The Geometry of the Derivative

I.  Quiz

II.  Homework

III.  Increasing and Decreasing Functions

Definition:  Let y = f(x) be a function, then we say that f is decreasing at x = c if for all nearby points a < b (points within epsilon of x for some small epsilon) that f(a) > f(b) and that f is increasing if f(b) < f(a).  We use the adjective strictly to say the same thing without the equality.  We say that f is increasing in an interval I if it is increasing for every point in the interval.  

Theorem:  Let f be a differentiable function then if f'(c) > 0 then f is strictly increasing at x = c.  

If f'(c) < 0, then f is strictly decreasing at x = c.  

If f'(c) = 0 on an interval I, then f is constant there.

Pictures will be given.

IV.  Critical, Stationary, Maximum and Minimum Points.

Definition:  A number is x = c called a critical value  of f if either f'(c) = 0 or does not exist and is called a stationary value of f if f'(c) = 0.  

Examples:  For the functions y = x³  and y = x² , 0 is both a critical value and a stationary value.  For y = |x|, 0 is a critical value.

Definition:  A number f(c) is called a local maximum is f(c) > f(x) for all x within some epsilon of c.  A number f(c) is called a local minimum is f(c) < f(x) for all x within some epsilon of c.  A number f(c) is called a global maximum is f(c) > f(x) for all x . A number f(c) is called a global minimum is f(c) < f(x) for all x . We use the term extremum to indicate either a max or a min.

V. Concavity  A function is concave up if the graph is an upward bowl and concave down if it is a downward bowl.  If the concavity of f changes at c, then we say that the graph has an inflection point at x = c.

Definition:  We define the second derivative as the derivative of the derivative. we denote it by f'' .  In liebnitz notation we will work on it.

We define the acceleration of s(t) as a(t) = s''(t)

Theorem:  A function with f'' < 0 is concave down and a function with f'' > 0 is concave up.