Day 7 - UH Department of Mathematics
Day 7 - UH Department of Mathematics
Day 7 - UH Department of Mathematics
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Math 1314<br />
Lesson 11: Exponential Functions as Mathematical Models<br />
x<br />
x<br />
Exponential models can be written in two forms: f ( x) = a ⋅ e or g ( x) a b<br />
= ⋅ . In the first<br />
model, the base <strong>of</strong> the exponent is the number e, which is approximately 2.71828…. In the<br />
second model, the base <strong>of</strong> the exponent is a positive number other than 1. In both cases, the<br />
variable is located in the exponent, and that’s why these are called exponential models.<br />
Exponential functions can be either increasing or decreasing.<br />
bx<br />
The function f ( x) = a ⋅ e is an exponential growth function and it’s increasing.<br />
bx<br />
The function f ( x) = a ⋅ e − is an exponential decay function and it’s decreasing.<br />
In both equations the variable a is called the initial amount and b is called the growth or decay<br />
constant, depending on the type <strong>of</strong> function.<br />
x<br />
For a function <strong>of</strong> the form g ( x) a b<br />
and is an exponential decay function if 0 < b < 1.<br />
= ⋅ , the function is an exponential growth function if b > 1<br />
We can also compute the rate at which an exponential function is increasing or decreasing.<br />
We’ll do this by finding a numerical derivative.<br />
We can use the regression feature to find an exponential equation for data that’s given.<br />
Example 1: Identify each function as a growth function or a decay function. Find the initial<br />
value. Calculate f (30) and f '(30) .<br />
a.<br />
0.285t<br />
= e −<br />
b. g( x ) = 38.6(1.0489) x<br />
f ( t) 12.8<br />
Example 2: Suppose the points (0, 10000) and (2, 10940) lie on the graph <strong>of</strong> a function. Find<br />
bx<br />
f x = a ⋅ e using GeoGebra, assuming that the<br />
the equation <strong>of</strong> the function in the form ( )<br />
function is exponential.<br />
Begin by entering the two given points in the spreadsheet, then make a list.<br />
Command:<br />
Answer:<br />
Lesson 11 – Exponential Functions as Mathematical Models 1
Uninhibited Exponential Growth<br />
Some common exponential applications model uninhibited exponential growth. This means<br />
that there is no “upper limit” on the value <strong>of</strong> the function. It can simply keep growing and<br />
growing. Problems <strong>of</strong> this type include population growth problems and growth <strong>of</strong> investment<br />
assets.<br />
Example 3: A biologist wants to study the growth <strong>of</strong> a certain strain <strong>of</strong> bacteria. She starts<br />
with a culture containing 25,000 bacteria. After three hours, the number <strong>of</strong> bacteria has grown to<br />
63,000. Assume the population grows exponentially and the growth is uninhibited.<br />
bx<br />
a. Find the equation <strong>of</strong> the function in the form f ( x) = a ⋅ e using GGB.<br />
State the two points given in the problem.<br />
Enter the two points in the spreadsheet, then make a list.<br />
Command:<br />
Answer:<br />
b. How many bacteria will be present in the culture 6 hours after she started her study?<br />
Example 4: The sales from company ABC for the years 1998 – 2003 are given below.<br />
Year 1998 1999 2000 2001 2002 2003<br />
pr<strong>of</strong>its in millions <strong>of</strong> dollars 51.4 53.2 55.8 56.1 58.1 59.0<br />
Rescale the data so that x = 0 corresponds to 1998. Begin by making a list.<br />
a. Find an exponential regression model for the data.<br />
Command:<br />
Answer:<br />
b. Find the rate at which the company's sales were changing in 2007.<br />
Command:<br />
Answer:<br />
Lesson 11 – Exponential Functions as Mathematical Models 2
Exponential Decay<br />
Example 5: At the beginning <strong>of</strong> a study, there are 50 grams <strong>of</strong> a substance present. After 17<br />
days, there are 38.7 grams remaining. Assume the substance decays exponentially.<br />
a. State the two points given in the problem.<br />
Enter the two points in the spreadsheet and make a list.<br />
b. Find an exponential regression model.<br />
Command:<br />
Answer:<br />
c. What will be the rate <strong>of</strong> decay on day 40 <strong>of</strong> the study?<br />
Command:<br />
Answer:<br />
Exponential decay problems frequently involve the half-life <strong>of</strong> a substance. The half-life <strong>of</strong> a<br />
substance is the time it takes to reduce the amount <strong>of</strong> the substance by one-half.<br />
Example 6: A certain drug has a half-life <strong>of</strong> 4 hours. Suppose you take a dose <strong>of</strong> 1000<br />
milligrams <strong>of</strong> the drug.<br />
a. State the two points given in the problem.<br />
Enter the two points in the spreadsheet and make a list.<br />
b. Find an exponential regression model.<br />
Command:<br />
Answer:<br />
c. How much <strong>of</strong> it is left in your bloodstream 28 hours later?<br />
