Calculus for Business UNIT- Review Material: Functions and Graphs Date:

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1 1 Calculus for Business UNIT- Review Material: Functions and Graphs Name: Date: Objective: To review prerequisite Precalculus content. LINEAR FUNCTIONS: Forms of Linear Equations: slope-intercept form: standard form: point-slope form: 1. Write the linear equation in slope intercept, standard, and point-slope form given the line passes through (5, ) and (7, 9). Write the equation of the horizontal line that passes through (-9, ) Intercepts Finding -intercepts: replace y with zero and solve Finding y-intercepts: replace with zero and solve 3. Find the intercepts of the following linear equations(if any): a. y = 3 + b y = Find the intercepts of the following non-linear equations(if any): a. y=( )( 6) b. y=9

2 Parallel & Perpendicular Lines Parallel lines have slopes. Perpendicular lines have slopes that are. 5. Write the linear equation in standard form given that the line passes through (-, 10) and is parallel to 4 the graph of y = Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of 4 y = Intersection of Linear Functions Methods Elimination Substitution Graphical 1. Cancel a variable and solve for the other.. Sub in result and solve for the other variable *Great method when neither equation is in y=m+b form 1. Plug one equation in the other and solve for the variable. Sub in result and solve for the other variable *Great method when given only one equation in y=m+b form 1. Graph both functions and see where they cross. *Great method when given two lines in y=m+b form + 3y = y = 9 Eamples: y = ½ y = 4 y = ½ y = Intersection of Other Functions Generally we stick with substitution and graphing here (may want to use separate paper for this one): Eample: y = + y =

3 3 Calculus for Business Lesson- Absolute Value Functions Name: Date: Objective: To learn how to graph absolute value functions Do Now: Find the intersection between y = + 1 & y = Absolute Value Functions: Standard form: f() = a - h +k; where ( h, k) is the verte and a is the slope if a is + then: if a is then: Graph the Following 1. f ( ) = 3. f ( ) = f ( ) = + 1 y y y 4. What are some things you notice about the graph of the absolute value function as a, h, k vary?

4 4 Calculus for Business Lesson- Functions Rules and Notation Name: Date: Objective: To review basic function rules and function notation IDENTIFYING FUNCTIONS Definitions: Relation: Function: Domain: Range: Eamples 1. State the domain and range of each relation, and state whether the relation is a function or not: a. {(-, 0), (3, ), (4, 5)} b. {(6, -), (3, 4), (6, -6), (-3, 0)}. Which relation is a function? Why? What is the shortcut rule for determining a function? (a) (b) (c) (d) 3. Find the Domain for each: a. + f ( ) = b f ( ) = 4 c. f ( ) = 3 d. f() =

5 5 Evaluating Functions: substitute numerical value or variable into function equation and simplify 4. find f(-1) if f() = 1 5. find g(m) if g() = find k(w + ) if k() = find h(a ) if h() = + 3 Operations with Functions: given functions f and g sum: ( f g )() = f() + g() + difference: ( f g )() = f() g() f f() quotient: () =, where g() 0 g g() product: ( f g )() = f() g() Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each 8. f ( ) = 5 + 4; g( ) = 1 9. f ( ) = 5 ; g( ) = + 1

6 6 Composition of Functions: given functions f and g notation: [ f g] ( ) = f ( g( ) ) Find [ f g ]( ) and [ f] () g for each f() and g(): 10. f ( ) = 1; g( ) = f ( ) = + 1; g( ) = 1. f ( ) = + 5; g( ) = f ( ) = + ; g( ) = f ( ) = 3; g( ) = 15. f ( ) = 1; g( ) = 4 One-to-one Functions A function is one-to-one when no two ordered pairs in the function have the same ordinate and different abscissas. The best way to check for one-to-oneness is to apply the vertical line test and the horizontal line test. If it passes both, then the function is one-to-one. (**Note: if a function is not one-to-one, it does not have an inverse**) Eamples: Determine whether the following functions are one-to-one. 16. f ( ) = f ( ) = f ( ) = 3 5

7 7 Inverse Functions Steps: 1. Write the equation in terms of and y.. Switch the with the y. 3. Solve for y. Eamples: 19. Find the inverse of y = Find the inverse of f ( ) = 5 1. Graph each of the lines in eamples one and two and their inverses on the same set of aes and identify any interesting characteristics. y 1. Given the function f() = 3 (a) Algebraically, find f -1 (). (b) Algebraically, verify your answer to part (a).

8 8 Calculus for Business Lesson- Evaluate functions with variables; difference quotient Name: Date: Objectives: evaluate functions as epressions that involve one or more variables eplore functions by evaluating and simplifying a difference quotient Evaluating & Simplifying a Difference Quotient: 1. For f() = , evaluate and simplify: a. ( h) f + b. f ( + h) f ( ) h, h 0 For each of the following, evaluate and simplify: a. f ( + h) f ( + h) f ( ) b. h, h 0 f() f(a) c., a 0 a. f() = 3 3. f() = 4 7

9 9 4. f( + h) f() For each given function, find and simplify:, h 0 h a. f() = - 4 b. f() = c. f() =

10 10 Calculus for Business Lesson- Symmetry, Odd/Even/Neither Functions Name: Date: Objectives: ~To review the algebraic method for testing for symmetry with respect to aes and origin ~To review the algebraic method for determining whether functions are even/odd/neither ~To learn shortcut approaches for the above Odd Functions Even Functions Symmetry Tests symmetric with respect to the origin TEST: f(-) = -f() symmetric with respect to the y-ais TEST: f() = f(-) symmetric with respect to the: the given equation is equivalent when: y-ais is replaced with - -ais y is replaced with y origin and y are replaced with and -y Eamples: 1. f() = - 4. f() = g() = f ( ) = 1

