Question 1

The graph of g(x) rises from x = 0 to x = 5, then falls for x > 5. State the sign of g′(x) for (a) 0 < x < 5 and (b) x > 5. State the value of g′(5).

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Question 2

The function f(x) = x³ − 2x. Which statement about f′(x) is correct?

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Question 3

Find f′(x) for f(x) = 4x³ − 3x + 7.

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Question 4

What is d/dx[−9]?

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Question 5

The graph of f(x) has a local minimum at x = 4. What must be true?

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Question 6

What is the derivative of 8x⁵?

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Question 7

Find dy/dx for y = 3x⁴ − 5x² + 2x − 8.

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Question 8

A function's graph is falling steeply at x = −2. Which is correct?

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Question 9

For which value of x is f′(x) = 0 certain to be true?

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Question 10

A company's monthly revenue R (thousands of dollars) is modelled by a curve. The graph of R(t) rises from t = 0 to t = 6, reaches a maximum at t = 6 with R(6) = 50, then falls.

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Question 11

h(x) = −x³ + 6x².

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Question 12

P(t) is the profit (thousands of dollars) after t years. P′(3) = 4. What does this mean?

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Question 13

The graph of a function passes through A(1, 4) and B(3, 4) with a smooth peak between them. A student says: "f′(1) = f′(3) because the function has the same y-value at both points." Is the student correct? Explain using the meaning of f′.

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Question 14

Find the gradient of y = x² − 5x at x = 3.

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Question 15

Differentiate y = x(3x − 4).

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Question 16

Find the gradient of f(x) = 2x³ − 6x at x = −1.

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Question 17

Expand and then differentiate y = (x + 2)(x − 3).

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Question 18

For g(x) = x³ − 12x, find the x-values where g′(x) = 0.

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Question 19

A car's position (metres) at time t (seconds) is s(t) = t² + 2t. (a) What does s′(t) represent? (b) Interpret s′(3) = 8 in context.

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Question 20

h(t) = −5t² + 20t is the height (metres) of a ball at time t (seconds). Without calculating, state what h′(t) = 0 tells you about the motion of the ball.

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Topic 5.3: Increasing, Decreasing and Stationary Points Questions

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