Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 1 Ingredients of Change: Functions and Linear Models

Copyright © by Houghton Mifflin Company, All rights reserved. Chapter 1 Key Concepts FunctionsFunctionsFunctions Discrete and Continuous FunctionsDiscrete and Continuous FunctionsDiscrete and Continuous FunctionsDiscrete and Continuous Functions Special FunctionsSpecial FunctionsSpecial FunctionsSpecial Functions Limits of FunctionsLimits of FunctionsLimits of FunctionsLimits of Functions Linear ModelsLinear ModelsLinear ModelsLinear Models

Copyright © by Houghton Mifflin Company, All rights reserved. Functions For every input there is exactly one output.For every input there is exactly one output. xy FunctionNon-Function xy

Copyright © by Houghton Mifflin Company, All rights reserved. Functions: Example FunctionNon-Function

Copyright © by Houghton Mifflin Company, All rights reserved. Functions: Exercise 1.1 #21 You want to finance a car for 60 months at 10% interest with no down payment. The graph shows the car value as a function of the monthly payment. Estimate the car value when the monthly payment is $200 and estimate the monthly payment for a $16,000 car.You want to finance a car for 60 months at 10% interest with no down payment. The graph shows the car value as a function of the monthly payment. Estimate the car value when the monthly payment is $200 and estimate the monthly payment for a $16,000 car. $200 payment   $9400 car $16,000 car   $340 payment

Copyright © by Houghton Mifflin Company, All rights reserved. Discrete and Continuous Functions Discrete - graph is a scatter plotDiscrete - graph is a scatter plot Continuous - graph doesn’t have any breaksContinuous - graph doesn’t have any breaks Continuous with Discrete Interpretation - continuous graph with meaning only at certain pointsContinuous with Discrete Interpretation - continuous graph with meaning only at certain points Continuous without Restriction - continuous graph with meaning at all pointsContinuous without Restriction - continuous graph with meaning at all points

Copyright © by Houghton Mifflin Company, All rights reserved. Discrete and Continuous: Example Discrete Continuous (without restriction)

Copyright © by Houghton Mifflin Company, All rights reserved. Discrete and Continuous: Example Continuous with Discrete Interpretation

Copyright © by Houghton Mifflin Company, All rights reserved. Discrete/Continuous: Exercise 1.2 #5 On the basis of data recorded in the spring of each year between 1988 and 1997, the number of students of osteopathic medicine in the U.S. may be modeled as O(t)=0.027t t thousand students in year t.On the basis of data recorded in the spring of each year between 1988 and 1997, the number of students of osteopathic medicine in the U.S. may be modeled as O(t)=0.027t t thousand students in year t. Is this function discrete, continuous with discrete interpretation or continuous without restriction?Is this function discrete, continuous with discrete interpretation or continuous without restriction? The function is continuous without restriction.The function is continuous without restriction.

Copyright © by Houghton Mifflin Company, All rights reserved. Special Functions Composite FunctionsComposite Functions Inverse FunctionsInverse Functions Piecewise Continuous FunctionsPiecewise Continuous Functions

Copyright © by Houghton Mifflin Company, All rights reserved. Composite Functions: Example t = time into flight (minutes) F(t) = feet above sea level ,500 7,500 13,000 19,000 26, ,500 7,500 13,000 19,000 26,000 F = feet above sea level A(F) = air temperature (Fahrenheit) F Min.Feet A FeetDegrees

Copyright © by Houghton Mifflin Company, All rights reserved. Composite Functions: Example A(F(t)) = air temperature (Fahrenheit) t = time into flight (minutes) A(F(t)) = F Minutes A FeetDegrees

Copyright © by Houghton Mifflin Company, All rights reserved. Composite Functions: Exercise 1.3 #9 Draw and label the input/output diagram for the composite function given P(c) is the profit from the sale of c computer chips and C(t) is the number of computer chips produced after t hours.Draw and label the input/output diagram for the composite function given P(c) is the profit from the sale of c computer chips and C(t) is the number of computer chips produced after t hours. C Hours (t) P Chips C(t) Profit P(C(t)) Profit = P(C(t))

Copyright © by Houghton Mifflin Company, All rights reserved. Inverse Functions: Example Year Average Home Sales Price ($) ,400 64,600 84, , ,900 Year Average Home Sales Price ($) ,400 64,600 84, , ,900 f YearPrice f -1 PriceYear

Copyright © by Houghton Mifflin Company, All rights reserved. Inverse Functions: Exercise 1.3 #25 Age % with Flex Work Schedule Determine if the tables are inverse functions.Determine if the tables are inverse functions. Age % with Flex Work Schedule FunctionNon-Function

Copyright © by Houghton Mifflin Company, All rights reserved. Piecewise Continuous: Example

Copyright © by Houghton Mifflin Company, All rights reserved. Yearly water ski sales (in millions of dollars) in the continental U.S. between 1985 and 1992 can be modeled byYearly water ski sales (in millions of dollars) in the continental U.S. between 1985 and 1992 can be modeled by Piecewise : Exercise 1.3 #43 Find S(85), S(88), S(89), and S(92).Find S(85), S(88), S(89), and S(92). S(85) = 123.1S(85) = S(88) = 159.4S(88) = S(89) = 97.79S(89) = S(92) = 53.39S(92) = 53.39

Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Functions The limit of f(x) as x approaches c is writtenThe limit of f(x) as x approaches c is written In order for the limit to exist, the graph of the function must approach a finite value, L, as x approaches c from the left and from the right.In order for the limit to exist, the graph of the function must approach a finite value, L, as x approaches c from the left and from the right. - right-hand limit - left-hand limit

Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Functions: Example

Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Functions: Exercise 1.4 #4

Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Functions: Example t  4 - r(t) t  4 + r(t)

Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Functions: Exercise 1.4 #27 t  -1/3 + f(t) t  -1/3 - f(t)

Copyright © by Houghton Mifflin Company, All rights reserved. Linear Models Constant rate of changeConstant rate of change Equation of form f(x) = ax + bEquation of form f(x) = ax + b Graph is a lineGraph is a line a is the constant rate of change (slope)a is the constant rate of change (slope) b is the vertical intercept of the graph of fb is the vertical intercept of the graph of f

Copyright © by Houghton Mifflin Company, All rights reserved. Linear Models: Example t (years)P (%) The table shows the percent P of companies in business after t years of operation. Rate of change: -3 percentage points / year Model: P = -3t + 65 Note: This model assumes 5  t  8.

Copyright © by Houghton Mifflin Company, All rights reserved. Linear Models: Exercise 1.5 #1 Estimate slope: -0.5 Find rate of change of profit: $0.5 million dollar decrease per year What is the significance of vertical intercept? Profit at year 0 is about 2.5 million dollars.