11 11 Calculus for Business Review of Factoring Name: Date: Objective: Review methods of factoring including: Perfect square trinomials Difference of two squares Sum of two squares * Difference of two cubes Sum of two cubes Factoring by grouping Undefined terms Reducing to lowest terms Multiplying Rational Epressions Dividing Rational Epressions Adding and Subtracting Rational Epressions Solving Rational Equations Simplifying Comple Fractions Perfect Square Trinomial To factor a perfect square trinomial follow the steps below: 1) Create two empty binomials as indicated on the right ( )( ) ) Take the square root of the first term of the given trinomial 3) Take the square root of the last term of the given trinomial 4) Take the result of step and put in the 1 st position in each binomial 5) Take the result of step 3 and put in the nd position in each binomial 6) The signs in the binomials should be the same as the middle term of the binomial Eample 1: factor : Difference of Two Squares a b To factor a difference of two squares follow the steps below: 1) Create two empty binomials as indicated on the right ( )( ) ) Take the square root of the first term of the given difference of two cubes 3) Take the square root of the last term of the given difference of two cubes 4) Take the result of step and put in the 1 st position in each binomial 5) Take the result of step 3 and put in the nd position in each binomial 6) Alternate the signs Eample : 9 y 16 Sum of Two Squares a + b The sum of squares is not factorable

12 Sum of Two Cubes To factor a sum of two cubes follow the steps below: 1) Create an empty binomial and an empty trinomial as indicated on the right ( )( ) ) Take the cube root of the first term of the given sum of two cubes **note- ignore all signs until the last step** 3) Take the cube root of the last term of the given sum of two cubes 4) Take the result of step and put in the 1 st position in the binomial 5) Take the result of step 3 and put in the nd position in the binomial 6) Take the result of step 4, square it and put it in the first position of the trinomial 7) Take the result of step 4 and multiply by the result of step 5 and put it in the middle position of the trinomial 8) Take the result of step 5, square it and put it in the last position of the trinomial 9) Arrange the signs as follows ( + )( + ) 1 Eample 3: Difference of Two Cubes The only difference between a difference of two cubes and a sum of two cubes is the sign arrangement. Arrange the signs as follows ( )( + + ) Eample 4: z Factoring by Grouping Steps: 1) Find a convenient point in the polynomial to partition ) Factor within each group 3) Factor across the groups Eample 5: Eample 6: + y + y z Eample 7: r + 8s + r + rs + s

13 13 Eample 8. Find the value of that makes the epression undefined: Reducing- to reduce (or simplify) a rational epression means to write the answer in lowest terms. This only applies when working with a single fraction, a product of fractions or a quotient of fractions (you cannot reduce across a sum or difference of two or more fractions c d Eample 9. Reduce to lowest terms: c d If there is an addition or subtraction sign in the numerator we must factor the numerator first and then cancel with like factors in the denominator!!! Eample 10. Reduce to lowest terms: y y y Sometimes you may also have to factor the denominator in order to reduce Eample 11. Reduce to lowest terms: 1 4a a + a 1 Multiplying Rational Epressions: Steps: Factor each numerator and denominator completely Cancel any like factor in any numerator with any like factor in any denominator Multiply the remaining epressions in each numerator Multiply the remaining epressions in each denominator Reduce if possible Eample 1. h h 3 h h 9 + 5h + 6 h 1 Dividing Rational Epressions: Steps: Multiply the first fraction by the reciprocal of the second (KCF) Factor each numerator and denominator completely Cancel any like factor in any numerator with any like factor in any denominator Multiply the remaining epressions in each numerator Multiply the remaining epressions in each denominator Reduce if possible Eample 13. b + b b

14 Adding and Subtracting Rational Epressions Steps: Find the least common denominator among all fractions (if none already eists) Multiply each denominator by an appropriate factor to make it equivalent to the LCD Multiply each numerator by the same factor that you multiplied its denominator by Combine all numerators (make sure the signs are placed appropriately) and simplify and put over LCD Reduce if possible 14 Combine each set of rational epressions and simplify Eample 14: y 6 y Eample 15: y + 3y 4y Solving Rational Equations Steps: Find the LCD Multiply each fraction by this LCD Cancel all denominators Solve for the variable Solve each rational equation Eample 16. = Eample = Simplifying comple fractions Steps: Find the grand LCD Multiply the numerator and denominator by the GLCD (will cancel out denominators within the subfractions) Follow the method for reducing fractions Eample

15 15 Calculus for Business Lesson- Polynomial Functions including quadratics Name: Date: Do Now: 1. State the degree and leading coefficient of each polynomial: a. 3b 7b 5 b b. 6a 4 + a 3 a. For each polynomial equation, sketch the graph and state the number of -intercepts. Based on your answers, do you notice a pattern? a. y = + 3 b. y = + c. y = Write the polynomial equation with the given roots: Recall: Imaginary Roots Theorem: Imaginary roots occur in comple conjugate pairs. 8 and -9 3, 4i, and -4 i, -i, 5i, and -5i 1, 0, and i

16 16 4. Write P() as a product of first-degree factors using the given zero (use synthetic division): a. P() = ; is a zero b. P() = ; 1 is a double zero c. P() = ; -3 is a zero 5. Solve the following quadratic equations by completing the square: 5 1 = = = 0 4

17 17 6. Complete the square to change the following quadratic functions to the form: f ( ) = a( h) + k and state the verte for each parabola: 1 f() = + 3 g() = h() = Rational Roots Theorem: Possible roots represents factors of the leading coefficient. p =, where p represents factors of the constant term and q q Process: a. Use Rational Roots Theorem to find potential rational roots. b. Use synthetic division, or long division, to find an actual root. c. Repeat step until the polynomial is of degree. d. Factor the remaining quadratic polynomial (it is possible to get imaginary answers). e. List all factors in simplest factored form Find all of the factors of each:

18 18 Calculus for Business Lesson- Graphing Rational Functions Name: Date: Objective: Objectives: To review graphing a rational function Graphing Rational Functions without the graphing calculator by: a) determining zeroes of the rational function b) determining vertical and horizontal asymptotes c) determining slant asymptotes d) describing the behavior of the function around vertical asymptotes Do Now: Given the function f() = (1) What is the degree of this polynomial? () What is the leading coefficient? (3) Determine if 4 is a zero of this function f(): Graph each (on separate graph paper) of the following functions (and check with the graphing calculator) by: 1. a) determining zeroes of the rational function b) determining vertical and horizontal asymptotes c) determining slant (oblique) asymptotes d) describing the behavior of the function around vertical asymptotes (increasing/decreasing) y f ( ) = f ( ) = + 4 y

