MAT 1234
Calculus I
Section 1.4
The Tangent and Velocity
Problems
http://myhome.spu.edu/lauw
WebAssign



Homework 1.4
Good News…Only 2 problems…
Bad News...No tutoring today
(If you need help over the weekend, I
posted a HW guide online)
Two Worlds and Two Problems
The Tangent Problem
The Velocity Problem
?
Real World
Abstract World
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
60 miles/hour at t=40s
30 ml/s at t=5s
-30ml/s at t=5s
-$5/min at t=8:05am
Context
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
Slope=?
y
y  f ( x)  0.5 x 2
1
x
3
Example 1 The Tangent Problem
Slope=?
y
We are going to use an
“limiting” process to
“find” the slope of the
tangent line at x=1.
y  f ( x)  0.5 x 2
1
x
3
Example 1 The Tangent Problem
Slope=?
y
First we compute the
slope of the secant line
between x=1 and x=3.
y  f ( x)  0.5 x 2
1
x
3
Example 1 The Tangent Problem
Slope=?
y
Then we compute the
slope of the secant line
between x=1 and x=2.
y  f ( x)  0.5 x 2
1
2
x
3
Example 1 The Tangent Problem
Slope=?
y
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.
y  f ( x)  0.5 x 2
1
2
x
3
Example 1 The Tangent Problem
Slope=?
y
First we compute the
slope of the secant line
between x=1 and x=3.
y  f ( x)  0.5 x 2
f (3)  f (1)
f (3)  f (1)

3 1
2
0.5  32  0.5 12

2
2
Slope 
1
x
3
Example 1 The Tangent Problem
Slope=?
y
First we compute the
slope of the secant line
(1  h)  f (1)
between x=1 andfx=3.
y  f ( x)  0.5 x 2
f (3)  f (1)
f (3)  f (1)

3 1
2
0.5  32  0.5 12
h

2
2
Slope 
x
1
3
2
Example 1 The Tangent Problem
Let us record the results
in a table.
y
f (1  h)  f (1)
slope 
h
y  f ( x)  0.5 x 2
x
1
3
h
h
2
1
0.1
0.01
slope
2
Example 1 The Tangent Problem
We see from the table
that the slope of the
tangent line at x=1 should
be _________.
y
y  f ( x)  0.5 x 2
x
1
3
h
Limit Notations
When h is approaching 0,
approaching ___.
We say as h0,
Or,
f (1  h)  f (1)
h
f (1  h)  f (1)

h
f (1  h)  f (1)
lim

h 0
h
is
Definition
For the graph of y  f ( x) , the slope of the
tangent line at x  a is
f ( a  h)  f ( a )
lim
h 0
h
if it exists.
Two Worlds and Two Problems
The Velocity Problem
The Tangent Problem
y  f ( x)
xa
f ( a  h)  f ( a )
lim
h 0
h
Real World
Abstract World
Example 2 The Velocity Problem
y = distance dropped (ft)
t = time (s)
2
Displacement Function y  f (t )  16t
(Positive Downward)
t 2
Find the velocity of the ball at t=2.
Example 2 The Velocity Problem
Again, we are going to use the same
“limiting” process.
t 2
Find the average velocity of the ball
from t=2 to t=2+h by the formula
t  2h
Average velocity 

distance traveled
time elapsed
f (2  h)  f (2)
h
f (t )  16t 2
Example 2 The Velocity Problem
t
2 to 3
h
1
t 2
t  2h
2 to 2.1
2 to 2.01
Average Velocity
(ft/s)
f (3)  f (2)

1
0.1
f (2.1)  f (2)

0.1
0.01
f (2.01)  f (2)

0.01
2 to 2.001 0.001
f (2.001)  f (2)

0.001
Example 2 The Velocity Problem
We see from the table that
velocity of the ball at t=2 should
be _________ft/s.
t 2
t  2h
Limit Notations
When h is approaching 0,
approaching____.
We say as h0,
Or,
f (2  h)  f (2)
h
f (2  h)  f (2)

h
f (2  h)  f (2)
lim

h 0
h
is
Definition
For the displacement function y  f (t ) , the
instantaneous velocity at t  a is
f ( a  h)  f ( a )
lim
h 0
h
if it exists.
Two Worlds and Two Problems
The Velocity Problem
t 2
y  f (t )
ta
lim
h 0
f ( a  h)  f ( a )
h
Real World
The Tangent Problem
y  f ( x)
xa
f ( a  h)  f ( a )
lim
h 0
h
Abstract World
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.