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5.3 Modeling with Power FunctionsPowerPoint Presentation

5.3 Modeling with Power Functions

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Power Data & Linear Data Connection

- If f is a power function of x, then ln(f) is a linear function of ln(x).

Getting a Power Model from Data

Ex. 1) Volume Inside a Sphere

a.)Plot the graph of ln(V) versus ln(r) to determine whether or not it is reasonable to think that the volume is related to the radius by a power function.

b.) Use regression to find a formula for the line formed by the plot of ln(V) as a function of ln(r). Then, use that formula to write a formula for the volume of a sphere. Round parameters to 2 decimal places.

c.) Use regression to find a formula for the volume of a sphere as a function of the radius. Round to 2 decimal places.

d.) Plot both the function from b and c and the original data point on the same graph.

e.) What is the radius of a sphere with a volume of 453.25? (Use the function from part c.

Ex. 2) Generation Time as a Power Function of Length

- a.) Plot ln(T) against ln(L). Determine whether it is reasonable to model T as a power function of L.
- b.) Find a formula for the regression line of Ln(T) against ln(L).
- c.) Find a formula that models T as a power function of L, and plot the function along with the given data. Round to 2 decimal places.
- d.) If one generation time is 70% more than another, how much greater would the length be (given as a percentage)?

Ex. 3) Length vs. Flying Speed

- a.) Verify that it is appropriate to model the data given in the table using a power function.
- b.) Using regression, find a formula that models F as a power function of L.
- c.) Is the average flying speed for a Flying fish more or less than would be expected from the model?
- d.) If one animal’s flying speed is 35% less than another, how much shorter is it’s length (given as a percentage)?

Practice:

a.) Make a power model for weight versus height.

b.) According to the model from part a, what percentage increase in weight can be expected if height is increased by 10%?

5.4: Composition of Functions

- Composing functions is a way of manipulating equality relationships in order to change the independent variable in an equation.
- It is useful because it can directly link functions that were previously linked by one or more other variables. This allows us to show clearer relationships between variables.

5.4: Composition of Functions

- Composing Functions: Real world application
- If a boy’s height depends upon his age according to h(t)=1.9t + 40, and his weight depends on his height according to w(h) = 4.4h – 15, how can we write an equation for weight that depends on age?

Ex.) Combining Functions: Adding, Subtracting, and Multiplying.

Practice: Multiplying.

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