Question 1Practice
The graph of , the derivative of a function , is shown. It is negative for , crosses the -axis at , and touches the -axis at without crossing it. Which statement about the graph of is correct?
SACE Methods calculus: derivatives and integrals
Differentiate with the product, quotient and chain rules, then use the first and second derivatives to find and classify stationary points and points of inflection. Integrate by reversing differentiation, and use definite integrals for areas under and between curves and for total change.
| The examination | 130 minutes, 100 marks, 30% of the subject |
|---|---|
| Booklets | Two question booklets of 50 marks, about 65 minutes each |
| Allowed | A graphics calculator throughout, two A4 sheets of handwritten notes, a formula sheet |
| Answers | Working shown; three significant figures unless a question says otherwise |
Derivatives in order
- Name the form first: a product (x^2 e^(3x)), a quotient, or a composite such as sin(2x + 1), then apply its rule.
- Stationary points: solve f'(x) = 0; classify with f''(x) or a sign diagram, and give the coordinates.
- Optimisation: write the quantity as a function of one variable, find where its derivative is zero, check the endpoints, and justify the maximum or minimum.
Integrals and area
- An anti-derivative of f(ax + b) divides by a; an indefinite integral needs + c, and an initial condition fixes c.
- Where a curve is below the x-axis its integral is negative: split at a root before you find an area.
- The area between two curves is the integral of the upper minus the lower between their intersections.
Try 2 questions
Question 2Practice
The velocity against time graph shows a cyclist's velocity for the first 16 seconds of a ride: it rises steadily from 0 to 12 m/s over 4 seconds, stays at 12 m/s until , then falls steadily to 0 at . How far, in metres, does the cyclist travel in the 16 seconds?
Next step
Questions people ask
When do I need the second derivative?
To classify a stationary point (negative means a local maximum, positive a local minimum), to find points of inflection, and to describe concavity or acceleration.
Why is my area negative?
Part of the curve is below the x-axis there. Find the roots, integrate each piece and add the absolute values.