19 19 5. f ) = 4 ( y f ( ) = + 1 y

20 0 Calculus for Business Lesson- Graphing Piecewise Functions Name: Date: Objective: To review graphing a piecewise function Do Now: State the domain in interval notation and determine any asymptotes for the f ( ) = 1 Piecewise functions This is a graph that is eactly what it sounds like. It is a graph that is basically in pieces. Graph the following: if 0 f ( ) = if < 0 y The procedure is to graph each part of the function separately. Graph the Following 3 if f ( ) = 3 if 0 < 0 y

21 1 (1) > + = if if 4 () f () > + < = 0 if 3 0 if 3 f() (3) > + = if 4 1 if 1 3 () f (4) < = 0 if 1 0 if 1 () f y y y y

22 (5) > = < = 0 if 0 if 0 if () f (6) > = < + = 0 if 0 if 0 if 5 1 () f (7) < < = if 1 1 if 1 if () f (8) + < + = 1 if 1 if 1 f() 3 y y y y

23 3 Calculus for Business Lesson- Regression and Modeling Name: Date: Correlation and Regression Correlation- When one variable is related to another in some way Scatterplot- A plot on an -y plane, where (, y) are paired data plotted as a single point Types of plots: Linear Cases Perfect + Strong + Moderate + Perfect - Strong - Moderate - Non-Linear Cases- Only sketch the Perfect Correlations Eponential Quadratic Cubic Quartic Natural Logarithmic None

24 Linear Correlation Coefficient (r) measures the strength of the linear relationship between the given variables (AKA Pearson s Product Moment Correlation Coefficient) n y ( )( y) r = ; where n is the number of ordered pairs n n y y ( ) ( ) ( ) ( ) r (in percent form) is the percent of variation in the y variable that can be eplained by variation in the variable. r Type of correlation 1 Perfect +.75 r < 1 Strong +.50 r <.75 Moderate < r <.50 None -.75 < r -.50 Moderate - -1 < r -.75 Strong - -1 Perfect - E 1: Construct a scatterplot and compute the linear correlation coefficient between and y y Linear Regression AKA- Least Squares Line, Line of Best Fit, Linear Regression Equation This is the equation that best fits the data set given. Calculator uses: y = a + b where a = slope and b = y-intercept n slope = a = y int ercept ( y) ( )( y) n( ) ( ) ( y)( ) ( )( y) = b = n( ) ( ) E : Using the data from E 1, write the equation of the line of best fit. Plot this line on your scatterplot as verification.

25 5 Using the Graphing Calculator: Note: You should only need to do steps -8, 10 once. 1. Enter values in L 1 and y values in L nd. Press y = 3. Make sure all plots are off, if not, Press 4 (Plotsoff) Enter nd 4. Press y = 5. Select 1 6. Turn this plot on by highlighting On and hitting Enter 7. Press the Down Arrow once and Press Enter 8. Verify that L 1 and L are listed. 9. Press Zoom 9 (On your graph you should see a scatterplot) 10. nd 0 Press DiagnosticsOn Press Enter twice 11. Press Stat, Right Arrow, 4 (LinReg(a + b)) 1. L 1, L, Y 1 Enter (Correlation and Regression data is now presented) 13. Zoom 9 (Line should appear on the graph verifying that equation is correct) Making Predictions To determine y given an you can either: Sub the given into the best fit model and solve for y or Enter the best fit equation into your calculator and hit: nd Trace Value. Type in the given and hit enter. To determine given a y you can either: Sub the given y into the best fit model and solve for or Enter the best fit equation into your calculator. Enter the given y as an equation. Then find the intersection ( nd Trace Intersection and follow the instructions on the screen). Other Regressions There are other regressions that can be determined using the graphing calculator Quadratic: y = a + b + c Cubic: y = a + b + c + d Quartic: y = a + b + c + d + e Natural Logarithmic: b y = a + b ln Eponential: y = a b Power Regression: y = a Sine Regression: y = Asin( B + C) + D

26 6 Eamples: 1. A study was conducted to determine if there is a relationship between the number of cookies eaten on a daily basis and an individual male s weight. The data are given below. # of cookies daily Weight in lb a. Construct a scatterplot (complete, with appropriate labels) b. Visually determine the best fit model. c. Determine, using your calculator, the best regression model. Why is the one you chose the best? Justify your answer by showing all work. d. Determine the equation of best fit. e. What weight would you epect a person who eats 100 cookoes per day to be? f. How many cookies per day would you epect a person who is 150 lb to eat?. The table below gives the average monthly temperature for Omaha, Nebraska, over a 30 year period. Jan Feb Ma Apr Ma Jun Jul Aug Sep Oct Nov Dec a. Put the data into your calculator and eamine the resulting scatter plot. b. Use the sinreg function to regress the data and generate an equation. Plot the equation with the scatterplot. c. Write the equation here: d. Amplitude: Period: Frequency:

27 7 Calculus for Business Lesson- Continuity & Increasing/decreasing/constant functions Name: Date: Objectives: determine whether a function is continuous or discontinuous determine whether a function is increasing, decreasing, or constant within an interval Continuous graphs: Discontinuous graphs: Point Discontinuity: y Eample: 1 f ( ) = + 1 if if < 0 > 0 Jump Discontinuity: y Eample: f ( ) = Infinite Discontinuity: y Eample: f ( ) = + 1

28 8 e) For each graph: (a) Find the domain (b) Find the range (c) Find the intervals over which the function is increasing (d) Find the intervals over which the function is decreasing (e) Find the intervals over which the function is constant (f) State any points of discontinuity (1) y () y

29 9 Calculus for Business Lesson- Limits of Sequences Name: Date: Definition: The limit of a sequence is the unique real number that the terms of the sequence are approaching. The terms of the sequence may or may not actually reach the limiting number, but they will eventually be etremely close to that limit. The limit must be unique; that is, there must only be one value for the limit. If the terms do not approach a unique real number, we say the limit does not eist (DNE) If the term grows without bound, the limit is (or - ) Eamples: Find the limit of each of the following sequences: 1. 16, -8, 4, -,., 5, 8, 11, 3. 5,5,5,5, , 0.33, 0.333, 5. 1,,1,, 6. 1,4,16, ,3,9,7, 8.,,, ,,1,,0,, 1,, ,6,9,1,15,