Command:<br />
Answer:<br />
Lesson 11 – Exponential Functions as Mathematical Models 3
Example 7: The half-life <strong>of</strong> Carbon 14 is 5770 years.<br />
a. State the two points given in the problem.<br />
Enter the two points in the spreadsheet and make a list.<br />
b. Find an exponential regression model.<br />
Command:<br />
Answer:<br />
c. Bones found from an archeological dig were found to have 22% <strong>of</strong> the amount <strong>of</strong> Carbon 14<br />
that living bones have. Find the approximate age <strong>of</strong> the bones.<br />
Command:<br />
Answer:<br />
Limited Growth Models<br />
Some exponential growth is limited; here’s an example:<br />
A worker on an assembly line performs the same task repeatedly throughout the workday. With<br />
experience, the worker will perform at or near an optimal level. However, when first learning to<br />
do the task, the worker’s productivity will be much lower. During these early experiences, the<br />
worker’s productivity will increase dramatically. Then, once the worker is thoroughly familiar<br />
with the task, there will be little change to his/her productivity.<br />
The function that models this situation will have the form Q( t)<br />
= C − Ae −kt .<br />
This model is called a learning curve and the graph <strong>of</strong> the function will look something like<br />
this:<br />
The graph will have a y intercept at C – A and a horizontal asymptote at y = C. Because <strong>of</strong> the<br />
horizontal asymptote, we know that this function does not model uninhibited growth.<br />
Lesson 11 – Exponential Functions as Mathematical Models 4
Example 8: Suppose your company’s HR department determines that an employee will be able<br />
0.5t<br />
to assemble Q( t) = 50 − 30e − products per day, t months after the employee starts working on<br />
the assembly line. Enter the function in GGB.<br />
a. How many units can a new employee assemble as s/he starts the first day at work?<br />
b. How many units should an employee be able to assemble after two months at work?<br />
c. How many units should an experienced worker be able to assemble?<br />
d. At what rate is an employee’s productivity changing 4 months after starting to work?<br />
Logistic Functions<br />
The last growth model that we will consider involves the logistic function. The general form <strong>of</strong><br />
A<br />
the equation is Q( t)<br />
= and the graph looks something like this:<br />
kt<br />
1 + Be −<br />
If we looked at this graph up to around x = 2 and didn’t consider the rest <strong>of</strong> it, we might think<br />
that the data modeled was exponential. Logistic functions typically reach a saturation point – a<br />
point at which the growth slows down and then eventually levels <strong>of</strong>f. The part <strong>of</strong> this graph to<br />
the right <strong>of</strong> x = 2 looks more like our learning curve graph from the last example. Logistic<br />
functions have some <strong>of</strong> the features <strong>of</strong> both types <strong>of</strong> models.<br />
Note that the graph has a limiting value at y = 5. In the context <strong>of</strong> a logistic function, this<br />
asymptote is called the carrying capacity. In general the carrying capacity is A from the<br />
formula above.<br />
Lesson 11 – Exponential Functions as Mathematical Models 5
Logistic curves are used to model various types <strong>of</strong> phenomena and other physical situations such<br />
as population management. Suppose a number <strong>of</strong> animals are introduced into a protected game<br />
reserve, with the expectation that the population will grow. Various factors will work together to<br />
keep the population from growing exponentially (in an uninhibited manner). The natural<br />
resources (food, water, protection) may not exist to support a population that gets larger without<br />
bound. Often such populations grow according to a logistic model.<br />
Example 9: A population study was commissioned to determine the growth rate <strong>of</strong> the fish<br />
population in a certain area <strong>of</strong> the Pacific Northwest. The function given below models the<br />
population where t is measured in years and N is measured in millions <strong>of</strong> tons. Enter the<br />
function in GGB.<br />
2.4<br />
N( t)<br />
=<br />
0.338t<br />
1 + 2.39e −<br />
a. What was the initial number <strong>of</strong> fish in the population?<br />
b. What is the carrying capacity in this population?<br />
c. What is the fish population after 3 years?<br />
d. How fast is the fish population changing after 2 years?<br />
Lesson 11 – Exponential Functions as Mathematical Models 6
Math 1314<br />
Lesson 12<br />
Curve Analysis (Polynomials)<br />
This lesson will cover analyzing polynomial functions using GeoGebra.<br />
Suppose your company embarked on a new marketing campaign and was able to track sales<br />
based on it. The graph below gives the number <strong>of</strong> sales in thousands shown t days after the<br />
