**12.3 The Tangent Line Problem**

The Slope of the graph of a function can be used to analyze rates of change at particular points on the graph.

The Slope of a line indicates the rate at which a line rises or falls.

- For a straight line, this rate (or slope) is the same at every point on the line.

- For other graphs than lines, the rate at which the graph rises or falls changes from point to point.

Recall: slope formula = (the change in y's)/(the change in x's) = Δy/Δx

Review: With circles, you were taught about tangent lines. A tangent line is a line that intersects a circle in exactly one point, called the point of tangency. Using this process, to determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at that point.

The tangent line to a graph of a function "f" at a point P(x

_{1}, y

_{1}) is the line that best approximates the slope of the graph at that point.

**Example1:**f(x) = x

^{2}- 2x + 1

find the tangent line at the point (1, f(x))

f(1) = 1

^{2}-2(1) + 1 = 0

so the tangent line at (1,0) is a straight line and the equation of the tangent line at (1, f(x)) is y=0.

A more precise method then "eyeballing" the tangent line

is making use of the secant line through the point of tangency

and a second point on the line where (x, f(x)) is the first point

and (x + h, f(x + h)) is a second point on the graph of "f", the

slope of the secant line through these two points is:

M

_{secant}= (f(x + h) - f(x))/h

Using the limit process, you can find the

exact slope of the tangent line at (x, f(x)).

**Definition of the Slope of a Graph**:

The slope m of the graph of "f" at the point (x, f(x)) is equal to the slope

of its tangent line at (x, f(x)) and is given by:

M =

lim

*m*

_{sec}=

^{h®0}

lim (f(x+h) - f(x))/h

^{h®0}

provided the limit exists!

**Example 2**: given f(x) = 2x + 5, find the slope of the tangent line at (-1, -3)

M

_{sec}= (f(x+h) - f(x))/h

= (f(-1 + h) - f(-1))/h

= (2(-1 + h) + 5 - (2(-1) + 5))/h

= (-2 + 2h + 5 +2 - 5)/h

= (2h)/h

= 2

The graph has a slope of 2 at the point (-1, -3)

To find the equation of the tangent line:

y = mx + b

(-3) = 2(-1) + b

-1 = b

so the equation of the tangent line at point (-1, -3) is

y = 2x -1

**Example 3**: given f(x) = 10x - 2x

^{2}at point (3, 12)

M

_{sec}= [10(x+h) - 2(x + h)

^{2}- (10x - 2x

^{2})]/h

M

_{sec}= [10x + 10h -2(x

^{2}+2xh + h

^{2}) -10x + 2x

^{2}]/h

M

_{sec}= [10h - 2x

^{2}- 4xh - 2h

^{2}+ 2x

^{2}]/h

M

_{sec}= [10h - 4xh - 2h

^{2}]/h

M

_{sec}= 10 - 4x - 2h

Slope of the tangent line = M =

lim M

_{sec}=

^{h®0}

lim 10 - 4x - 2h = 10 - 4x so at point (3, 12), M = 10 - 4(3) = -2

^{h®0}

Therefore the slope of the tangent line at the point (3, 12) is -2.

To find the equation of the tangent line, use the point and the slope:

y = mx + b

12 = (-2)(3) + b

12 = -6 + b

18 = b

y = 2x + 18 is the equation of the tangent line at the point (3, 12)

Finding a Formula for the slope of a Graph:

**Example4:**g(x) = x

^{3}at points (1, 1) and (-2, -8)

M

_{sec}= [(x + h)

^{3}- (x

^{3})]/h

M

_{sec}= [x

^{3}+ 3x

^{2}h + 3xh

^{2}+ h

^{3}- x

^{3}]/h

M

_{sec}= [3x

^{2}h + 3xh

^{2}+ h

^{3}]/h

M

_{sec}= 3x

^{2}+ 3xh + h

^{2}, where h¹0

Next take the limit of M

_{sec}as h approaches 0.

