Ratio of increments and the tangent
Answers are for parents and teachers.
1
Find the average rate of change of f(x) = x² − 3x on [2, 5].
4
2
A point moves by s(t) = t³ (m). Express its average speed on [1, 1 + h] in terms of h and compute it for h = 0.01. Which number does it approach?
((1 + h)³ − 1)/h = 3 + 3h + h². For h = 0.01 it is 3.0301. The ratio approaches 3.
3
The graph of a linear function passes through (0, 2) and (4, 14). Find the ratio of increments and write the function.
(14 − 2)/4 = 3, so y = 3x + 2.
4
The volume of water in a tank is V(t) = 40 + 6t litres (t in minutes). Is the rate of change the same on every interval? Explain.
Yes: V is linear, so the ratio of increments is always 6 L/min.
5
On y = x², write the slope of the secant through A(2, 4) and B(2 + h, (2 + h)²) in terms of h. Compute it for h = 0.1 and 0.01 and state the approximate slope of the tangent.
((2 + h)² − 4)/h = 4 + h. For h = 0.1 this is 4.1, for h = 0.01 it is 4.01; the tangent slope is ≈ 4.