30 30 Calculus for Business Lesson- Limits and Vertical Asymptotes Name: Date: Objectives: determine the limit as the graph approaches a vertical asymptote Limits: Left-hand limits: Right- hand limits: (1) Graph (a) 1 f() = and find the following limits: 1 lim y (b) lim + 1 (c) 1 lim () Graph (a) f() = and find the following limits: + 3 lim y (b) (c) lim lim 3 3 +

31 (3) Graph f() = and find the following limits: 4 y 31 (a) lim 4 (b) lim + 4 (c) lim 4 1 (4) Graph f () = and find the following limits: y 1 (a) lim 0 1 (b) lim (c) lim 0 (5) Graph (a) 1 () = and find the following limits: 15 f 1 lim 5 15 y (b) (c) 1 lim lim 3 15

32 3 Calculus for Business Lesson- Determining limits graphically Name: Date: Objectives: determine the limit of function graphically that don t necessarily involve vertical asymptotes (1) Graph f() = and find the following limits: + 3 y (a) lim (b) (c) lim lim () Graph (a) (b) (c) (d) (e) (f) 3 () = and find the following limits: 3 f 3 lim lim lim lim lim lim 3 3 y

33 33 Given the value of c, use the graph of f to find each of the following values: (a) f (c) (b) lim f() c (c) lim f() + c (d) lim f() c (1) () (3) (4)

34 34 Calculus for Business Lesson - Limits and horizontal asymptotes Name: Date: Objectives: determine the limit as the graph approaches a horizontal asymptote Limits where approaches infinity: Limits where approaches a constant Find each of the following limits: (1) lim () lim 4 + (3) lim sin X + (4) + 1 lim (5) lim 1 + (6) 3 lim

35 35 Find the limit for each of the following algebraically: (7) lim ( + 4) 3 (8) lim ( 1) 5 (9) lim ( ) (10) lim (11) lim (1) lim 0 ( + ) 4 (13) 1 lim 1 1 (14) lim + 6

36 36 Calculus for Business Lesson- Limits of Piecewise Defined Functions Name: Date: Objective: write piecewise functions to model a situation to learn how to find the limit of a piecewise defined function DO NOW: a. 1 lim 49 7 b. lim 1 49 Recall: A piecewise function is a function that is graphed one piece at a time Eample 1: f + if < 1 ) = 3 if 1 ( Find each limit, if it eists. a. lim f ( ) b. lim f ( ) 3 0 c. lim f ( ) 1 d. lim f ( ) + 1 e. lim f ( ) 1 Eample : Ta tables for the 005 US federal income ta show that the income ta due for those filing as head of household is calculated as follows: 0.10d $ ( d $10,450) $5, ($39,000) T ( d) = $1, ( d $10,800) $39, ( d $166,450) $91, ( d $36,450) if if if if if if d $10,450 $10,450 < d $39,000 $39,800 < d $10,000 $10,800 < d $166,450 $166,450 < d $36,450 d > $36,450 where T(d) is income ta due and d is taable income. a. Find the income ta due for a person filing as head of household whose taable income is $85,40. b. Determine lim T ( d) and interpret your answer. d $39,800

37 Eample 3: A beauty supply store sells eyeliner pencils for $4.00 each. For an order of dozen or more pencils, the price per pencil for all pencils ordered is reduced to $3.50. For an order of 5 dozen or more pencils, the price per pencil for all pencils ordered is reduced to $3.5. a. Mary decides to go shopping and buys the following number of eyeliner pencils. Fill in the chart to determine the total cost for each order she placed: 37 Number of Pencils Total Cost b. Write a piecewise function C() that relates the total cost in dollars for an order of pencils. c. Find: i. lim f ( ) 4 ii. lim f ( ) + 4 iii. lim f ( ) 4 Eample 4: On weekends and holidays, Gunning Plumbing s emergency plumbing repair service charges $.00 per minute for the first 30 minutes of a service call and $1.00 per minute for each additional minute. a. Write a piecewise function P(t) which epresses the total cost of a service call in terms of time t in minutes. b. Find: i. lim f ( ) 30 ii. lim f ( ) + 30 iii. lim f ( ) 30

38 38 Calculus for Business Lesson- Continuity of Piecewise Defined Functions Name: Date: Objective: to learn how to determine if a piecewise function is continuous at a given point Requirements for Continuity at a Point: 1. f(a) must be defined. lim f ( ) must eist a 3. f(a) must equal lim f ( ) Eample 1: Determine if the function is continuous at the given point, a. a 3 f ( ) = 4 8 if if if < < 3 3 a. a = - b. a = 0 c. a = 0 Eample : Cab fares in Las Vegas are $.0 for the first mile and $1.50 for each additional mile. Graph the function for the first 6 miles and identify the points of discontinuity. Give an eample of how this function fails to satisfy the above rules. Eample 3: Package fees for parcel airlift are given in the following table. Weight not more than Fee pounds $ pounds $ pounds $ pounds $1.70 a. Sketch a graph of the function. b. Identify the discontinuities of the function.

39 39 Calculus for Business Lesson- Rates of Change and Slope Name: Date: Objective- To learn about rates of change, slope, and velocity. Velocity of an object in motion: Average rate of change: speed with direction amount of change in the quantity over the interval divided by the amount of change in an interval. (also known as the slope of the secant line) Rule: f ( b) f ( a) b a Eample: The height of a rock t seconds after being thrown upward in the air from a platform 50 feet above the ground is given by the function h ( t) = 16t + 40t + 50, where h(t) is the height above the ground in feet. Find the average rate of change in the height of the rock between: a. 0.5 and 1 second b. and 3 seconds Instantaneous rate of change: amount of change that is occurring at a particular moment. This occurs when the two points that make up the secant have a distance between them that is equal to 0 making it a tangent. Rule: lim0 h f ( a + h) h f ( a) Eample: What is the instantaneous rate of change (velocity) of the rock in the previous eample one second after its release?

