2/13/11

Alright, sooo were still doing AP stuff.. with nothing new learned I think i'm just gonna review some old topics. I know questions similar to these showed up on the practice AP's we took, and I forgot how to do them. So i'll explain it and give some examples.

EXAMPLES OF SLOPE:
1. Find the slope m of the line tangent to the graph of the function g(x) = 5 - x^2 at the point (2,1).
*To do this, you must find the derivative, then plug in your x-value.
Derivative = -2x
Plug in: -2 (2) = -4
The slope is -4

2. Find the slope of the graph of the function at the given value.
f(x) = 2 (4x + 6)^2 when x = 5
*Find the derivative and then plug in.
Derivative:
2 * (2 (4x + 6)) * 4
16(4x + 6)
16(4(5) + 6)
16(20 + 6)
16 (26) = 416
416 is the slope of the graph.

EXAMPLE: Mean Value Theorem
1. Check if the function is continuous.
2. Check for differentiability or non-differentiability.
3. Slope
f ' (c) = f(b) - f(a) / b - a
* It means on an interval [a,b]. There must be some x-value where the derivative = the slope between a & b.
slope of a secant line = slope of a tangent line between a & b

1. Given f(x) = 5 - 4/x, find all values of c in the open interval (1,4) such that
f ' (c) = f(4) - f(1) / 4-1
yes it is both continuous and differentiable
= 4-1/4-1 = 1
5 - 4x^-1
4/x^2 = 1
4 = x^2
x = +/- 2
c = +/- 2
*By the mean value theorem, the function is continuous and differentiable, therefore some value c=2 where f ' (x) = the slope between (1,4).