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1-6 Modeling with functions

1-6 Modeling with functions. Modeling with Functions. After today’s lesson you should be able to Identify appropriate basic functions with which to model real-world problems Produce specific functions to model data, formulas, graphs, and verbal descriptions. Why develop models for functions?.

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1-6 Modeling with functions

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  1. 1-6 Modeling with functions

  2. Modeling with Functions After today’s lesson you should be able to • Identify appropriate basic functions with which to model real-world problems • Produce specific functions to model data, formulas, graphs, and verbal descriptions

  3. Why develop models for functions? • Think of two different ways to find the circumference of a circle. • Which method is more efficient?

  4. Ex 1: Write the area of a circle as a function of its • Radius r • Diameter d • Circumference C

  5. Ex 2: A square of size x inches is cut out of each corner on an 8 in x 15 in piece of cardboard and the sides are folded up to form an open topped box. • Write the volume of the box as a function of x. • Find the domain of x.

  6. Ex 3: A small satellite dish is packaged with a cardboard cylinder for protection. Suppose the parabolic dish has a 32 in. diameter and is 8 in. deep. How tall must the 12 in diameter cylinder be to fit in the middle of the dish and be flush with the top of the dish?

  7. Ex 4: Water is stored in a conical tank with a faucet at the bottom. The tank has depth 24 inches and radius 9 in., and it is filled to the brim. If the faucet is opened to allow the water to flow at a rate of 5 cubic inches per second, what will the depth of the water be after 2 minutes?

  8. Functions from Data Given a set of data points of the form (x,y), to construct a formula that approximates y as a function of x: • Make a scatter plot of the data points. The points do not need to pass the vertical line test. • Determine from the shape of the plot whether the points seem to follow the graph of a familiar type of function (line, parabola, cubic, sine curve, etc.) • Transform a basic function of that type to fit the points as closely as possible.

  9. Regression Equations • The effectiveness of a data-based model is highly dependent on the number of data points and on the way they were selected. • Regression line – the line of best fit used to describe a set of data. • Correlation coefficient – measure of how well a line models a set of data • Denoted r • Note: Used to describe linear data only • Coefficient of determination – measure of how well a non-linear equation models a set of data • Denoted R2

  10. Ex 5 The estimated number of U.S. children that were home-schooled in the years from 1992 to 1997 were: • Produce a scatter plot of the number of children home-schooled in thousands (y) as a function of years since 1990 (x). • Find the linear regression equation. Does the value of r2 suggest that the linear model is appropriate? • Find the quadratic regression equation. Does the value of R2 suggest that a quadratic model is appropriate? • Use both curves to predict the number of U.S. children that are home-schooled in the year 2005. How different are the estimates?

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