40 40 For a circle: For the graph of a function? secant tangent f() = (-)(-4) A tangent -touches but does not cross -touches at only one point Slope of a curve at any point To do this we need to define a tangent line to the graph of y=f() at the point (a, f(a)) Step 1: Find the slope of the secant line through the point (a, f(a)) and choose a second point on the curve: Step : Slope of the tangent line at the point (c, f(c))

41 41 Slope of a curve at a particular point Eample: Find the slope of the tangent line to the graph of y = ² at the point (3, 9). f ( a + h) f ( a) Note: If lim = ± 0 h then the tangent line at (a, f(a)) is vertical. 3 Eample: Find the slope of the tangent line to the graph y = at = 0. y = 3 We can think of a Derivative as the slope of the line tangent to a curve at a particular point. But since it could f ( + h) f ( ) be any point on the curve we must generalize to: lim h 0 h Eample: Find the derivative with respect to of the function y = /.

42 4 Calculus for Business Lesson- Derivative Shortcut Rules Name: Date: Objective: To learn the most common derivative rules Constant Rule: The derivative of a constant is 0. d d n n 1 Power Rule: ( ) Eamples: 5 f( ) = = n f( ) = 10 f( ) 1 3 = f( ) = d d Constant Multiple Rule: [ cf ( )] = c [ f ( )] d d Eamples: f( ) 4 3 = f( ) = 5 f( ) = f( ) 0.47 k ( ) = = m ( ) = 5 ( 3)

43 43 Sum and Difference Rules: The derivative of the sum (or difference) equals the sum (or difference) of the derivatives. d Eample: Find [ f ( ) ± g( )] d 3 f ( ) = g( ) = 5 5 More Derivative Practice: Check each using nd CALC 6 1. Find the equation of the tangent line to the graph of y = 3(5 ) at the point (5, 0).. Determine the points, if any, at which the function has a horizontal tangent line. 3 1 (a) y = (b) y =

44 44 Calculus for Business Lesson- Position Function and Velocity Name: Date: Objective: To learn how to use derivatives as a means to solve projectile motion problems. Velocity: Derivative of the Position Function = Instantaneous Velocity Function Position Function s(t) = Eample 1 A ball is dropped from a height of 100 feet. It s height, in feet, t seconds later is given by s(t) = -16t² What is the average velocity of the ball between 1 and seconds after it is dropped? change in dis tan ce Note: Average velocity is also Negative velocity means the object s speed is in a change in time downward direction What is the average velocity of the ball between 1 and 1.5 seconds after it is dropped? What is the average velocity of the ball between 1 and 1.1 seconds after it is dropped? What is the instantaneous velocity at eactly 1 second after it is dropped? Note: Instantaneous velocity is found by finding the first derivative then subbing in a given value of t. v(t) =

45 45 Eample s(t)= A ball is thrown straight down from the top of a 0-foot building with an initial velocity of -ft/sec. What is its velocity after 3 seconds? v(t)= v(3)=. What is its velocity after falling 108 feet? Find the position (height above ground) after falling 108 feet. Find the time t at which the ball is at this position. Find the velocity at this particular t. Eample3 A ball is thrown straight down from the top of a 0-foot building with an initial velocity of -ft/sec. What is its velocity after 3 seconds?

46 46 Calculus for Business Lesson Marginal Analysis Name: Date: Objective: To learn about cost, revenue, profit, marginal revenue, and marginal productivity. Terms: Productivity- ratio of output units to input units Marginal Productivity- additional output resulting from adding one more unit of input Cost (C())- may be fied or variable. Fied- not reliant upon number of units Variable- depends on number of units produced Total cost- Fied cost plus variable cost Marginal Cost (C ())- additional cost for producing one more input unit Revenue (R())- amount received in sale of units produced. Marginal Revenue (R ())- additional amount received when producing one more input unit Profit (P())- revenue minus cost: R() C() Marginal Profit- P () = R () C () Eample 1: The cost and revenue functions for production of table saws are: C() = & R()= , where is the number of saws produced. a. find the marginal cost b. find the marginal revenue c. evaluate R (1500) and R (4500) and interpret your answers d. graph R() and C() on the same set of aes. Determine the break-even points and the regions of gain and loss. e. Find the profit function, P() f. Find the marginal profit. g. evaluate P (1500) and P (4500) and interpret your answers

47 Eample : Union membership as a percentage of the labor force can be modeled by 3 M ( ) = , where is the number of years after 1900 and M is membership as a percentage of the labor force. a. Find the rate at which membership is changing in b. Find the rate at which membership is changing in c. Find the average rate of change between 1960 and Eample 3: A cell phone manufacturing company finds that the cost of producing phones is modeled by c( ) = and the profit from producing phones is modeled by p ( ) = dollars. a. find the marginal cost function b. find the marginal cost when 100 phones are produced c. find the marginal profit function d. find the marginal profit when 100 phones are produced e. find the marginal revenue function f. find the marginal revenue when 100 phones are produced

48 48 Calculus for Business Lesson Product and Quotient Rules Name: Date: The Product Rule If f and g are two differentiable functions, then their product fg is also a differentiable function and d [ f ( g ) ( )] = f ( g ) ( ) + g ( ) f ( ). d In words, The derivative of a product is the 1st function times the derivative of the nd plus the nd function times the derivative of the 1st. 3 Eample 1 Differentiate: h ( ) = ( 6+ 5)( 3) Eample If g ( ) = sin, find g ( ). Eample 3 sin cos y = Find dy d.

49 49 The Quotient Rule If f and g are two differentiable functions, then their quotient f/g is also a differentiable function (wherever g ( ) 0) and d f( ) g( ) f ( ) f( ) g ( ) d g( ) =. g ( ) [ ] In words, The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared. Eample 4 If + f( ) = 7, find f ( ). Eample 5 Differentiate sin t ( ) =. Eample 6 y = Find dy d.

50 50 Eample 7 y =. Find dy d. 5 Eample 8 Find f ( ) if f( ) = 4 4. Eample 9 Differentiate 1 y = Eample 10 f( ) = ( + + 1)

51 51 Higher Order Derivatives The derivative of the first derivative of f is called the second derivative of f. The derivative of the second derivative of f is called the third derivative of f. And so on for the nth derivative of f. Notation: Function y f() First Derivative y dy d [ ] D y f ( ) d d [ f( ) ] Second derivative y d y d [ ] D y f ( ) d ( ) d [ f ] Third derivative y 3 d y d 3 D [ y ] f ( ) [ f ] 3 3 d ( ) 3 d Fourth derivative (4) y 4 d y d 4 (4) D [ y ] f ( ) [ f ] 4 4 d ( ) 4 d n th derivative ( n) y n d y d n ( ) D [ y ] f n ( ) [ f( ) ] n n d d n 3 Eample 1 Find the second derivative of f( ) = 1. Eample When our function is a position function, y = s(t), then the first derivative is the the second derivative is the function = s (t) function =s (t)

52 5 90t Eample 3 An auto s velocity starting from rest is vt () = where v is measured in feet per second. 4t + 10 Find the vehicle s velocity and acceleration at each of the following times: (a) 1 second (b) 5 seconds (c) 10 seconds Eample 4 The graphs of f, f and f are shown on the same set of coordinate aes. Which is which? Eplain your reasoning.

53 53 Calculus for Business Lesson- Chain Rule Name: Date: Chain Rule Let y be a differentiable function of u, y = f(u), and u be a differentiable function of, u = g(). d ( ) Then, f g ( ) d d = [ fu ( )] [ g ( )]. d du d Think: (derivative of the outside) times (derivative of the inside) In other symbols: dy dy du = d du d Find the first derivative of each: 4 y = sin ( ) y = sin = (sin ) 3. y = y = 1 ( ) y cos ( 4t ) = 6. ( ) 3 f( ) = y = g ( ) = 3 5 tan( ) 7 9. cos g ( ) = csc 10. Find the derivative of the function 1 f( ) = + cos at the point π, π.

54 Find an equation of the tangent line to the graph of ( ) Graph the function and its tangent. f( ) 9 3 = at the point (1,4). 1. Determine the points at which the graph of f( ) = 1 has a horizontal tangent. 13. The sum of $8000 is deposited into an annuity paying r% interest compounded quarterly. After four 16 years, the value of the annuity is V ( r) = 8000( r). Evaluate V(5) and V (5), and interpret your answers.

55 PROPERTIES OF EXPONENTIAL FUNCTIONS AND LOGS 55

56 56 Calculus for Business Lesson- properties, equations with eponents and power and eponential functions Name: Date: Objectives: use the properties of eponents solve equations containing rational eponents eamine power and eponential functions Do Now: Use the eponential properties to simplify and rewrite the following epressions: (1) a a y = y () ( a ) = (3) ( ab ) = (4) a b = (5) a = y a (6) a = 0 (7) a = Using Eponential Function Properties to Solve for : Process 1 Process Eamples (each relates to Process 1 ): = = = 9

57 57 More Eamples (each relates to Process ): = = 4 6. ( 1) = 3 Power function: eponential function: Small Group Activity On your graphing calculator, simultaneously graph: y = 0.5, y = 0.75, y =, y = 5 (1) What is the range of each eponential function? () What is the behavior of each graph? (3) Do the graphs have any asymptotes? (4) (a) What point is on the graph of each function? (b) Why? Characteristics of graphs of y = n n > 1 0 < n < 1 domain range y-intercept behavior horizontal asymptote vertical asymptote

58 58 Graphing eponential functions, eponential growth and decay Objectives: graph eponential functions use eponential functions to determine growth and decay half-life Using Eponential Functions for Real World Applications: Eponential Growth or Decay: N = N 0 (1 + r) t (1) A researcher estimates that the initial population of honeybees in a colony is 500. They are increasing at a rate of 14% per week. What is the epected population in weeks? () Write a formula that represents the average growth of the population of a city with a rate of 7.5% per year. Let represent the number of years, y represent the most recent total population of the city, and A is the city s population now. What is the epected population in 10 years if the city s population now is,750 people? Graph the function for 0 0. (3) Meico has a population of about 100 million people, and it is estimated that the population will double in 1 years. If population growth continues at the same rate, what will be the population in: (a) 15 years (b) 30 years (c) graph the population growth for 0 time 50

59 59 (4) In 1990, Eponential City had a population of 700,000 people. The average yearly rate of growth is 5.9%. Find the projected population for 010. (5) Find the projected population of each location in 015: (a) (b) In Honolulu, Hawaii, the population was 836,31 in The average yearly rate of growth is 0.7%. The population in Kings County, New York has demonstrated an average decrease of 0.45% over several years. The population in 1997 was,40,384. (6) The population of Los Angeles County was 9,145,19 in If the average growth rate is 0.45%, predict the population in 010. Graph the equation for 0 time 0. (7) Radioactive gold 198 ( 198 Au), used in imaging the structure of the liver, has a half-life of.67 days. If the initial amount is 50 milligrams of the isotope, how many milligrams (rounded to the nearest tenth) will be left over after: (a) ½ day (b) 1 week

60 60 (8) If a farmer uses 5 pounds of insecticide, assuming its half-life is 1 years, how many pounds (rounded to the nearest tenth) will still be active after: (a) 5 years (b) 0 years (9) In 000, the chicken population on a farm was 10,000. The number of chickens increased at a rate of 9% per year. Predict the population in 005. Graph the equation for 0 time 15. (10) If Kenya has a population of about 30,000,000 people and a doubling time of 19 years and if the growth continues at the same rate, find the population (rounded to the nearest million) in: (a) 10 years (b) 30 years

61 61 Compound Interest Objectives: use eponential functions to determine compound interest (1) Rates can be compounded in different increments per year. Eponential growth occurs how often if the rate is compounded: annually: bi-annually: quarterly: monthly: weekly: daily: The general equation for eponential growth is modified for finding the balance in an account that earns compound interest. Compound Interest: A = P 1 + r n n t (1) If Charlie invested $1,000 in an account paying 10% compounded monthly, how much will be in the account at the end of 10 years? () Mike would like to have $0,000 cash for a new car 5 years from now. How much should be placed in an account now if the account pays 9.75% compounded weekly?

62 6 (3) Suppose $,500 is invested at 7% compounded quarterly. How much money will be in the account in: (a) ¾ year (b) 15 years (4) Suppose $4,000 is invested at 11% compounded weekly. How much money will be in the account in: (a) ½ year (b) 10 years (5) How much money must Cindy invest for a new yacht if she wants to have $50,000 in her account that earns 5% compounded quarterly after 15 years? (6) Carol won $5,000 in a raffle. She would like to invest her winnings in a money market account that provides an APR of 6% compounded quarterly. Does she have to invest all of it in order to have $9,000 in the account at the end of 10 years? Show your work and eplain your answer.

