Download Lesson 7

Lesson 7 – Derivatives at a Point, Numerical Derivatives, and Applications
Many problems are all about a “rate of change” at a specific number. The number
will express problem dependent meaning, of course. To answer the question and
provide the requisite management information, we’ll take a derivative and evaluate
it at the requested x value.
Example: Productivity on the assembly line
A study conducted for a manufacturing company showed that the number of lawn
chairs assembled by the average worked t hours after starting work at 6 a.m. is
given by:
N (t )  t 3  7t 2  18t
How did they get this function?
What does N’(t) tell us?
The rate at which the average worker will be assembling chairs at t hours
What is the derivative?
What is the rate at noon?
(rescaling!)
Of course GGB does this! Use “derivative”, and then evaluate it at the correct x
value.
1
Let’s call up GGB and find
f '(4)
f ( x)  x3/2  2 x
Popper 05
Question 1
2
Example
C ( x)  0.0000003x3  x 2  .019 x  .0022
Find the numerical derivative when x = 2550.
Change the sig digits: Options/Rounding/10places
3
What’s going on with the graphics view?
Endpoints? The function’s motions are WAY too big for the screen!
Popper 05
Question 2
4
Another mechanical example:
Find the slope of the tangent line at x = − 1 for f ( x) 
x x2
s
(2 x  3)2
Definitely a GGB thing.
Ans: 4.5
see below
5
Equations of tangent lines – a 3 step process.
A
Get the point – both coordinates as numbers, use the function
B
Get the derivative and evaluate it at x, this is the slope
C
Use the point slope formula to get the tangent line equation
Note: a tangent line shares ONE POINT with the graph!
Example:
1
f ( x)  x3  2 x 2  7 x
3
f '( x)  x 2  4 x  7
Find the equation of the tangent line to f (x) at x = 3.
A
the point is (3, 12)
how come 12?
B
the slope is 4
how come 4?
C
y  y0  m( x  x0 )
6
On your own:
f ( x) 
A
5 3 8
x 
3
3
x=−1
Find the equation of the tangent line there
the point pair is
B
f '( 1) =
C
y  y0  m( x  x0 )
Ans: y  5 x  6
7
Popper 05
Question 2
8
Another GGB one
Write an equation of the line that is tangent to the function at x = 3.75
f ( x)  1.6 x3  2.31x 2  6.39 x  2.81
The point is (3.75, 73.043125)
Slope of the tangent line is 56.565
Equation
y − 73.043125 = 56.565(x – 3.75)
ugly!
What graphs are in the graphics view? Where would the line show up?
9
Popper 05
Question 4
10
Application problems:
Sometimes you need a time or quantity at which the graph changes direction from
increasing to decreasing or vice versa.
At these points, the slope of the tangent line is 0.
So, for example, productivity stops going down at 0 and makes a turn around…
Now, to find these turn around points, set the derivative = 0 and solve for x.
We will be finding the ROOTS of the derivative for this exercise and applying the
information to the function!
11
Let’s find the derivative and solve for the roots to find the x at which the parabola
makes a turnaround:
Example
Given
f ( x) 
1 3
x  4 x 2  15 x  19
3
Find out where the grapf of f has turn around points!
12
Popper 05
Question 5
13
Example
1
3
3
2
Find all points on the graph of f ( x)  x3  x 2  7 x  3 for which the derivative is 3
(i.e. f '( x)  3 ).
Get the derivative, set it equal to 3 and solve for x.
Example
14
A country’s GDP in millions of dollars is modeled by the function
G( x)  2 x3  45x 2  20 x  600
with x = 0 corresponding to the beginning of 1997.
What is the AROG (average rate of growth) over the period 1999 – 2004?
Remember for average use the Difference Quotient!
Initial value:
Final value:
G (7)  G (2) 7659  6204

 291
5
5
At what rate was GDP changing at the beginning of 2002?
Remember this is then the numerical derivative
f '( x)  6 x 2  90 x  20
f '(5)  320
Popper 05
Question 6
15
Example
Let’s identify WHICH equation to use for each question!
The equation
N (t )  34.4(1  0.32125t )0.15 gives the number of people in the US
who are between the ages of 45 and 55. N(t) is in millions and t = 0 corresponds to
the beginning of 1995.
How large is this segment of the population projected to be at the beginning of
2011?
How fast will this segment of the population be growing at the beginning of 2011?
Popper 05
Question 7
16
Example
Velocity
Velocity is a rate of change and is a common application of the derivative of
distance. The following two formulas are for velocity. Note that v0 is initial
velocity and h0 is the initial height in each formula. The first formula is used if the
distances are measured in feet and the second formula is used if the distances are
measured in meters.
h(t )  16t 2  v0t  h0
h(t )  4.9t 2  v0t  h0
Suppose you are standing on top of a building that is 28 meters high. You throw a
ball up in the air with an initial velocity of 10.2 meters per second. Write the
equation that gives the height of the ball at time t.
Which equation? What numbers get filled in where?
Find the velocity of the ball at t = 2.
What are the steps?
On your own:
−9.4 m/s is the answer!
17
Higher order derivatives
Sometimes we need to find the rate at which the rate is changing…in other words,
we need the derivative of the derivative! Since the first derivative is itself a
function finding the second derivative is not hard to do. We call this f ''( x) .
Example
Given
f ( x)  4 x 5  x 2  7 x  5
Find
f ''(3)
Ans: −2162
Example
Find the value of the second derivative when x = 5 if f ( x) 
x 2 ln x
1
( x 2  3) 3
GGB!
18
Answer: .829666212
rounded above in GGB (use options to change the number of digits)
Popper 05
Question 8
19
Popper 05
Question 9
Popper 05
Question 10
20