What is the average rate of change of the function y = 4x^3 - 2 between x = x1 and x = x2

Question

Answers: a. 4(x2^2 + x2x1 − x1^2)
b. 4(x2^2 − x2x1 + x1^2)
c. 4(x2^2 − x2x1 − x1^2)
d. 4(x2^2 + x2x1 + x1^2)

Answer 1

Ralph,

 

If the function is f(x), the average rate of change on the interval [x₁,x₂] is:

 

f_ave = (f(x₂) - f(x₁))/(x₂ - x₁).

 

What you must do is use f(x) = 4x³ - 2 to calculate with this formula.

 

f(x₁) = 4x₁³ - 2 and f(x₂) = 4x₂³ - 2

 

Subtracting, f(x₂) - f(x₁) = 4x₂³ - 4x₁³ = 4(x₂³ - x₁³) = 4(x₂ - x₁)(x₂² + x₁x₂ + x₁²).

 

After you divide by x₂ - x₁ and reduce, you obtain the correct answer, d.