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Calculus/Solving usi)ng Definition of Derivative

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Expert: - 9/29/2011


Question
I have the following question on a homework assignment from my Calc 110 class, really struggling to solve it. I know its easily solvable using the power rule, but we are supposed to solve it using the equation:

(f(x)-f(a))/(x-a)

and then again using:

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

Find the slope of the tangent line to the parabola y= 4x-x^2 at the point (1,3)

Not looking for just an answer to homework here, really hoping for some help on how to do this stuff.

Thanks a lot!

Answer
Let a be close to x.  Since x = 1, a could be like 0.9.
It was given that f(x) = 3, and it f(0.9) can be found.
It is known that f(0.9) = 4(0.9) - 0.9² = 3.6 - 0.81 = 2.79.

What we have is then (f(x)-f(a))/(x-a) = (3 - 2.79)/(1-0.9) = 0.21/0.1 = 2.1.


If we think of h as 0.1 and a as 1, then we have (f(1.1) - f(1))/0.1.
From f(x) = 4x - x², f(1.1) = 4.4 - 1.21 = 3.19.
This gives (f(a+h)-f(a))/h as (f(1.1)-f(1))/0.1 = (3.19-3)/0.1 = 1.9.


Note that the first one approximated the derivative at 1 using 0.9 and 1 and
the second one approximated the derivative using 1 and 1.1.
The true derivative, of course, can be seen to be 4 - 2x = 4 - 2 = 2.

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