Question 2(b) Achieved

A parametric derivative, a velocity from a log displacement, a second-derivative verification, an inflection point on (ln x)/x, and a turning point that survives an unknown constant.

An object is travelling in a straight line. Its displacement, in metres, is s(t) = ln(3t² + 5t + 2), where t > 0 and t is time in seconds.

Find the velocity of this object when t = 1 second.

The 2024 paper, page 5 ↗ Marking schedule, page 3 ↗

Work it out

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

1. 1 Velocity is the rate of change of displacement. What do you differentiate?

   v = ds/dt

2. 2 How do you differentiate ln of a function?

   The inside’s derivative over the inside: v = (6t + 5)/(3t² + 5t + 2)

3. 3 Put t = 1 in.

   v = (6 + 5)/(3 + 5 + 2) = 11/10 = 1.1

What the marker wanted

• Achieved The correct derivative and v = 1.1. Units are not needed.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

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