Studdy AI Logo
Math  /  Calculus

QuestionFor the given function, find f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}. f(x)=2x2+4f(x)=2 x^{2}+4 f(x+h)−f(x)h=\frac{f(x+h)-f(x)}{h}= □\square (Simplify your answer.)

Studdy Solution

STEP 1

1. The function given is f(x)=2x2+4 f(x) = 2x^2 + 4 .
2. We need to find the difference quotient f(x+h)−f(x)h \frac{f(x+h)-f(x)}{h} .
3. The difference quotient is a fundamental concept in calculus that approximates the derivative of a function as h h approaches 0.

STEP 2

1. Substitute x+h x+h into the function f(x) f(x) .
2. Calculate f(x+h)−f(x) f(x+h) - f(x) .
3. Divide the result by h h and simplify the expression.

STEP 3

Substitute x+h x+h into the function f(x) f(x) .
f(x+h)=2(x+h)2+4 f(x+h) = 2(x+h)^2 + 4

STEP 4

Expand the expression (x+h)2 (x+h)^2 .
(x+h)2=x2+2xh+h2 (x+h)^2 = x^2 + 2xh + h^2

STEP 5

Substitute the expanded form back into the function.
f(x+h)=2(x2+2xh+h2)+4 f(x+h) = 2(x^2 + 2xh + h^2) + 4

STEP 6

Distribute the 2 in the expression.
f(x+h)=2x2+4xh+2h2+4 f(x+h) = 2x^2 + 4xh + 2h^2 + 4

STEP 7

Calculate f(x+h)−f(x) f(x+h) - f(x) .
f(x+h)−f(x)=(2x2+4xh+2h2+4)−(2x2+4) f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 + 4) - (2x^2 + 4)

STEP 8

Simplify the expression by canceling out the common terms.
f(x+h)−f(x)=4xh+2h2 f(x+h) - f(x) = 4xh + 2h^2

STEP 9

Divide the simplified expression by h h .
f(x+h)−f(x)h=4xh+2h2h \frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2}{h}

STEP 10

Simplify the fraction by dividing each term by h h .
f(x+h)−f(x)h=4x+2h \frac{f(x+h) - f(x)}{h} = 4x + 2h
Solution: f(x+h)−f(x)h=4x+2h \frac{f(x+h) - f(x)}{h} = 4x + 2h

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord