Pre-Calculus

Limits

Calculus

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

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