AP CALCULUS

1006: Secants and Tangents

Average Rates of ChangeThe AVERAGE SPEED (average rate of change) of a quantity over a period of time is the amount of change divided by the time it takes.

In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval.

Therefore, the average rate of changecan be thought of as the slope of a secant line to a curve.

2 1( )

2 1avg

d ddSpeedt t t

2 1( )

2 1avg

y yyfx x x

Average Rate of Change

<Pre – Cal> Known Formula 

Average Rate of Change :

secantm (in function notation)

a x

Slope of a Tangent

 

Calculus I -The study of ___________________________________ 

Slope :

secantm (in function notation)

tan gentm

<Calculus – with the Limit>

a x

A. THE DERIVATIVE (AT A POINT)

A. THE DERIVATIVE (AT A POINT) 

   WORDS: Layman’s description:

The DERIVATIVE is a __________________ function.

Let T(t) be the temperature in Dallas( in o F) t hours after midnight on June 2, 2001. The graph and table shows values of this function recorded every two hours,. What is the meaning of the secant line (units)? Estimate the value of the rate of change at t = 10.

x

y

t T0 732 734

706

698 7210 81

EX: THE DERIVATIVE AT A POINT

2( ) 2f x x x EX 1: at a = 4

 

 Notation:

Words:

Equation of the Tangent

To write the equation of a line you need:a)b)

Point- Slope Form:

2( ) 2 at 4f x x x a

2( ) 2f x x x

4(4) 8

af

Tangent8 6( 4)

6 16

y x

y x

Normal to a Curve

The normal line to a curve at a point is the line perpendicular to the tangent at the point.

Therefore, the slope of the normal line is the negative reciprocal of the slope of the tangent line.

EX 1(cont): at a = 4

Find the equation of the NORMAL to the curve

2 2y x x

EX: THE DERIVATIVE AT A POINT

EX 2: at x = -2

  Method:

2( ) 2 4 1f x x x

At a Joint PointPiece Wise Defined Functions:

The function must be CONTINUOUS

Derivative from the LEFT and RIGHT must be equal.

The existence of a derivative indicates a smooth curve; therefore,

3 , 1( )

5 , 1x x

f xx x

Last Update:

• 08/12/10