Studdy Solution
STEP 1
What is this asking?
We're finding the difference between the function's value at *x* + *h* and its value at *x*.
Watch out!
Don't forget to distribute the negative sign correctly when subtracting f(x)!
STEP 2
1. Evaluate *f*(*x* + *h*)
2. Evaluate *f*(*x* + *h*) - *f*(*x*)
3. Simplify
STEP 3
We **start** by substituting (x+h) into our function f(x)=4x2−4.
This gives us:
f(x+h)=4(x+h)2−4
STEP 4
Now, let's **expand** (x+h)2.
Remember, it's (x+h)⋅(x+h), not x2+h2!
So, using the distributive property (sometimes called FOIL), we get:
(x+h)⋅(x+h)=x⋅x+x⋅h+h⋅x+h⋅h=x2+2⋅x⋅h+h2
Substituting this back into our expression for f(x+h):
f(x+h)=4(x2+2⋅x⋅h+h2)−4
STEP 5
Next, we **distribute** the **4** across the terms in the parentheses:
f(x+h)=4⋅x2+4⋅2⋅x⋅h+4⋅h2−4=4x2+8⋅x⋅h+4h2−4
STEP 6
Now, we **subtract** f(x) from f(x+h):
f(x+h)−f(x)=(4x2+8⋅x⋅h+4h2−4)−(4x2−4)
STEP 7
Remember to **distribute** that negative sign correctly!
f(x+h)−f(x)=4x2+8⋅x⋅h+4h2−4−4x2+4
STEP 8
Let's **combine** those like terms!
Notice that 4x2−4x2 adds to zero, and −4+4 also adds to zero.
What's left?
f(x+h)−f(x)=8⋅x⋅h+4h2
STEP 9
f(x+h)−f(x)=8⋅x⋅h+4h2
