← Differentiation methods: past papers

AS 91578 · 2023 paper · trig, quotients, the helicopter

Question 2(d) Merit

Practises Tangents, normals and curve features

A quotient with cos, a cot gradient, a stationary point on an exponential quotient, inflection on x2 ln x, and a related-rates chase where the rate given is an angle.

Find the x-value(s) of any points of inflection on the graph of f(x) = 3x2 ln(x). You can assume that your point(s) found are actually point(s) of inflection.

The 2023 paper, page 7 ↗ Marking schedule, page 4 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 Find f′(x) with the product rule.

    f'(x) = 6x ln x + 3x2 · 1/x = 6x ln x + 3x

  2. 2 Differentiate again. 6x ln x is another product.

    f''(x) = 6 ln x + 6x · 1/x + 3 = 6 ln x + 9

  3. 3 Set f″(x) = 0 and solve.

    6 ln x = −9, so ln x = −1.5 and x = e−1.5 ≈ 0.223.
Answer x = e−1.5 ≈ 0.223.

What the marker wanted

  • Achieved A correct f′(x).
  • Merit Correct f′(x) and f″(x), and the right value of x.

In our words, from NZQA’s 2023 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.