WACE 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 signed area, the area between curves and total change.

Section One	Calculator-free, 35%, 50 minutes working
Section Two	Calculator-assumed, 65%, 100 minutes working
Questions	5 to 10 in Section One, 8 to 13 in Section Two
Section Two allows	Up to three calculators (CAS assumed), two A4 sheets of notes, a formula sheet

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 test, 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.
A definite integral is a signed area: split at a root before you find the area between a curve and the x-axis.
The area between two curves is the integral of the upper minus the lower between their intersections.

Try 2 questions

Question 1Practice

The graph shows

y

= f'(x)

, the derivative of a function

f

: a parabola that crosses the

x

-axis at

x

= -1

and

x

= 3

and has its lowest point at

(1, -4)

. Which statement about the graph of

y

= f(x)

is correct?

Question 2Practice

The graph shows

y

= 3 + 2x - x^2

. What is the area of the shaded region, between the curve and the

x

-axis from

x

= 0

x

= 3

Next step

Take the free test Try the lesson on the second derivative and optimisation

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.