AS 91578 · 2023 paper · derivatives, tangents, a proof
Question 1(e) Merit to Excellence
Practises Optimisation
A chain-rule root, a product rate of change, a second tangent parallel to a given one, a parametric circle tangent to be proved in general form, and an optimisation whose answer is a fraction of a given area.
The graph of y = x(x − 2m)2, where m > 0, is shown. The total shaded area between the curve and the x-axis from x = 0 to x = 2m is A = 4m4/3. A right-angled triangle is constructed with one vertex at (0, 0) and another on the curve.
Show that the maximum area of such a triangle is 3/8 of the total shaded area. You do not have to prove that the area you have found is a maximum.
| y | x(x − 2m)2, m > 0 |
|---|---|
| total shaded area A | 4m4/3 |
Work it out
Try it on paper first. Stuck? Take one hint at a time.
-
1 The right angle is on the x-axis, under the point on the curve. If that point is (x, y), what are the base and height?
The base is x and the height is y = x(x − 2m)2. -
2 Write the area as a function of x.
A = (1/2) x · x(x − 2m)2 = (1/2) x2(x − 2m)2 -
3 Differentiate, then factorise.
Product rule: dA/dx = x(x − 2m)2 + x2(x − 2m)
= x(x − 2m)(2x − 2m) = 2x(x − 2m)(x − m) -
4 Which root gives the biggest triangle?
dA/dx = 0 at x = 0, x = m and x = 2m. At 0 and 2m the triangle has no height, so the maximum is at x = m. -
5 Find that area, and compare it with the shaded area.
A = (1/2) m2(m − 2m)2 = (1/2) m4
((1/2) m4)/((4/3) m4) = 1/2 × 3/4 = 3/8
What the marker wanted
- Achieved The correct derivative.
- Merit The correct derivative, and x = m.
- Excellence The comparison made: ½m⁴ shown to be 3/8 of the shaded area (E8). Reaching ½m⁴ without the comparison, or a correct solution with one minor error, is E7.
In our words, from NZQA’s 2023 marking schedule.
✕Common mistake
Stopping at ½m⁴. The question is to show that it is 3/8 of the shaded area, and only that last line earns the top Excellence score. NZQA’s 2023 report names this part among the proofs students did not finish formally.How did it go?
Got it moves it further out. Review brings it back tomorrow.