campaign began.<br />
Now suppose you are assigned to analyze this information. We can use calculus to answer the<br />
following questions:<br />
When are sales increasing or decreasing? (Note that the graph stops at t = 40.)<br />
What is the maximum number <strong>of</strong> sales in the given time period?<br />
Where does the growth rate change?<br />
Etc.<br />
Calculus can’t answer the “why” questions, but it can give you some information you need to<br />
start that inquiry.<br />
There will be several features <strong>of</strong> a polynomial function that we’ll need to find. Let’s start with a<br />
few College Algebra topics.<br />
An example <strong>of</strong> a polynomial function is<br />
f ( x) ( x 2)( x 1) ( x 1)<br />
3 2<br />
= − − + . Its graph looks like:<br />
*The domain <strong>of</strong> any polynomial function is ( ) , −∞ ∞ . Polynomial functions have no asymptotes.<br />
*To find the roots/zeros/x-intercepts/solutions <strong>of</strong> any function, set the function equal to zero and<br />
solve for x. Or you may simply use the root or roots command in GGB.<br />
*To find the y-intercept for any function, set x = 0 and calculate.<br />
*The range <strong>of</strong> any polynomial is easier found by looking at the graph <strong>of</strong> the function.<br />
Lesson 12 – Curve Analysis (Polynomials) 1
3 2<br />
Example 1: Let f ( x) = x − 3x − 13x<br />
+ 15 . Enter the function in GGB.<br />
a. Find any x-intercepts <strong>of</strong> the function.<br />
Command:<br />
Answer:<br />
b. Find any y-intercept <strong>of</strong> the function.<br />
Command:<br />
Answer:<br />
Intervals on Which a Function is Increasing/Decreasing<br />
Definition: A function is increasing on an interval (a, b) if, for any two numbers x1<br />
and x2<br />
in<br />
(a, b), f ( x1 ) < f ( x2<br />
) , whenever x<br />
1<br />
< x2<br />
. A function is decreasing on an interval (a, b) if, for<br />
any two numbers x1<br />
and x<br />
2<br />
in (a, b), f ( x1 ) > f ( x2<br />
) , whenever x<br />
1<br />
< x2<br />
.<br />
In other words, if the y values are getting bigger as we move from left to right across the graph <strong>of</strong><br />
the function, the function is increasing. If they are getting smaller, then the function is<br />
decreasing. We will state intervals <strong>of</strong> increase/decrease using interval notation. The interval<br />
notation will consists <strong>of</strong> corresponding x-values wherever y-values are getting bigger/smaller.<br />
Example 2: Given the following graph <strong>of</strong> a function, state the intervals on which the function is:<br />
a. increasing. b. decreasing.<br />
We can use calculus to determine intervals <strong>of</strong> increase and intervals <strong>of</strong> decrease. A function can<br />
change from increasing to decreasing or from decreasing to increasing at its critical numbers, so<br />
we start with a definition <strong>of</strong> critical numbers:<br />
The critical numbers <strong>of</strong> a polynomial function are all values <strong>of</strong> x that are in the domain <strong>of</strong> f<br />
where f ′( x) = 0 (the tangent line to the curve is horizontal).<br />
A function is increasing on an interval if the first derivative <strong>of</strong> the function is positive for every<br />
number in the interval. A function is decreasing on an interval if the first derivative <strong>of</strong> the<br />
function is negative for every number in the interval.<br />
Lesson 12 – Curve Analysis (Polynomials) 2
Example 3: The graph given below is the first derivative <strong>of</strong> a function, f. Find:<br />
a. any critical numbers.<br />
b. any intervals where the function is increasing/decreasing.<br />
5 4<br />
Example 4: Let f ( x)<br />
= x − x , find:<br />
a. any critical numbers.<br />
b. any intervals where the function is increasing/decreasing.<br />
Lesson 12 – Curve Analysis (Polynomials) 3
To find the intervals on which a given polynomial function is increasing/decreasing using<br />
GGB:<br />
1. Use GGB to graph the derivative <strong>of</strong> the function.<br />
′ = ;<br />
2. Find any critical numbers. (Recall that the critical numbers occur whenever f ( x) 0<br />
hence, simply find the zeros <strong>of</strong> f ′ .)<br />
2. Create a number line, subdividing the line using the critical numbers.<br />
3. Use the graph <strong>of</strong> the derivative (or compute the value <strong>of</strong> a test point) to determine the sign<br />
(positive or negative) <strong>of</strong> the y values <strong>of</strong> the derivative in each interval and record this on your<br />
number line.<br />
4. In each interval in which the derivative is positive, the function is increasing. In each interval<br />
in which the derivative is negative, the function is decreasing.<br />
5 2<br />
Example 5: Let f ( x) = x − 16x + 4x<br />
. Using GGB find:<br />
a. any critical numbers.<br />
Command:<br />
Answer:<br />
b. any intervals where the function is increasing/decreasing.<br />
