1 Chapter 2 Differentiation: Tangent Lines. tangent In plane geometry, we say that a line is tangent to a circle if it intersects the circle in one point.

1

Chapter 2Chapter 2

Differentiation:Differentiation:

Tangent LinesTangent Lines

Tangent LinesTangent Lines In plane geometry, we say that a line is tangenttangent

to a circle if it intersects the circle in one point. But, for more general curves, we need a better

definition. The concept of tangent lines is essentialessential to your

understanding of differential calculus so we must have an accurate idea of it’s meaning. There are many “rough” ideas of what tangent lines are – many of which are not only rough, but wrong.

Write down what your definition of a tangent line to a curve by completing the sentence:

A line is tangent to a curve _____________________

Tangent LinesTangent Lines There are several incorrect definitions about

tangent lines that we must be careful to avoid:

Definition #1:Definition #1: “A line is tangent to a curveA line is tangent to a curve if it crosses the if it crosses the

curve in exactly one pointcurve in exactly one point.” It’s possible for a line to touch at one point and

NOTNOT be tangent.

WRONGWRONG!!

This line is NOTNOT a tangent line.

x

y

x

y

Tangent LinesTangent Lines There are several misconceptions about

tangent lines that we must be careful to avoid:

Definition #2:Definition #2: “A line is tangent to a curveA line is tangent to a curve if it touches the if it touches the

curve only oncecurve only once.” It’s possible for a tangent line to touch a curve at

multiple points.

WRONGWRONG!!

This is a tangent line.

But it crosses in two places

x

y

Tangent LinesTangent Lines There are several misconceptions about

tangent lines that we must be careful to avoid:

Definition #3:Definition #3: “A line is tangent to a curveA line is tangent to a curve if it touches the if it touches the

curve at only one point but does not cross curve at only one point but does not cross the curvethe curve.”

It’s possible for a line segment to touch a curve at only one point and not cross and still NOTNOT be a tangent.

WRONGWRONG!!

This is NOTNOT a tangent line

x

y

Tangent LinesTangent Lines There are several misconceptions about

tangent lines that we must be careful to avoid:

Definition #4:Definition #4: “A line is tangent to a curveA line is tangent to a curve if it ‘grazes’ the if it ‘grazes’ the

curve at one point but does not cross at curve at one point but does not cross at that pointthat point.”

It’s possible for a tangent to cross our curve at one point.

WRONGWRONG!!

This is a tangent line

An Informal DefinitionAn Informal DefinitionAt this point, it’s difficult for us to get a clear definition of a tangent line to a curve. So we need to rely on our general knowledge of a tangent line. It’ll take some time before we can understand it’s true definition.However, since I always have students that

must see the true definition, here it is:

A straight line is said to be a A straight line is said to be a tangenttangent of of a curve a curve y y = = f f ((xx) at a point ) at a point xx = = cc on the on the curve if the line passes through the curve if the line passes through the point P(point P(cc, , f f ((cc)) on the curve and has )) on the curve and has slope slope f f ‘(c) where ‘(c) where ff '' is the derivative of is the derivative of ff……now, I know you feel much better…

Example:Example:

1point at the 2)( xxxf

Try to sketch the tangent line to the curves at the indicated point:

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

x

y

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

x

y

Example:Example:

0point at the 1)( 2 xxxf

Try to sketch the tangent line to the curves at the indicated point:

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

x

y

Example:Example: Try to sketch the tangent line to the curves

at the indicated point:

1point at the 133)( 23 xxxxxf

Example:Example: Try to sketch the tangent line to the curves

at the indicated point:

xxxf point at the sin)(

-1

1

x

y

The Key to the Tangent Line to The Key to the Tangent Line to a Curvea Curve

Although it may be difficult for us to come up with a good definition for the tangent line to a curve, we get a little help from Newton and Leibniz and we just need to use their definition.

There are two key components to the definition of the tangent line: limits and limits and the slopethe slope!

But before we dive into studying the tangent line of a curve, we need to make sure we remember the slope.

Don’t forget the basic formula for slope between two points: run

rise

12

12

x

y

xx

yym

Practice ProblemsPractice Problems Determine the slope of the line that passes

through the following two points:1. (-4, 0) and (8, 0)2. (-2, -5) and (3, 15)3. (3, -2) and (-1, 1)

Sketch the graph, then approximate the slope of the tangent line for each function at the given point (you may use your calculator to graph the function):1. f (x) = (x – 3)2 – 1 at the x = 32. f (x) = x4 – x2 at the x = -13. f (x) = ln x at the x = e

HomeworkHomework Tangent Lines

Worksheet: Tangent Lines