The slope of the tangent line M =

lim 3x

^{2}+ 3xh + h

^{2}= 3x

^{2}

^{h®0}

Now use this M = 3x

^{2}for the slope at (1,1)

M = 3(1)

^{2}= 3

now y = mx + b

1 = 3(1) + b

b = -2

so the equation of the tangent line at point (1,1) is y = 3x - 2

Again now at point (-2, -8)

Now use this M = 3x

^{2}for the slope at (-2,-8)

M = 3(-2)

^{2}= 3(4) = 12

now y = mx + b

-8 = 12(-2) + b

-8 = -24 + b

b = 16

so the equation of the tangent line at point (-2,-8) is y = 12x +16

The formula that you derived from the function f(x) = x

^{3}and used the limit process to get M = 3x

^{2}represents the slope of the graph of "f" at the point (x, f(x)).

The Derived function is called the derivative of f at x. It is donoted by f ' (x), which is reas as "f prime of x".

Definition of the Derivative:

The Derivative of "f" at "x" is

f ' (x) = lim

_{h®0}(f(x+h) - f(x))/h

provided the limit exists!

**Example 5**: f(x) = x

^{2}- 3x + 4

f ' (x) = lim

_{h®0}[(x + h)

^{2}- 3(x + h) + 4 - (x

^{2}- 3x + 4)]/h

f ' (x) = lim

_{h®0}[x

^{2}+ 2xh + h

^{2}- 3x -3h + 4 - x

^{2}+ 3x -4]/h

f ' (x) = lim

_{h®0}[2xh + h

^{2}- 3h]/h

f ' (x) = lim

_{h®0}2x + h - 3 = 2x - 3

f '(x) = 2x - 3

So therefore the derivative of f(x) = x

^{2}- 3x + 4 is f ' (x) = 2x - 3

NOTE that in addition to f ' (x) , other notations can be used to denoted the derivatives of y = f(x). The most common are:

(dy)/(dx)

y'

(d/(dx))[f(x)]

and D

_{x}[y]

Using the derivative of f(x) = x

^{3}- x, find the slope of the tangent line at point (2,6).

f'(x) = lim

_{h®0}[(x + h)

^{3}- (x + h ) - (x

^{3}- x)]'h

f'(x) = lim

_{h®0}[x

^{3}+ 3x

^{2}h + 3xh

^{2}+ h

^{3}- x - h - x

^{3}- x]/h

f'(x) = lim

_{h®0}[3x

^{2}h + 3xh

^{2}+ h

^{3}- h]/h

f'(x) = lim

_{h®0}(3x

^{2}+ 3xh + h

^{2}- 1)

f ' (x) = 3x

^{2}- 1

So at the point (2, 6), the slope is

f ' (2) = 3(2)

^{2}- 1 = 3(4) - 1 = 12 - 1 = 11

So the equation of the tangent line at point (2,6) is

6 = 11(2) + b

6 = 22 + b

b = -16

so the equation of the tangent line at point (2,6) is y = 11x - 16

**Example 6:**Find the derivative of f. Use the derivative to determine any points on the graph of f where the tangent line is horizontal.

f(x) = x

^{3}- 3x

f ' (x) = lim

_{h®0}[(x + h)

^{3}- 3(x + h) - (x

^{3}- 3x)]/h

f ' (x) = lim

_{h®0}[x

^{3}+ 3x

^{2}h + 3xh

^{2}+ h

^{3}- 3x -3h - x

^{3}+ 3x]/h

f ' (x) = lim

_{h®0}[3x

^{2}h+ 3xh

^{2}+ h

^{3}-3h]/h

f ' (x) = lim

_{h®0}[3x

^{2}+ 3xh+ h

^{2}-3]

f ' (x) = 3x

^{2}- 3

Recall a horizontal line has a slope of zero so

f ' (x) = 0 so

0 = 3x

^{2}- 3

3 = 3x

^{2}

1 = x

^{2}

x = ±1

putting these values in for x:

So "f" has horizontal tangents at (1, 0) adn (-1, 0)

**If you need extra help, watch the Chapter 2 lesson 1 tutorial at**

http://www.calculus-help.com/funstuff/phobe.html