Average value of the derivative Suppose ƒ ' is a continuous function for all real numbers. Show that the average value of the derivative on an interval [a, b] is ƒ⁻' = (ƒ(b) ―ƒ(a))/ (b―a) . Interpret this result in terms of secant lines.

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(c) The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b] .

Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).

Find the average value of ƒ(𝓍) = e²ˣ on [0, ln 2] .

Find the average value of the function on the interval $[2,5]$.

$F\left(x\right)=x^3-\frac{2}{\sqrt{x}}$

Find the average value of the function on the interval $[4,9]$.

$F\left(x\right)=\sqrt{x}+6$

Find the average value of the function on the interval $[0,1]$.

$G\left(x\right)=\frac{2}{x^2+1}$