Methods in calculus

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

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

WB B1 Find the mean value of 𝑎) 𝑓 𝑥 = 4 2+3𝑥 over the interval [2, 6] = 1 6−2 2 6 4 2+3𝑥 −1/2 𝑑𝑥 = 2 6 2+3𝑥 −1/2 𝑑𝑥 area = 2 2+3𝑥 1/2 × 1 3 6 2 = 2 3 20 − 8 = 4 3 5 − 2

WB B2 part 1 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 9 7 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 6 4 1+ 𝑒 𝑥 −1 𝑑𝑥 = 4 ln 3 ln 2 ln 6 1+ 𝑒 𝑥 −1 𝑑𝑥 area = 4 ln 3 2 6 4 1+𝑢 × 1 𝑢 𝑑𝑢 Integration by substitution = 4 ln 3 2 6 1 𝑢 − 1 1+𝑢 𝑑𝑢 Partial fractions = 4 ln 3 ln 𝑢 − ln (𝑢+1) 6 2 = 4 ln 3 ln 6 − ln 7 − ln 2 − ln 3 = 4 ln 3 ln 9 7 QED

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 9 7 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 9 7 ln 3 + 4 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 9 7 ln 3 NOW DO EX 3B

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 –

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