63 63 Eponential Functions with base e Objective: use eponential functions with base e Euler Savings Bank provides a savings account that earns compounded interest at a rate of 100%. You may choose how often to compound the interest, but you can only invest $1 over the course of one year. Eponential Growth or Decay (in terms of e): N = N 0 e kt (1) According to Newton, a beaker of liquid cools eponentially when removed from a source of heat. Assume that the initial temperature T i is 90 F and that k = (a) Write a function to model the rate at which the liquid cools. (b) Find the temperature T of the liquid after 4 minutes (t) (c) Graph the function and use the graph to verify your answer in part (b)

64 64 () Suppose a certain type of bacteria reproduces according to the model B = 100 e 0.71 t, where t is the time in hours. (a) At what percentage rate does this type of bacteria reproduce? (b) What was the initial number of bacteria? (c) Find the number of bacteria (rounded to the nearest whole number) after: (i) 5 hours (ii) 1 day (iii) 3 days (3) A city s population can be modeled by the equation y = 33,430e t, where t is the number of years since (a) Has the city eperienced a growth or decline in population? (b) What was the population in 1950? (c) Find the projected population in 010.

65 65 (4) If you invest $5,50 in an account paying 11.38% compounded continuously, how much money will be in the account at the end of: (a) 6 years 3 months (b) 04 months (5) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to pay for the note now if the note gains value at a rate of 9% compounded continuously?

66 66 Continuous Compound Interest Objective: use eponential functions to determine continuously compounded interest Continuously Compounded Interest: A = Pe rt (1) Tim and Kerry are saving for their daughter s college education. If they deposit $1,000 in an account bearing 6.4% interest compounded continuously, how much will be in the account when she goes to college in 1 years? () Paul invested a sum of money in a certificate of deposit that earns 8% interest compounded continuously. If Paul made the investment on January 1, 1995, and the account was worth $1,000 on January 1, 1999, what was the original amount in the account? (3) Compare the balance after 30 years of a $15,000 investment earning 1% interest compounded continuously to the same investment compounded quarterly. (4) Given the original principal, the annual interest rate, the amount of time for each investment, and the type of compounded interest, find the amount at the end of the investment: (a) P = $1,50; r = 8.5%; t = 3 years; compounded semi-annually (b) P = $,575; r = 6.5%; t = 5 years 3 months; compounded continuously

67 67 Properties of a logs, rewriting Eponential functions as logarithms, log graphs Objective: To learn what a logarithm is To learn the properties of logs To learn to rewrite an eponential function as a logarithm What is a logarithm? logarithm 1610s, Mod.L. logarithmus, coined by Scottish mathematician John Napier ( ), lit. "ratio-number," from Gk. logos "proportion, ratio, word" (see logos) + arithmos "number" (see arithmetic). Related: Logarithmic. eponent 1706, from L. eponentem (nom. eponens), prp. of eponere "put forth" (see epound). A mathematical term at first; the sense of "one who epounds" is 181. As an adjective, from 1580s. Logarithms are eponents. In some ways they can be inverses of eponential functions. Logarithms are functions because eponential functions are one-to-one functions. We cannot solve an equation like: must try an alternative technique. y = using the algebraic techniques we have learned so far. Therefore, we y Rule: = b is equivalent to y = logb The log to the base b is the eponent to which b must be raised to obtain. Properties of Logs log b 1 = 0 log b b = 1 logb b = log b b =, where > 0 logb MN = logb M + logb N M logb = logb M logb N N p logb M = p logb M Eample: Convert each into logarithmic form Convert each into eponential form 1 1. y = 4. log 5 5 =. 3 = 9 5. log a b = c = 6. log 3 = 5 9

68 68 What is a Natural Logarithm? y Rule: = b is equivalent to y = logb The log to the base b is the eponent to which b must be raised to obtain. Properties of Logs ln 1 = 0 ln b = 1 ln e =, where > 0 ln e = ln MN = ln M + ln N M ln = ln M ln N N ln M p = p ln M Eample: Convert each into logarithmic form Convert each into logarithmic form 1. y = e 4. ln 5 =. e = 5. ln b = c = e e 6. ln y = Eample: Graph each of the following on the same set of aes using the graphing calculator. 1. y = y. = 3. log y = 4. log = y 5. y = e y 6. = e 7. ln y = 8. ln = y y

69 69 Simplify log epressions, common logs, evaluate Objectives: simplify epressions using the properties of logarithmic functions define common logarithms evaluate epressions involving logarithms Problem Set: write the following epressions in simpler logarithmic forms: (1) log 7 log b u v () b a 1 (3) 3 m log b 1 (4) n log b u vw 3 n (5) log b (6) log b 3 p q (7) Use logarithmic properties to find the value of (without using a calculator): 1 logb = logb 9 + logb 8 logb 6 3

70 70 Write each epression in terms of a single logarithm with a coefficient of one: (8) 5logb + 4logb y (9) logb logb y u ie : logb u logb v = logb v 1 (10) 3logb + logb y logb z (11) 8logb c 4 3 (1) logb w logb u (13) 1 log (a b 3 b + ) 3 Common Logarithm: log 10 = log Change of Base Formula: log a ln a log log b a = = = logb ln b log p p a b Given log a n, evaluate each logarithm to four decimal places: (14) log 8 17 (15) log (16) log

71 71 Properties of Logarithmic Functions, Simplifying logarithmic epressions Objectives: eamine properties of logarithmic functions simplify epressions using the properties of logarithmic functions Use the properties of logarithmic functions to solve for : (1) log 5 = () log 4 64 = (3) log 8 = 3 (4) log 8 = 3 Use the properties of logarithmic functions to simplify each epression: (5) log 8 8 (6) log (7) log 10 1, 000 (8) log 64 (9) log (10) log (11) log e e (1) 3 log 5 5

72 7 Write the following epressions in simpler logarithmic forms: (13) log b y (14) log b 8 u v (15) mn 1 log b (16) log 4 pq b a (17) log 3 (18) log y 5 b b

73 73 Application Questions Objectives: solve real-world applications with natural logarithmic functions (1) Ana is trying to save for a new house. How many years, to the nearest year, will it take Ana to triple the money in her account if it is invested at 7% compounded annually? () At what annual percentage rate (to the nearest hundredth of a percent) compounded continuously will $6,000 have to be invested to amount to $11,000 in 8 years. (3) In 1990, Eponential City had a population of 14,000 people. In what year will the city have a population of about 00,000 people if it was growing at an eponential rate of k = 0.014? (4) If $5,000 is invested at an annual interest rate of 5% compounded quarterly, how long will it take the investment to double? (5) What was the annual interest rate (to the nearest hundredth of a percent) of an account that took 1 years to double if the interest was compounded continuously and no deposits or withdrawals were made during the 1-year period?

