Download Section 4 - Introduction Handout

Calculus
The Derivative
Chapter 3
Section 4
Definition of a Derivative
Essential Question: How do you calculate a derivative? When can a derivative not exist?
Student Objectives: The student will determine the derivative of a function at a given point.
The student will state the four conditions of a function that may not be
differentiable.
The student will determine the tangent and normal line of a function at a
given point.
The student will determine the equation of the derivative of a function to
be used for any given point.
Terms:
Corner
Cusp
Derivative
Discontinuous
nDeriv
Normal line
Secant line
Slope
Tangent line
Vertical Tangent
Key Concepts:
Existance of a Derivative
A derivative of a function exist if the function is continuous at the point.
You must show the following to prove continuity at point a:
lim f ( a ) = f ( a ) = lim+ f ( a )
x→a −
x→a
The function does not have a derivative at point a if all three of these
expression are not equal, because the function is not continous at point a.
The function has a derivative a point a if the function is continuous at point a
and
f '( a− ) = f '( a+ )
lim−
x→a
f ( x) − f (a)
f ( x) − f (a)
= lim+
.
x→a
x−a
x−a
The function fails to have a derivative at point a if:
1. The function is not continuous at x = a. The function is discontinuous.
2. The function has a corner at x = a. A corner exist if the f ' ( a − ) ≠ f ' ( a + )
and the derivatives from the left and right of point a are both numeric
3
but not equal. Ex: 4 ≠ - .
5
3. The function has a vertical tangent at x = a. The vertical tangent exists
if f ' ( a − ) = f ' ( a + ) and the derivatives from the left and right of point a
are both either ∞ or –∞.
4. The function has a cusp at x = a. The cusp exists if f ' ( a − ) ≠ f ' ( a + ) and
the derivatives from the left and right of point a are both ∞ and –∞ or
they are equal to -∞ and ∞. The slopes are both infinities but not the
same infinity.
Average Rate of Change (Slope of a Secant Line)
The average rates of change of f ( x ) with respect
to x for a function f as x changes from a to b is:
f (b ) − f ( a )
.
b−a
The secant line is a line through two points of a function.
m=
Instantaneous Rates of Change
The instantaneous rates of change for a function
f when x = a is:
f (a + h) − f (a)
h→0
h
provided the limit exists.
f ' ( x ) = lim
Instantaneous Rates of Change ( Alternate Form )
The instantaneous rates of change for a function
f when x = a is:
f ( x + a) − f (a)
x−a
provided the limit exists.
f ' ( x ) = lim
x→a
Slope of a Tangent Line
The slope of a tangent line of a grapgh of y = f ( x ) at a point ( a, f ( a ))
is a line through this point having the slope of m, where
f ( x + h) − f ( x)
h
provided the limit exists. If the limit does not exist, then there is no
tangent line at this point.
m ( x ) =lim
h→0
Slope of a Normal Line
The normal line is perpendicular to the tangent line.
Graphing Calculator Skills:
Determine the derivative of a function at a point x = a.
Place the function into y1 in the calculator.
Graph the function in an appropriate window.
Press the 2nd button and then the CALC button.
Select option 6 (dy/dx).
Type in the desired x value.
dy/dx will be displayed at the bottom of the screen.
Determine the derivative of a function at point a. (2nd option).
Start in the HOME screen.
Press Math.
Select option nDeriv(
Older version
nDeriv( Function, x , x-value)
nDeriv (7x +3, x, 4) = 7
Newer version
d
3x 2 − 5x + 2 )
(
dx
= 13
x=3
Drawing a tangent line
Place the function into y1 in your calculator.
Graph the function in an appropriate window.
Press the 2nd key and the PRGM key to activate the DRAW options.
Select option 5 Tangent(.
Enter you desired x-value and Press ENTER
The calculator will draw the tangent line for you and it will display the
equation at the bottom of the screen.
Sample Questions:
f ( x ) = −x 2 + 2x + 8
1. Determine the slope of the tangent line at any given x-value.
2. Determine the equation of the tangent line at x = 3.
3. Use the alternative form of the derivative to determine the slope of the tangent line
at x = -1.
4. Place the equation of the function into the calculator. Draw the graph of the function in
an appropriate window. Make the calculator draw and calculate the tangent line of the
function at x = 1. What is the equation of the tangent line?
5. Use the built-in derivative function on the calculator to determine the value of the
derivative at x = 5. Write down what you typed into your calculator as well as the
answer from the calculator.
6. Repeat steps 1 through 5 for the function given below.
1
f ( x ) = ( 2x 3 − 7x 2 − 139x + 360 )
50
Homework:
Pages 176 - 179
Exercises: #1, 7, 13, 17, 23, 29, 37, 41, 51, and 61
Exercises: #2, 10, 16, 20, 26, 30, 38, 40, 52, and 56