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MAT 1234Calculus I
Section 1.4
The Tangent and Velocity Problems
http://myhome.spu.edu/lauw
WebAssign
Homework 1.4 (Only 2 Problems!) Have a nice weekend!
Two Worlds and Two Problems
Real World Abstract World
The Velocity Problem The Tangent Problem
?
What do we care?
How fast “things” are going• The velocity of a particle
• The “speed” of formation of chemicals
• The rate of change of a population
What is “Rate of Change”?Rate of Change Context
60 miles/hour at t=40s
30 ml/s at t=5s
-30ml/s at t=5s
-$5/min at t=8:05am
What is “Rate of Change”?
We are going to look at how to understand and how to find the “rate of change” in terms of functions.
The Problems
The Tangent Problem The Velocity Problem
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y Slope=?
1 3
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
We are going to use an “limiting” process to “find” the slope of the tangent line at x=1.
Slope=?
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
First we compute the slope of the secant line between x=1 and x=3.
Slope=?
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
Then we compute the slope of the secant line between x=1 and x=2.
Slope=?
2
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
As the point on the right hand side of x=1 getting closer and closer to x=1, the slope of the secant line is getting closer and closer to the slope of the tangent line at x=1.
Slope=?
2
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
First we compute the slope of the secant line between x=1 and x=3.
Slope=?
2 2
(3) (1) (3) (1)Slope
3 1 2
0.5 3 0.5 1
22
f f f f
Example 1 The Tangent Problem
First we compute the slope of the secant line between x=1 and x=3.
Slope=?
2 2
(3) (1) (3) (1)Slope
3 1 2
0.5 3 0.5 1
22
f f f f
2( ) 0.5y f x x
x
y
1 32
(1 ) (1)f h f
h
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
Let us record the results in a table.
h
h slope2 21
0.10.01
(1 ) (1)f h f
h
slope
Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
We see from the table that the slope of the tangent line at x=1 should be _________.
h
Limit Notations
When h is approaching 0, is approaching ___.
We say as h0,
Or,
(1 ) (1)f h f
h
0
(1 ) (1)limh
f h f
h
(1 ) (1)f h f
h
Definition
For the graph of , the slope of the tangent line at is
if it exists.
h
afhafh
)()(lim
0
( )y f x
x a
Two Worlds and Two Problems
Real World Abstract World
The Velocity Problem The Tangent Problem
h
afhafh
)()(lim
0
( )y f x
x a
Example 2 The Velocity Problem
y = distance dropped (ft)t = time (s)Displacement Function(Positive Downward)
Find the velocity of the ball at t=2.
2( ) 16y f t t
2t
Example 2 The Velocity Problem
Again, we are going to use the same “limiting” process.
Find the average velocity of the ball from t=2 to t=2+h by the formula
2t
2t h
(2 ) (2)f h f
h
distance traveledAverage velocity
time elapsed
Example 2 The Velocity Problem
2t
2t h
t h Average Velocity (ft/s)
2 to 3 1
2 to 2.1 0.1
2 to 2.01 0.01
2 to 2.001
0.001
2( ) 16f t t
(3) (2)
1
f f
(2.1) (2)
0.1
f f
(2.01) (2)
0.01
f f
(2.001) (2)
0.001
f f
Example 2 The Velocity Problem
2t
2t h
We see from the table that velocity of the ball at t=2 should be _________ft/s.
Limit Notations
When h is approaching 0, is approaching____.
We say as h0,
Or,
(2 ) (2)f h f
h
0
(2 ) (2)limh
f h f
h
(2 ) (2)f h f
h
Definition
For the displacement function , the instantaneous velocity at is
if it exists.
h
afhafh
)()(lim
0
( )y f t
t a
Two Worlds and Two Problems
Real World Abstract World
The Velocity Problem The Tangent Problem
h
afhafh
)()(lim
0
( )y f x
x a
2t ( )y f t
t a
h
afhafh
)()(lim
0
Review and Preview
Example 1 and 2 show that in order to solve the tangent and velocity problems we must be able to find limits.
In the next few sections, we will study the methods of computing limits without guessing from tables.
