SOLUTION: Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. (Simplify your answers completely.) f(x) = 4x^2 + 3x Step 1

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. (Simplify your answers completely.) f(x) = 4x^2 + 3x Step 1      Log On


   

Question 1160763: Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. (Simplify your answers completely.)
f(x) = 4x^2 + 3x
Step 1: f(x + h)=
Step 2: f(x + h) − f(x)=
Step 3: f(x + h) − f(x)/h=
Step 4: f '(x) = lim h→0 f(x + h) − f(x)/h=
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Let's find f(x+h)

f%28x%29+=+4x%5E2+%2B+3x

f%28x%2Bh%29+=+4%28x%2Bh%29%5E2+%2B+3%28x%2Bh%29 Replace every x with (x+h)

f%28x%2Bh%29+=+4%28x%5E2%2B2xh%2Bh%5E2%29+%2B+3%28x%2Bh%29 FOIL rule

f%28x%2Bh%29+=+4x%5E2%2B8xh%2B4h%5E2+%2B+3x%2B3h Distribute. This is one way to write the result for step 1.

---------------------------------------------------------
Now subtract off f(x)

f%28x%2Bh%29+-+f%28x%29+=+4x%5E2%2B8xh%2B4h%5E2+%2B+3x%2B3h+-+%28+f%28x%29+%29

f%28x%2Bh%29+-+f%28x%29+=+4x%5E2%2B8xh%2B4h%5E2+%2B+3x%2B3h+-+%28+4x%5E2%2B3x+%29

f%28x%2Bh%29+-+f%28x%29+=+4x%5E2%2B8xh%2B4h%5E2+%2B+3x%2B3h+-++4x%5E2+-+3x+

f%28x%2Bh%29+-+f%28x%29+=+%284x%5E2+-++4x%5E2%29%2B8xh%2B4h%5E2+%2B+%283x-3x%29%2B3h

f%28x%2Bh%29+-+f%28x%29+=+8xh%2B4h%5E2%2B3h This is the result for step 2.

Notice how each term has an 'h' in it

---------------------------------------------------------

That common h will be factored out to help cancel and simplify things as shown below

%28f%28x%2Bh%29+-+f%28x%29%29%2Fh+=+%288xh%2B4h%5E2%2B3h%29%2Fh

%28f%28x%2Bh%29+-+f%28x%29%29%2Fh+=+%28h%288x%2B4h%2B3%29%29%2Fh



%28f%28x%2Bh%29+-+f%28x%29%29%2Fh+=+%28cross%28h%29%288x%2B4h%2B3%29%29%2Fcross%28h%29

%28f%28x%2Bh%29+-+f%28x%29%29%2Fh+=+8x%2B4h%2B3 The result for step 3.

---------------------------------------------------------

As h approaches infinity, the expression 8x%2B4h%2B3 will approach 8x%2B4%2A0%2B3. All I did here was plug in h = 0. Afterward, you simplify to go from 8x%2B4%2A0%2B3 to 8x%2B3

Therefore, if f(x) = 4x^2+3x, then f'(x) = 8x+3 is the derivative.

Side note: To find the slope of the tangent line at any given point (x,y), you would plug the x coordinate into f'(x) = 8x+3 and simplify.