Expert: Paul Klarreich - 9/1/2011

Question
Hi I have a homework question:

Find the numerical derivative of f(x)=5x^3-12x^2+3x-10 at x=4 by using the formula: f(x+h) - f(x)/h.

I came up with the answer -103 but I don't think the answer is supposed to be a negative number so I just wanted to make sure.

Answer
Questioner:Alice
Country:New Jersey, United States
Category:Calculus
Private:No
Subject:Numerical derivative help
Question:

Hi I have a homework question:

Find the numerical derivative of f(x)=5x^3-12x^2+3x-10 at x=4 by using the
formula: f(x+h) - f(x)/h.

I came up with the answer -103 but I don't think the answer is supposed to be a
negative number so I just wanted to make sure.
..................................................
There is no reason the derivative can't be negative -- many things in New Jersey are.

You want:
    f(4 + h) - f(4)  << I think this is what
lim  ---------------  << your teacher is calling the
h->0        h         << numerical derivative.


    5(4+h)^3 - 12(4+h)^2 + 3(4+h) - 10 - [5(4)^3 - 12(4)^2 + 3(4) - 10]
lim  -------------------------------------------------------------------
h->0          h

Let's do it in pieces:

5(4+h)^3 - 12(4+h)^2 + 3(4+h) - 10 =
5(64 + 48h + 12h^2 + h^3) - 12(16 + 8h + h^2) + 12 + 3h - 10  =
320 + 240h + 60h^2 + 5h^3 - 192 - 96h - 12h^2 + 2 + 3h  =
130 + 147h + 48h^2 + 5h^3

5(4)^3 - 12(4)^2 + 3(4) - 10 =
5(64) - 12(16) + 12 - 10 =
320 - 192 + 2  =
130

Put them back:
     130 + 147h + 48h^2 + 5h^3 - 130
lim  ---------------------------------
h->0        h

     147h + 48h^2 + 5h^3
lim  ---------------------
h->0        h


lim   147 + 48h + 5h^2
h->0        

= 147

Now let's check (by cheating, of course -- you are not supposed to know how to do this.)  This is important; there was so much algebra we could easily blow a sign or arithmetic.

f(x) = 5x^3 - 12x^2 + 3x - 10

f'(x) = 15x^2 - 24x + 3

f'(4) = 15(4)^2 - 24(4) + 3

f'(4) = 15(16) - 96 + 3

f'(4) = 240 - 96 + 3

f'(4) = 144 + 3 = 147

Whew!