Relative Extrema<br />
The relative extrema are the high points and the low points <strong>of</strong> a function. A relative maximum<br />
is higher than all <strong>of</strong> the points near it; a relative minimum is lower than all <strong>of</strong> the points near it.<br />
A relative maximum or a relative minimum can only occur at a critical number.<br />
Lesson 12 – Curve Analysis (Polynomials) 4
You can use the same number line that you created to determine intervals <strong>of</strong> increase/decrease to<br />
find the x coordinate <strong>of</strong> any relative extrema. Use these three statements to determine if a critical<br />
number generates a relative extremum. Once you find that x = c generates a relative extremum,<br />
you can find the y coordinate <strong>of</strong> the relative extremum by computing f ( c ) or for a more<br />
accurate answer use the extremum command in GGB.<br />
1. If the sign <strong>of</strong> the derivative changes from positive to negative at a critical number, x = c , then<br />
c,<br />
f c .<br />
the function has a relative maximum at the point ( ( ))<br />
2. If the sign <strong>of</strong> the derivative changes from negative to positive at a critical number, x = c , then<br />
c,<br />
f c .<br />
the function has a relative minimum at the point ( ( ))<br />
3. If the sign <strong>of</strong> the derivative does not change sign at a critical number, x = c , then the function<br />
c,<br />
f c .<br />
has neither a relative maximum nor a relative minimum at the point ( ( ))<br />
Example 6: Find any relative maximum and relative minimum for<br />
5 4<br />
= − .<br />
f ( x)<br />
x x<br />
Example 7: Find any relative extrema:<br />
Command:<br />
f x = x + x +<br />
Answer:<br />
6<br />
( ) 0.25 4<br />
Lesson 12 – Curve Analysis (Polynomials) 5
Concavity<br />
In business, for example, the first derivative might tell us that our sales are increasing, but the<br />
second derivative will tell us if the pace <strong>of</strong> the increase is increasing or decreasing.<br />
From these graphs, you can see that the shape <strong>of</strong> the curve change differs depending on whether<br />
the slopes <strong>of</strong> tangent lines are increasing or decreasing. This is the idea <strong>of</strong> concavity.<br />
Example 8: The graph given below is the graph <strong>of</strong> a function f. Determine the interval(s) on<br />
which the function is concave upward and the interval(s) on which the function is concave<br />
downward.<br />
We find concavity intervals by analyzing the second derivative <strong>of</strong> the function. The analysis is<br />
very similar to the method we used to find increasing/decreasing intervals.<br />
1. Use GeoGebra to graph the second derivative <strong>of</strong> the function. Then find the zero(s) <strong>of</strong> the<br />
second derivative.<br />
2. Create a number line and subdivide it using the zeros <strong>of</strong> the second derivative.<br />
3. Use the graph <strong>of</strong> the second derivative to determine the sign (positive or negative) <strong>of</strong> the y<br />
values <strong>of</strong> the second derivative in each interval and record this on your number line.<br />
4. In each interval in which the second derivative is positive, the function is concave upward. In<br />
each interval in which the second derivative is negative, the function is concave downward.<br />
Lesson 12 – Curve Analysis (Polynomials) 6
Example 9: State intervals on which the function is concave upward and intervals on which the<br />
1 5 2 2<br />
function is concave downward: f ( x) = x − x −8x<br />
− 1<br />
2 3<br />
Command:<br />
You’ll also need to be able to identify the point(s) where concavity changes. A point where<br />
concavity changes is called a point <strong>of</strong> inflection.<br />
You can use the same number line that you created to determine concavity intervals to find the x<br />
coordinate <strong>of</strong> any inflection points. Use the following two statements to determine if a zero <strong>of</strong><br />
the second derivative generates an inflection point.<br />
1. If the sign <strong>of</strong> the second derivative changes from positive to negative or from negative to<br />
positive at a number, x c<br />
c,<br />
f c .<br />
= , then the function has an inflection point at the point ( ( ))<br />
2. If the sign <strong>of</strong> the second derivative does not change sign at a number, x = c , then the function<br />
c,<br />
f c .<br />
does not have an inflection point at the point ( ( ))<br />
Once you find that x = c generates an inflection point, you can find the y coordinate <strong>of</strong> the<br />
inflection point by computing f ( c ).<br />
Example 10: Given<br />
1 5 2 2<br />
f ( x) = x − x −8x<br />
− 1, find any inflection points.<br />
2 3<br />
Example 11: Given<br />
Command:<br />
3 2<br />
f ( x)<br />
= x − 3x<br />
− 24x<br />
+ 32 , find any inflection points.<br />
Answer:<br />
Lesson 12 – Curve Analysis (Polynomials) 7