1. Methods in calculus

2. FM Methods in Calculus: mean value of a function
KUS objectives
BAT evaluate the mean value of a function using integration
Starter: find
𝑒 3𝑥+1 𝑑𝑥
= 1 3 𝑒 3𝑥+1 +𝐶
4𝑥 𝑒 𝑥 2 𝑑𝑥
=2 𝑒 𝑥 2 +𝐶
1 5𝑥+3 𝑑𝑥
= 1 5 𝑙𝑛 5𝑥+3 +𝐶

3. Notes 𝑜 𝑦 𝑥 𝑦=𝑓(𝑥) 𝑎 𝑏 𝑏−𝑎 𝑓 𝑓 = 1 𝑏−𝑎 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 𝐴𝑟𝑒𝑎= 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
You find the mean average of discrete numbers by adding them and dividing by the number of values
To find the mean value of a function in interval [𝑎, 𝑏] we represent their sum by integrating the function between a and b, and represent the number of values as the interval 𝑏−𝑎
In the diagram the area of the rectangle is (𝑏−𝑎) 𝑓 but this equals the area under the curve. So
(𝑏−𝑎) 𝑓 = 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
𝑓 = 1 𝑏−𝑎 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥

4. WB B1 Find the mean value of
𝑎) 𝑓 𝑥 = 𝑥 over the interval [2, 6]
= 1 6− 𝑥 −1/2 𝑑𝑥
= 𝑥 −1/2 𝑑𝑥
area
= 𝑥 1/2 ×
= − 8 = − 2

5. WB B2 part 𝑓 𝑥 = 4 1+ 𝑒 𝑥
a) Show that the mean value of f(x) over the interval ln 2 , ln 6 is 4 ln ln 3
Use (a) to find the mean value over the interval ln 2 , ln 6 of 𝑓 𝑥 +4
Use geometric considerations to find the mean value of –𝑓(𝑥) over ln 2 , ln 6
= 1 ln 6 − ln 2 ln 2 ln 𝑒 𝑥 −1 𝑑𝑥
= 4 ln ln 2 ln 𝑒 𝑥 −1 𝑑𝑥
area
= 4 ln 𝑢 × 1 𝑢 𝑑𝑢
Integration by substitution
= 4 ln 𝑢 − 1 1+𝑢 𝑑𝑢
Partial fractions
= 4 ln ln 𝑢 − ln (𝑢+1) 6 2
= 4 ln ln 6 − ln 7 − ln 2 − ln = 4 ln ln QED

6. NOW DO EX 3B WB B2 part 2 𝑓 𝑥 = 4 1+ 𝑒 𝑥
a) Show that the mean value of f(x) over the interval ln 2 , ln 6 is 4 ln ln 3
Use (a) to find the mean value over the interval ln 2 , ln 6 of 𝑓 𝑥 +4
Use geometric considerations to find the mean value of –𝑓(𝑥) over ln 2 , ln 6
b) Mean of 𝑓 𝑥 +4 is mean of f(x) plus 4
every value has gone up 4 so the mean has gone up 4 = 4 ln ln
You can check by going through the integration from the start!
𝑐) −𝑓(𝑥) is a reflection in the x-axis of f(x)
so the mean value over the interval will be the negative area = − 4 ln ln 3
NOW DO EX 3B

7. One thing to improve is –
KUS objectives
BAT evaluate the mean value of a function using integration
self-assess
One thing learned is –
One thing to improve is –

8. END