that common value is the:Read the answer and worked solution →If the two one sided limits differ, the limit at that point:Read the answer and worked solution →To find a limit at 5, one looks at values of x that are:Read the answer and worked solution →A limit describes the behaviour of a function:Read the answer and worked solution →The derivative of a function at a point is defined using a:Read the answer and worked solution →The definition of derivative applies to a function that is:Read the answer and worked solution →

Worked practice questions

If f(x) = 3x² + 5x + 7, then f′(x) is:

  1. 6x + 5
  2. 3x + 5
  3. 6x + 7
  4. 3x² + 5
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Correct answer: A. 6x + 5

1. Apply power rule to each term: d/dx (3x²) = 6x; d/dx (5x) = 5; d/dx (7) = 0. 2. Sum: f′(x) = 6x + 5. _Source: syllabus Maths Ch 13 §13.4 Derivatives of Polynomials_

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d/dx [(x² + 1)(x + 1)] using product rule equals:

  1. (2x + 1)(x + 1)
  2. (2x)(x² + 1) + (x² + 1)
  3. 2x(x + 1) + (x² + 1)
  4. (x² + 1)(2x + 1)
Show answer and solution

Correct answer: C. 2x(x + 1) + (x² + 1)

1. Product rule: (uv)′ = u′v + uv′. 2. Let u = x² + 1 (u′ = 2x); v = x + 1 (v′ = 1). 3. (uv)′ = 2x · (x + 1) + (x² + 1) · 1 = 2x(x + 1) + (x² + 1). 4. Expanded: 2x² + 2x + x² + 1 = 3x² + 2x + 1. _Source: syllabus Maths Ch 13 §13.4 Product Rule_

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lim(x→0) (tan x) / x equals:

  1. 0
  2. 1
  3. x
  4. Undefined
Show answer and solution

Correct answer: B. 1

1. tan x / x = (sin x / x) · (1 / cos x). 2. As x → 0, sin x / x → 1 and cos x → 1. 3. So tan x / x → 1 · 1 = 1. _Source: syllabus Maths Ch 13 §13.2 Standard Trig Limits_

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