2.1 The Derivative and the Tangent Line Problem(Part 1)

Great Sand Dunes National Monument, ColoradoGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

Objectives

• Find the slope of the tangent line to a curve at a point.

• Use the limit definition to find the derivative of a function.

Origins of CalculusCalculus grew out of four major problems that European mathematicians were working on during the 17th century:

1.Tangent line problem

2.Velocity and acceleration problem

3.Minimum/maximum problem

4.Area problem

How do you define a tangent line to a curve?

Can cross the curve

Can touch at more than one point

Slope of Tangent Line

As Q approaches P, the slope of the secant line approaches the slope of the tangent line.

tan0

0

( ) ( )lim

( )

( ) ( )lim

x

x

f c x f cm

c x c

f c x f c

x

P (c,f(c))

Q (c+∆x,f(c+∆x))

∆x

Slope of Tangent Line

tan

( ) ( )limx c

f x f cm

x c

P (c,f(c))

Q (x,f(x))

Slope of Tangent Line

tan

( ) ( )'( ) lim

x c

f x f cm f c

x c

tan 0

0

( ) ( )'( ) lim

OR

( ) ( )lim

x

h

f c x f cm f c

x

f c h f c

h

Standard definition of derivative at a point

Alternate definition of derivative at a point

Definition of Tangent Line

The definition of a tangent line to a curve doesn’t cover a vertical tangent line. If m=∞ or -∞ or

Then there is a vertical tangent line at x=0.

0

( ) ( )limh

f c h f c

h

Definition of the Derivative of a Function

If you let (x,f(x)) represent an arbitrary point on the graph, the definition of the derivative of a function is

0

0

( ) ( )'( ) lim

( ) ( )lim

x

h

f x x f xf x

xf x h f x

h

Derivative at a Point (c,f(c)) Derivative of a function

Derivative

0

( ) ( )'( ) lim

h

f c h f cf c

h

0

( ) ( )'( ) lim

h

f x h f xf x

h

( ) ( )'( ) lim

x c

f x f cf c

x c

( ) ( )

'( ) limt x

f t f xf x

t x

Standard

formA

lternate form

Terminology

• The process of finding a derivative is called differentiation.

• A function is differentiable at x if its derivative exists at x.

• A function is differentiable on (a,b) if it’s differentiable at every point in the interval.

• f ′ (x) is read “f prime of x”.

f x “f prime x” or “the derivative of f with respect to x”

y “y prime”

dy

dx“dee why dee ecks” or “the derivative of y with

respect to x”

df x

dx“dee dee ecks uv eff uv ecks”

or “the derivative of f of x”( of of )d dx f x

dx does not mean d times x !

dy does not mean d times y !

dy

dx does not mean !dy dx

(except when it is convenient to think of it as division.)

df

dxdoes not mean !df dx

(except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.)

df x

dxdoes not mean times !

d

dx f x

In the future, all will become clear.

ExampleFind the slope of the graph of f(x)=2x-3 at the point (2,1).

Examplef(x)=x3+2x Find f ′(x).

Exampley = 2 / t Find dy/dt.

0 0

( ) ( ) ( ) ( )If lim then lim .

h h

f x h f x f t h f tdy dydx dth h

Homework

2.1 (page 103)

#5-9 odd

11,13, 17-21 odd

25-29 odd (part a only)

( ) ( )Use '( ) lim

x c

f x f cf c

x c

0

( ) ( )Use '( ) lim

h

f x h f xf x

h