Find and simplify the following for the function. ( Assume h not equal 0).

Question

f(x) = 4x^3-x^2+1

f(a+h)-f(a)/h

Answer 1

Hi Gabrielle,

First, to expand f(a + h), we need to substitute (a+h) and into f(x) wherever x appears.

f(a+h) = 4(a+h)³ - (a+h)² + 1

Now to expand (you can FOIL this in two steps):

f(a+h) = 4(a³+ h³ + 3a²h + 3ah²) - (a² + 2ah + h²) + 1

f(a+h) = 4a³ + 4h³ + 12a²h + 12ah² - a² - 2ah - h² + 1

Then, to expand f(a)/h, we will first substitute a into f(x) wherever x appears.

f(a) = 4a³ - a² + 1

Note here I've assumed that we're solving for (f(a+h) - f(a))/h; let me know if this isn't the case.

We solve for the above quantity by substituting the two terms above:

f(a+h) - f(a) =

4a³ + 4h³ + 12a²h + 12ah² - a² - 2ah - h² + 1 - (4a³ - a² + 1)

= 4a³ + 4h³ + 12a²h + 12ah² - a² - 2ah - h² + 1 - 4a³ + a² - 1

= 4h³ + 12a²h + 12ah² - 2ah - h²

Finally, we divide this all by h:

(f(a+h) - f(a))/h =

(4h³ + 12a²h + 12ah² - 2ah - h²)/ h

= 4h² + 12a² + 12ah - 2a - h

And this is the final, simplified result.

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Answer 2

[4(a + h)³ - (a + h)² + 1 - (4a³ - a² + 1)]

As a Calculus student what is preventing you from multiplying out and simplifying?