Let f(x) = 2x^2 - 4x. show your work to justify all solutions a. Find the avg. r

Question

Let f(x) = 2x^2 - 4x. show your work to justify all solutions

a. Find the avg. rate of change of f over the interval [3,3+h]. How is this related ot the secant line passing through the points (3,6) and (3+h, f(3+h)?

b. Use part (a) to ifnd the instantaneous rate of change of f with respect to x at 3. How is this related to the tangent line passing through the point (3,6)?

c. Find an equation of the tangent line to the graph of f at the point (3,6).

d. Verify that the equation y= -8x-2 is an equation of the tangent line to the graph of f at the (-1,6).

Explanation / Answer

a.) f(x) = 2x^2 - 4x

f'(x) = 4x - 4

f'(x) = [f(x+h) - f(x)]/h

f'(3) = [f(3+h) - f(3)]/h

But f(3) = 2(3)^2 - 4(3) = 18 - 12 = 6

So f'(3) = [f(3+h) - 6]/h

the avg. rate of change of f over the interval [3,3+h] = f'(3) = 4(3) - 4 = 8

Secant line has slope = [f(3+h) - 6]/h = 8

b.) f'(3) = 8

the instantaneous rate of change of f with respect to x at 3 = 8

Tangent at (3,6) has slope f'(3) = 8

c.) slope = 8

point = (3,6)

y - 6 = 8(x-3)

y - 6 = 8x - 24

8x - y - 18 = 0 is equation of tangent at (3,6)

d.) At (-1,6), slope = f'(-1) = 4(-1) - 4 = -8

y - 6 = -8(x+1)

y - 6 = -8x - 8

8x + y + 2 = 0 is the equation of tangent at (-1,6)

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.