Product Functions
Explanation
A product function is where two variables that are functions are being multiplied together to find the result. An obvious example is Revenue is equal to the price of a product times the quantity sold. The quantity sold is a function because the higher or lower the price is, can determine how many get sold. This is often used to optimize revenue in business. Essentially one can take one equation and multiply it to another variable. In order to find the derivative one must then use the Product Rule.
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Product Rule
The derivative of the product of f and g is equal to the derivative of f times g plus f times the derivative of g
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Simple Examples
Differentiate:
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Real World Example
United States Population Growth
This is a graph of the US Population, because it looks linear, a linear equation would fit best. Using the points (as years after 1990 and millions of people) (0, 250) and (17,300) use the slope formula and the y intercept to make the equation for this graph. Note that the unit for y is people in millions
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Average American Yearly Income
This is a graph of the average salary of Americans in years after 1990. The points were from data collected by the US census. To find a line for these points pick two such as (0,14.387) and (18 , 26.964) and use the slope formula along with the y intercept. This will result in the equation on the graph. Note that the g(x) term is dollars in thousands.
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Below are a few examples of how to use this equation. Say someone wanted to find out at what rate was the national populations income increasing in 2002 (12 years after 1990) in thousand of dollars. All they would have to do is substitute in 12 into the equation and solve for f'(12).
Below is another example, where one is finding out the rate of change for the US income per year in the year 2010 (20 years after 1990).
Quotient Functions
ExplanationA quotient function is a function where a variable that is a function is being divided by another variable that is a function. It is useful for dealing with functions that cannot be simplified into product form. It is used to find the derivatives of quotients and thus the method used is called the Quotient Rule. Essentially one student defines the numerator is f(x) and the denominator as g(x). Then take the derivative of f and multiply it by g then subtract the product of f and g prime, all over g(x) squared
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Quotient Rule |
Simple Examples |
Real World Examples
The units on the y-axes are millions of people. The x axes increases by 5 years for each tick mark, 0 being 1950.
1985:
1985 -1950 = 35/5 =7
h'(y) =17/(y-11)2
h'(7) = 17/((7)-11)2
h'(7) = 17/-42
h'(7) = 17/16
h'(7) = 1.0625
1.0625*5 = 5.3125
In 1985 the population of Seoul was growing at a rate of 5.3125 years per million people. (188.23529412 thousand people per year)
2020:
2020 -1950 = 70/5 =14
h'(y) = 17/(y-11)2
h'(14) = 17/((14)-11)2
h'(14) = 17/32
h'(14) = 17/9
h'(14) = 1.8888888889
1.8888888889*5= 9.4444444445
In 2020 the population of Seoul is projected to be growing at a rate of 9.4444444445 years per million people. (105.88235294 thousand people per year)
The rate of change for 1985 does not seem realistic because the slope of the doted line does not match that of the data at all, however, the projection for 2020 seems correct.