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AP Intermediate 2nd Year Continuity, Differentiability & Derivatives — practice questions

Explore 165 free Continuity, Differentiability & Derivatives questions in AP Intermediate 2nd Year Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Continuity, Differentiability & Derivatives questions and exercises with answers

Showing questions 1–30 of 165.

The derivative of $\sin(3x)$ is:Read the answer and worked solution →A function f(x) is INCREASING on an interval if for all x in the interval:Read the answer and worked solution →The tangent to y = x² at the point (1, 1) has slope:Read the answer and worked solution →Rolle's theorem requires that on [a, b], in addition to continuity + differentiability:Read the answer and worked solution →At a critical point c where f'(c) = 0, if f''(c) > 0, then c is a:Read the answer and worked solution →To find the dimensions of a rectangle with maximum area + perimeter 20, set up:Read the answer and worked solution →The critical points of f(x) = x³ − 3x + 1 are found by solving f'(x) = 0:Read the answer and worked solution →For a particle moving along x(t) = t³ − 3t² + 5, the velocity at t = 2 is:Read the answer and worked solution →Match each function to its derivative.Read the answer and worked solution →Fill the calculus turning point ruleRead 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 →

Worked practice questions

The derivative of $\sin(3x)$ is:

  1. $\cos(3x)$, ignoring the chain rule factor here on chart
  2. $3\sin(3x)$, applying chain rule to wrong function on chart
  3. $\cos(3)$, treating $3$ as a separate variable on the chart
  4. $3\cos(3x)$, by chain rule the inner derivative is $3$ here
Show answer and solution

Correct answer: D. $3\cos(3x)$, by chain rule the inner derivative is $3$ here

$(\sin(3x))' = 3\cos(3x)$ by chain rule.

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A function f(x) is INCREASING on an interval if for all x in the interval:

  1. $f''(x) > 0$ (positive second derivative)
  2. $f'(x) < 0$ (negative first derivative)
  3. $f'(x) > 0$ (positive first derivative)
  4. $f(x) > 0$ (positive function value)
Show answer and solution

Correct answer: C. $f'(x) > 0$ (positive first derivative)

f is INCREASING on an interval if f'(x) > 0 there. f''(x) > 0 means concave up (different concept). f' < 0 means decreasing.

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The tangent to y = x² at the point (1, 1) has slope:

  1. $1$ (the y-value at x = 1)
  2. $2$ (since dy/dx = 2x, at x = 1)
  3. $0.5$ (incorrect inverse calculation)
  4. $0$ (a horizontal tangent)
Show answer and solution

Correct answer: B. $2$ (since dy/dx = 2x, at x = 1)

dy/dx = 2x. At x = 1: slope = 2(1) = 2. Tangent line: y − 1 = 2(x − 1) → y = 2x − 1. Normal slope = -1/2.

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