Pre-Calculus Limits Calculus. Objectives: 1.Discuss slope and tangent lines. 2.Be able to define a...
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Transcript of Pre-Calculus Limits Calculus. Objectives: 1.Discuss slope and tangent lines. 2.Be able to define a...
Objectives:1. Discuss slope and tangent lines.2. Be able to define a derivative.
3. Be able to find the derivative of various functions.
Critical Vocabulary:Slope, Tangent Line,
Derivative
I. Slopes of Graphs
What are the slopes of the following linear functions?
How have you defined slope in the past?
I. Slopes of Graphs
What about other functions?
In these functions the slopes vary from point to point
I. Slopes of Graphs
To find the rate of change (slope) at a single point on the function, we can find a tangent line at that point
Recall Circles: A line was tangent to a circle if it intersected the circle ONCE.
I. Slopes of Graphs
To find the rate of change (slope) at a single point on the function, we can find a tangent line at that point
With Curves (functions), it is a little different. We can be concerned with the line of tangency at a specific point, even if the line would intersect the function someplace else.
Is the slope the same at each of these arbitrary
points?
Even though the tangent line is touching the
graph someplace else, we are only
describing the slope at the point
of tangency.
II. Defining a Derivative
We will be using the idea of limits to help us define a tangent line. Important things to know:
1. (x, y) is the same as (x, f(x))
2. What does Δx represent? Change in x-values
Let’s look at the slope formula:
12
12
xx
yym
Define our points:
A: (x, f(x))
B: (x + Δx, f(x + Δx))
Find the slope between points A and B
xxx
xfxxfm
)()(
x
xfxxfm
)()(
This is the Difference Quotient
x
xfxxfx
)()(lim
0
II. Defining a Derivative
The slope of the tangent line at any given point on a function is called a derivative of the function and is defined by:
x
xfxxfx
)()(lim
0
III. Finding the Derivative
Example 1: Find the derivative of f(x) = -2x + 4 using the definition of the derivative.
x
xfxxfx
)()(lim
0
x
xxxx
)42(4)(2lim
0
x
xxxx
42422lim
0
x
xx
2lim
0
2lim0
x
2
This is the general rule to find the slope at any
given point on the graph.
III. Finding the Derivative
Example 2: Find the derivative of f(x) = x2 + 1 using the definition of the derivative.
x
xfxxfx
)()(lim
0
x
xxxx
)1(1)(lim
22
0
x
xxxxxx
112lim
222
0
x
xxxx
2
0
2lim
xxx
2lim0
x2
This is the general rule to find the slope at any
given point on the graph.
What is the slope of the tangent line at the point (-1,
2)? (2, 5)?(-1, 2): Slope would be -2(2, 5): Slope would be 4
III. Finding the Derivative
1. 2x was the DERIVATIVE of f(x) = x2 + 1. This means it is the general rule for finding the slope of the tangent line to any point (x, f(x)) on the graph of f.
2. We write this by saying f’(x) = 2x. We say this “f prime of x is 2x”
3. The process of finding derivatives is called DIFFERENTIATION
Notations: xxf 2)('
xdx
dy2
xy 2'
xxfdx
d2)(
xx
yx
2lim0
III. Finding the Derivative
Example 3: Find the derivative of f(x) = 3x2 – 2x using the definition of the derivative.
x
xfxxfx
)()(lim
0
x
xxxxxxx
)23()(2)(3lim
22
0
x
xxxxxxxxx
)23()(2)2(3lim
222
0
x
xxxxx
236lim
2
0
236lim0
xxx
26 x This is the general rule to find the slope at any given point on the
graph.
x
xxxxxxxxx
2322363lim
222
0
III. Finding the Derivative
Example 4: Find the slope of g(x) = 5 - x2 at (2, 1)
x
xfxxfx
)()(lim
0
x
xxxx
)5()(5lim
22
0
x
xxxxxx
)5()2(5lim
222
0
x
xxxx
2
0
2lim
xxx
2lim0
x2 This is the general rule to find the slope at any given point on the
graph.
x
xxxxxx
222
0
525lim
What is the slope of the tangent line at (2, 1)?
(2, 1): Slope would be -4
III. Finding the DerivativeExample 5: Find the equation of the tangent line to the
graph of f(x) = x2 + 2x + 1 at the point (-3, 4).
x
xfxxfx
)()(lim
0
x
xxxxxxx
)12(1)(2)(lim
22
0
x
xxxxxxxxx
)12(1)(2)2(lim
222
0
x
xxxxx
22lim
2
0
22lim0
xxx
22 x
x
xxxxxxxxx
121222lim
222
0
What is the slope of the tangent line at (-3, 4)?
-4
y = mx + b
4 = (-4)(-3) + b
4 = 12 + b
-8 = b
f(x) = -4x - 8