1 Chapter 2 Differentiation: Tangent Lines. tangent In plane geometry, we say that a line is tangent...
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Transcript of 1 Chapter 2 Differentiation: Tangent Lines. tangent In plane geometry, we say that a line is tangent...
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