1004
Secants and Tangents – The Derivative
The 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.
Speed ( avg ) 
d d 2  d1

t
t2  t1
Suppose you make a weekend trip, leaving at 10am and arriving at 12 nn. The trip is a total of 100 miles.
Find the average speed on the trip. _______________
You pass a police car at 11 am, how fast were you traveling at 11am? ______________________
You cannot find the speed at an instant (instantaneous rate of change). Why
________________________________________________________________________________________
The following chart gives the distance traveled on smaller and smaller time intervals.
The smallest time interval suggests that at 11am you were traveling at approximately __________ mph.
Mathematically this is modeled by the slope (rate of change) function.
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.
f ( avg ) 
y y2  y1

x x2  x1
We will say that as Q gets closer to P , that the secant line PQ gets closer to the tangent line through P and thus
the slope of the secant line (msec) approaches the slope of the tangent line (mtan).
Example: Suppose f(x) = x 2 . Find the slope of the secant line between x = 1 and the value of x closer and
closer to 1 ( Set your calculator to full float.)
1
1
What is your best guess for the slope of the tangent line at x = 1? _____________
The instantaneous rate of change  d  or the slope of the tangent line is called the DERIVATIVE. The


 0 
derivative can be approximated numerically by choosing ever smaller intervals ( as above ) or graphically.
Ex. 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 slope of the secant line (units)? ______________________
Estimate the value of the rate of change at t = 10.
t
T
y


0
73
2
73
4
70
6
69
8
72
10
81







x












10
.
Approximating the derivative using slopes.
y’(-4)
y




y’ (-1)
x













y’(3)
Approximating the derivative using tables.
Traffic speed S along a certain road ( in mph) varies as a function of traffic density Q (number of cars per
mile on the road). Use the data to estimate the derivative.
Q (density)
100
110
120
130
140
S (speed)
45
42
39.5
37
35
One sided (at an endpoint):
Symmetric (at an interior point):
S’(100)
S’ (130)
S’(140)
S’(115)
CALCULUS :
m secant
m tangent
Slope of a Tangent.
(in function notation)
< Calculus – with the Limit >
Layman’s Description of THE DERIVATIVE:
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
A. THE DERIVATIVE (AT A POINT)
EX 1: Find the THE DERIVATIVE AT A POINT
f ( x)  2 x 2  4 x  1
at
x2
Notation:
Words: (our new Mantra!)
What does the DERIVATIVE find ? ____________________________________________________________
EX 2: Find the Equation of the Tangent Line.
To write the equation of a line you need: a)
b)
Point- Slope Form:
Find the Equation of the Tangent Line to f ( x)  x 2  2 x
at
x4
The normal line to a curve at a point is the line perpendicular to the tangent at the point.
Find the Equation of the Normal.
Assignment 1004A:
FDWK p.105
Use the definition of Derivative at a Point ,
lim
x a
f ( x)  f (a)
xa
, to find the derivative of the given
function at the indicated point.
1
, a2
x
5)
f ( x) 
7)
f ( x)  x  1 , a  3
6)
f ( x)  x 2  4 , a  1
8)
f ( x)  2 x  3 , a  1
R p. 126
19)
Find the equation of the tangent line at x = 3 , assuming that f  3  5 and f (3)  2 .
Find the derivative at the given point and write the equation of the tangent line to the curve at the given point.
27)
f ( x)  2 x2 +10 , a  3
32)
f ( x)  2 x3 +4x , a  4
38)
f ( x)  3x  5 , a  1
43)
f ( x) 
1
, a0
x 1
2
THE DERIVATIVE FUNCTION:
Graphing f ’ (x)

y









y
x



















x











x=0
m = _______
x=2
m = _______
x=4
m = _______
x=6
m = _______
x=8
m = _______
x = 10
The Derivative is a _________________ function.
m = _______
FDWK #30/107 The elevations (y) at various distances (x) downriver from the start of a kayaking route
on the Bear Creek in Georgia are given in the table.
Distance 0
0.56
(miles)
Elevation 1577 1512
(feet)
0.92
1.19
1.30
1.39
1.57
1.74
1.98
2.18 2.41 2.64 3.24
1448
1384
1319
1255
1191
1126
1062 998
933
869
a) Sketch the graph of the elevation (y) as a function of distance downriver (x).
b) Using the midpoint of each interval and the slope on the interval to create a table of the derivative
relationship. What unit of measure would be appropriate for the derivative (y’)?
Distance
(miles)
c) Sketch the graph of the rate of change of elevation (y’) as a function of distance downriver (x).
d) How would you identify the most dangerous section of the river (ignoring rocks) by analyzing the
graph of (a).
e) How would you identify the most dangerous section of the river by analyzing the graph of (c).
805
THE DERIVATIVE FUNCTION
ANYWHERE on the graph!
*Not at a specified point
m secant
(in function notation)
Sometimes called: __________________________
m tangent
x
< Calculus – with the Limit >
THE DEFINITION OF DERIVITIVE
(function form):
The DERIVATIVE is a ____________________________ function built on the _________________ function.
VOCABULARY:
DIFFERENTIATE means to find the derivative.
EX: Find THE DERIVATIVE FUNCTION
f ( x)  x 2  2 x
< 4 Step Method >
Use the Definition and the 4 Step Method to find the Derivative Function.
Use the Derivative to evaluate at
x = -2,
x = 5,
x=0
f ( x)  x3  12 x
f ( x) 
1
x2
f ( x)  x
Differentiability at a Point insures ________________________ and __________________
Local Linearity
Very close to the point a curve behaves like a line
Where the Derivative DNE.
A) Where f ( a ) Does Not Exist.
y



B) At the Endpoint of an Interval.

REM: The Derivative is a Limit.


x

C) At a VERTICAL TANGENT.
f ( x)  3 x
at
(0, 0)




D) At a CUSP (Sharp Turn).
f ( x)  x
at
(0,0)
Assignment 1004B:
FDWK p. 105
The table gives the approximate distance traveled by a downhill skier after t seconds for 0  t  10 .
Time
0
(seconds)
Distance 0
(feet)
1
2
3
4
5
6
7
8
9
10
3.3
13.3
29.9
53.2
83.2
119.8 163.0 212.9 269.5 332.7
Use the table to approximate the derivative at the the following times – include the units.
a)
t = 10 seconds
b)
22) Given the graph of the function f below,
sketch a graph of the derivative of f .
Use the definition of Derivative Function,
lim
h 0
t = 4 seconds
c)
t = 6.5 seconds
26) Given the graph of the function f below,
made up if segments
a) Sketch a graph of the derivative of f .
b) At what values of x between x = -4 and x = 6 is the
function not differentiable?
f ( x  h)  f ( x )
h
, to find the derivative of the given
function.
R p. 128
28)
36)
f ( x)  4  x 2
f ( x) 
2
1 x
29)
f ( x)  x  2 x 2
37)
f ( x)  4  x
30)
f ( x)  8 x 3
Each limit represents a derivative f ‘ (x) . Find f (x) and the point, a.
51)
(5  h)3  125
lim
h0
h
53)
sin( 6  h)  12
lim
h0
h
55)
(5) 2h  25
lim
h0
h
FDWK p. 114
52x)
( x)3  27
lim
x3
x 3
54)
( x ) 1  4
lim1
x 4
x  14
56)
(5) x  1
lim
x 0
x