The Derivative Function
Bacteria Growth
From looking at the the average rate of change of this function from data collected
every two hours determine:
• Where is the function increasing/decreasing?
• Where is the function concave up/down?
Warming UP
Exercise 7 from Derivative at a Point
Consider the graph below. The domain of the function is all the
real numbers. Assume that outside the window the function
continues the same behavior as the one indicated in the window.
1. Where is f(x)
increasing?
1. Where is f(x)>0?
1. Where is f(x)
concave up?
i. Draw the tangent line at each of the given points and use the
grid to complete the table below. All the answers are estimates
ii. Use interval notation to complete the following information
a. Intervals where the derivative is negative
(solutions to f ‘ (x) < 0)
b. Intervals where f(x) < 0. Describe those points graphically.
c. Intervals where the derivative is positive
(solutions to f ‘ (x) > 0)
d. Intervals where f (x) > 0. Describe those points graphically.
Critical Points of A Continuous
Function
A critical points of a continuous function y=f(x)
is a point in its domain where either f ‘(x)=0 or
f ‘(x) is undefined.
f’(x)=0 when the tangent line is horizontal
f’(x) is undefined at a point in the domain where
the tangent line does not exist (cusp, corner, end
point), or when the tangent line is vertical..
If x0 is not a critical point its derivative exists and either
f ‘(x0) > 0, or f ‘(x) < 0
Exercise 1
The first coordinate of the critical points of each
of the functions below are identified at the top
of each graph. Refer to the definition of a critical
point to explain why it is a critical point. Identify
the type of critical point (f ’=0 or f ’ undefined)
Exercise
http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_as_a_function.html
• Identify all the critical points on the
given domain
• Determine the sign of the derivative
between any two critical points
• Estimate the derivative (draw tangent
lines to find them) at x=-2, 0, 2, 4, 6
• Click on the link above and produce the
graph of its derivative.
• Compare your results with the applet
Derivative Function
Given a function y=f(x) a new function is defined
the following way: to each point in the domain
of the function y = f(x) the value of the
derivative at that point is assigned, or what is
the same the value of the slope of the tangent
line.
This new function is called the derivative
function of y = f(x). The derivative function of
y=f(x) is denoted
dy
y = f '(x) or y =
dx
Derivatives of Basic Functions
1.
2.
3.
4.
Open the down menu (bottom) and choose one of the basic functions
From the graph of the function identify
a. Critical points and classify them
b. Intervals where the derivative is positive
c. Intervals where the derivative is negative
Use the applet (check all the boxes) to generate the derivative function
and verify your answers on part 2
Your next task is to produce the formula for the derivative function:
a. Conjecture the type of graph the derivative would be
b. By choosing points on the graph of the derivative function produce
its formula
More Functions To Find Their
Derivative
Explore the derivative function for each of the
following functions and produce their formulas
f (x) = 5
f (x) = -2.5
f (x) = x
f (x) = -2x
ìï x, x ³ 0
f (x) = x « f (x) = í
ïî -x, x < 0
f (x) = Ln x
f (x) = 3.2x -1
Deriving Basic Derivative Formulas
If f(x)=c, constant f ‘( c )=0
y=m x + b , y ‘=m
y(x)=x2, y ‘(x)=2x
Derivative of a Power Function
y(x) = x , k any constant
k
y¢(x) = k × x
k-1
Exercise 5
Rewrite each of the following functions as a power function.
Use the shortcut for the derivative of power functions to find
the derivative. Give the final answer with positive exponents.
1
a. y = x d. y = x e. y = 2
x
4
3
For each of the functions above find all their critical points
Basic Derivatives