A Catalog of Essential Functions PDF

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A Catalog of Essential Functions...

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c Kathryn Bollinger, September 2, 2010

1.2 - Mathematical Models: A Catalog of Essential Functions Steps of Mathematical Modeling 1. Given a real-world problem, identify (and name) the independent and dependent variables. 2. Use knowledge (or data) of the physical situation and mathematical skills to form equations relating the variables. 3. Apply mathematics (calculus) to the mathematical model in order to derive mathematical conclusions. 4. Interpret the mathematical conclusions (and possibly use the information to make predictions). 5. Test the predictions by checking against new real data. If the predictions don’t compare well with reality, refine the model or formulate a new model. Note: A mathematical model is an

of a physical situation.

Linear Models • A linear function is of the form f (x) = mx + b, where m is the slope and (0, b) is the y-intercept. • The slope is the rate of change of y with respect to x. =⇒ For every unit increase in x, the slope gives the corresponding change in y. • Linear functions have a constant rate of change. Ex: If f (x) =

3 x − 6, 2

(a) What is the y-intercept? (b) Find and interpret the slope.

(c) If x decreases by 5 units, how does y change?

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c Kathryn Bollinger, September 2, 2010

Slopes & Types of Lines Type

Slope

Example

Rising (Increasing)

Falling (Decreasing)

Horizontal

Vertical

Ex: KB & Co. is manufacturing insulated mugs. The company has a total monthly cost of $1800 when producing 100 mugs, and a total monthly cost of $2100 when producing 200 mugs. (a) Find the linear monthly cost function for KB & Co., C(x), as a function of the number of mugs produced, x.

(b) Find and interpret both the slope and the “y”-intercept of this cost function.

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c Kathryn Bollinger, September 2, 2010

If there is no physical law or principle to help formulate a model we can construct an empirical model which is based entirely on collected data. We will do so by using regression analysis. Linear Regression: A method for finding the best linear function fitting a given set of data. 1. Standardize the data, if necessary. 2. Enter the data into L1 and L2 by pressing STAT and selecting 1:Edit. 3. Graph the data points in a scatterplot. Ask yourself, “Does the data appear to be linear in nature?” 4. Find the equation of the best linear model. Press STAT , move right and select CALC and then go down to 4:LinReg(ax+b) for a linear model (y = ax + b). 5. Store your model in the calculator so you can make predictions with your model and compare models to one another. To do this, after making your model selection described above, press VARS , move right and select Y-VARS, choose option 1:Function, and then choose a function name. Note: A similar process can also be used for finding other types of models...QuadReg (quadratic), CubicReg (cubic), QuartReg (quartic), LnReg (logarithmic), ExpReg (exponential), PwrReg (power), SinReg (sine).

Ex: The following table shows the blood sugar levels and cholesterol levels for 5 different patients. Blood Sugar Level Cholesterol Level

130 170

142 173

159 181

165 201

210 240

(a) Make a scatter plot of the data, where the blood sugar level is considered the independent variable.

(b) Find the equation of the best fitting linear model to the data, where a person’s cholesterol level is given as a function of his/her blood sugar level. (Round each coefficient to four decimal places, if necessary.)

c Kathryn Bollinger, September 2, 2010

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(c) Use your unrounded model to predict the cholesterol level for a person whose blood sugar level is 180.

(d) Use your unrounded model to predict the blood sugar level for a person whose cholesterol level is 100.

When a model is used to estimate values between observed values, this is called If a model is used to estimate values outside the region of observed values, this is called . Question: What type of estimation was performed in parts (c) and (d) above?

.

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c Kathryn Bollinger, September 2, 2010

Polynomials • A polynomial is a function of the form P (x) = an xn +an−1 xn−1 +· · ·+ a2 x2 + a1 x + a0 , where n is a non-negative integer and the numbers a0 , a1 , . . . , an are constants called coefficients of the polynomial. • If the leading coefficient an 6= 0, then the degree of the polynomial is n. • The domain of any polynomial is (−∞, ∞). ◦ Quadratic Functions: a polynomial of degree 2 =⇒ P (x) = ax2 + bx + c Graphs are parabolas. a>0

a0

a0

a 1 ◦ Increasing function =⇒ exponential growth

◦ Examples: y = 2x , y = 10x , y = ex

• Case 2: 0 < a < 1 ◦ Decreasing function =⇒ exponential decay  x  x  x 1 1 1 ◦ Examples: y = ,y = ,y = 2 10 e

Logarithmic Functions • Logarithmic functions are functions of the form f (x) = logb x, where the base b is a positive constant, with b 6= 1. • Logarithmic functions are inverse functions of exponential functions. • Examples:

Ex: Classify the following functions: (a) y = 7x t7 t+1 √ r (c) y = 4 r +6

(b) y =

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