Consider the function fx5x2x3 and complete parts a through c

Consider the function f(x)=5x²-x-3 and complete parts (a) through (c)

(A) Find f (a+h)

(B) Find f(a+h) -f(a) divide by h

(c) Find the instantaneous rate of change of f when a=3

Solution

f(x) = 5x²-x-3

A) f(a +h) = 5(a +h)²-(a +h)-3 = 5(a² + 2ah + h²) -a -h -3

==> f(a +h) = 5a² + 10ah + 5h² - a - h -3

B) f(a) = 5a²-a-3

==> [f(a +h) - f(a)]/h = [5a² + 10ah + 5h² - a - h -3 - 5a² +a +3]/h

==> [f(a +h) - f(a)]/h = [10ah + 5h² - h]/h = 10a + 5h -1

C) instantaneous rate of change at a = 3 ==> f \'(3)

==> f \'(x) = 10x -1       since d/dx xⁿ = n x^n-1

==> f \'(3) = 10(3) -1 = 29