7.1 The derivative as a gradient and a rate of change — Practise

Work through these on your own. Sketch the curve or the chord first, and write down the meaning of each gradient or rate in the context. Give lengths to three significant figures where an answer is not exact.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

Find the gradient of the straight line through $(2, 5)$ and $(6, 17)$ .
The curve $y = x^2$ passes through $P(1, 1)$ and $Q(3, 9)$ . Find the gradient of the chord $PQ$ .
The curve $y = x^2$ passes through $P(1, 1)$ and $Q(1.5, 2.25)$ . Find the gradient of the chord $PQ$ .
Use the chord from $(2, 8)$ to $(2.1, 9.261)$ on the curve $y = x^3$ to estimate the gradient at $x = 2$ . Give your answer to two decimal places.
A curve has gradient function $f'(x) = 3x^2 - 6x$ . Find the two values of $x$ where the gradient is zero.
A curve has gradient function $f'(x) = 3x^2 - 6x$ . Work out the gradient at $x = 3$ .
A stone falls a distance $s = 5t^2$ metres in $t$ seconds. Find the average speed between $t = 1$ and $t = 3$ seconds.
The population $P$ of a village after $t$ years is modelled by $\dfrac{dP}{dt} = 40 - 4t$ . Work out the rate of change of the population at $t = 5$ .
A curve has gradient function $f'(x) = (x - 1)(x - 4)$ . Find the two values of $x$ where the curve is flat.
A drink cools so that its temperature after $t$ minutes is $\theta = 100 - 3t$ degrees Celsius. Work out the change in temperature over $4$ minutes.

Answers and concise solutions

1. The gradient

Answer: Gradient $= \dfrac{17 - 5}{6 - 2} = \dfrac{12}{4} = 3$ . This is the same at every point on the line.

2. The gradient of the chord

Answer: Gradient $= \dfrac{9 - 1}{3 - 1} = \dfrac{8}{2} = 4$ . The chord is steeper than the tangent at $P$ , because it reaches all the way to $Q$ .

3. The gradient of the chord

Answer: Gradient $= \dfrac{2.25 - 1}{1.5 - 1} = \dfrac{1.25}{0.5} = 2.5$ . Moving $Q$ closer to $P$ has brought the gradient from $4$ to $2.5$ , towards the tangent value $2$ .

4. The estimated gradient

Answer: Gradient $= \dfrac{9.261 - 8}{2.1 - 2} = \dfrac{1.261}{0.1} = 12.61$ . A shorter chord moves the estimate nearer to the true gradient, which is $12$ .

5. The two values of $x$

Answer: Factorise: $3x(x - 2) = 0$ , so $x = 0$ or $x = 2$ . At both points the tangent is horizontal.

6. The gradient at $x = 3$

Answer: Substitute $x = 3$ : $3(9) - 6(3) = 27 - 18 = 9$ . The gradient is positive, so the curve is rising at $x = 3$ .

7. The average speed

Answer: The stone falls $45 - 5 = 40$ m in $2$ s, so the average speed is $\dfrac{40}{2} = 20$ m/s. This is the average over the interval, not the speed at one instant.

8. The rate of change

Answer: Substitute $t = 5$ : $40 - 4(5) = 20$ . The population is still growing at $20$ people per year at $t = 5$ .

9. The two values of $x$

Answer: A flat point has gradient zero, so $(x - 1)(x - 4) = 0$ . Therefore $x = 1$ or $x = 4$ .

10. The change in temperature

Answer: The change is $\theta(4) - \theta(0) = 88 - 100 = -12$ degrees. The negative sign shows that the temperature falls, and the rate is $-3$ degrees per minute.