Average Value of a Function (Ex 11J)

Year 12 Mathematical Methods Unit 3 · Cambridge §11J · Mr Wong

Name

Class

Date

Learning intentions.

Use $\bar y = \dfrac{1}{b-a}\int_a^b f(x)\,dx$ to compute the average value of a function
Interpret $\bar y$ as the height of the rectangle with the same area as the region under $y=f(x)$
Use a graphical / period argument to short-cut averages of sinusoids over a whole period

Key formula

$\bar y = \dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx$

Geometrically: $\bar y$ is the height of the rectangle on $[a,b]$ with the same area as the region under the curve. Over a whole number of periods, $a\sin(\omega x) + k$ has average $k$.

Part A — Straight averages

1. Find the average value of each function over the given interval.

a $f(x) = x^2$ on $[0, 3]$

b $f(x) = 2x + 1$ on $[0, 4]$

c $f(x) = \sin x$ on $[0, \pi]$

d $f(x) = e^x$ on $[0, 1]$

2. Find the average value of $f(x) = \dfrac{1}{x}$ on $[1, e]$. Leave your answer in exact form.

3. Find the average value of $g(x) = \sqrt{x - 5}$ on $[25, 36]$, giving an exact simplified surd form and a decimal answer to 2 d.p.

Part B — Graphical / period arguments

4. Without integrating, write down the average value of $y = 3\sin(5x) + 4$ over the interval $\left[0, \tfrac{2\pi}{5}\right]$. Justify in one sentence.

5. Without integrating, find the average value of $y = 5\cos(3x) - 2$ over $\left[0, \tfrac{2\pi}{3}\right]$.

Average Value — continued

Part C (application modelling) · VCAA-style

Name

Part C — Application contexts

6. The depth of water (m) at a jetty at time $t$ hours after midnight is $D(t) = 2\sin\!\left(\tfrac{\pi t}{6}\right) + 5$, $0 \le t \le 12$. Find the average depth over the 12-hour period.

7. The temperature (°C) in a room $t$ minutes after the air-conditioning is turned on is $T(t) = t\sin\!\left(\tfrac{\pi t}{4}\right) + 20$ for $0 \le t \le 14$. Find the average temperature in the room over these 14 minutes (exact and decimal).

8. Two stocks are modelled over 8 months by
Orange: $S(t) = 0.4t + 1$ Blue: $B(t) = -0.3t^2 + 2.7t - 0.83 + 2\sin(2t)$.
Find the average price of each stock over $[0,8]$. Which performed better on average?

Part D — VCAA-style

9. Based on VCAA 2014 Tech-Active (adapted). The function $h$ is defined piecewise on $[1, 11]$ as $h(x) = \begin{cases} x + 1 & 1 \le x \le 5\\ 6 & 5 < x \le 8 \\ 14 - x & 8 < x \le 11 \end{cases}$

a Find the average value of $h$ over $[1, 11]$.

b Find the average rate of change of $h$ over $[1, 11]$ — compare it to the answer in (a).