74 74 (6) If a car originally costs $18,000 and the average rate of depreciation is 30%, find the value of the car to the nearest dollar after 6 years. (7) How many years, to the nearest year, will it take for the balance of an account to double if it is gaining 6% interest compounded semiannually? (8) When Rachel was born, her mother invested $5,000 in an account that compounded 4% interest monthly. Determine the value of this investment when Rachel is 5 years old t (9) The decay of carbon-14 can be described by the formula A = A 0 e. Using this formula, how many years, to the nearest year, will it take for carbon-14 to diminish to 1% of the original amount? (10) In 00, a farmer had 400 pigs on his farm. He estimated that this population of pigs will double in 15 years. If population growth continues at the same rate, predict the number of pigs in: a. 010 b. 030

75 75 (11) If the world population is about 6 billion people now and if the population grows continuously at an annual rate of 1.7%, what will the population be (to the nearest billion) in 10 years from now? (1) If $100 is invested in an account that has an interest of 7% compounded quarterly, how long will it take for the balance to reach a value of $1,000? (13) What interest rate (to the nearest hundredth of a percent) compounded monthly is required for an $8,500 investment to triple in 5 years? (14) An optical instrument is required to observe stars beyond the sith magnitude, the limit of ordinary vision. However, even optical instruments have their limitations. The limiting magnitude L of any optical telescope with lens diameter D, in inches, is given by the equation L = logd. Use this equation to find the following to the nearest tenth: a. the limiting magnitude for a homemade 6-inch reflecting telescope. b. the diameter of a lens that would have a limiting magnitude of 0.6.

76 76 Review- Eponential and Logarithmic Functions Write each epression in terms of simpler logarithmic forms: 4 5 s 1 5 (1) logb y () log (3) log 7 b 8 (4) c b u log b 5 m n p 3 Given log a n, evaluate each logarithm to four decimal places: (5) log 3 4 (6) log1 5 (7) log Solve each equation and round answers to four decimal places where necessary: (8) log = 3 (9) log5 4 + log5 = log5 36 (10) = 75e (11) = log 6 1 (1) log 7 = (13) 49 log 4 = 1 (14) 10 = 7. 5 (15) log log5 = log log( 3) (16) log + log = 1 (17) log 4 = 3 (18) log9 (5 ) = 3log9 (19) log 0 log = 1 (0) 4 = 1.00 (1) e = () log( + 10) + log( 5) = 1 (3) log 6 16 log 6 36 = log 6

77 77 (4) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $ million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he need for this investment? (5) The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially: a. Epress the number of guppies as a function of the time t. b. Use your answer from part (a) to find how many guppies are present after 1 week? c. Use your answer from part (a) to find, to the nearest day, when will there be,000 guppies? (6) The relationship between intensity, i, of light (in lumens) at a depth of feet in Lake Erie is given by i log = What is the intensity, to the nearest tenth, at a depth of 40 feet? 1 (7) Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula i D = 10log, where D is the decibel level of sound, i is the intensity of the sound, and i 0 = 10-1 watt i 0 per square meter is a standardized sound level. Use this information and formula to find the intensity of the sound at the concert. (8) How many years, to the nearest year, will it take the world population to double if it grows continuously at an annual rate of %.

78 78 Calculus for Business Lesson- Derivatives of Eponentials and Logs Name: Date: Objective: to learn how to find the derivative of an eponential or logarithmic function Finding the Derivative of f ( ) = e Using limit process: RULE: Eamples: Find f () for each: 1. f 3 ( ) = e. f ( ) 6 = e 3 3. e f ( ) = 4. f 3 ( ) = e 5. 3 f ( ) = ( e 4) 6. f ( ) = ( e ) f ( ) = e 8. Finding the Derivative of f ( ) = ln Derivation: 6 f ( ) = e 9. f ( ) = 7e

79 79 Eamples: 1. f() = 4 ln. f ( ) = + ln 3. f ( ) = (8 3ln ) 3 4. f ( ) = ln(5) 5. f ( ) = 3 ln(4) 6. f ( ) = (ln ) 4 7. For what values of does f ( ) = ln( ) have a horizontal tangent line? Putting it all together Find f () for f ( ) = 6 9. Find f () for f ( ) = log8(3 )

80 80 Application Problems: 1. The percent of information retained t months after being tested on that information is given by: f(t) = 8 14 ln (t + 1). Evaluate f() and f () and interpret your answers.. The cost and revenue functions for producing units of a certain product are C() = and R() = 1811 ln(1. + 1). a. Find the marginal cost if 00 units are produced. b. Find the marginal revenue if 00 units are produced. c. Find the marginal profit if 00 units are produced. d. Graph the two functions and use your graphing calculator to find the break-even points. e. To make a profit, how many units should be produced.

81 81 Calculus for Business Lesson- Etrema and Critical Number Name: Date: Etreme Value Theorem If f is continuous on a closed interval I = [a,b] then f has both a minimum and a maimum value on I. f has a relative maimum at c or f(c) is a relative maimum of f if there is some open interval I containing c on which f(c) is the maimum. f has a relative minimum at c or f(c) is a relative minimum of f if there is some open interval I containing c on which f(c) is the minimum. Relative Etrema Let I be any interval (closed or open) containing the -value c. The etreme values, or etrema, of a function f on I are defined by: The absolute maimum of f on I is f(c) if f() c f() for all in I. The absolute minimum of f on I is f(c) if f() c f() for all in I. NOTE: A function need not have a maimum value or a minimum value on a given interval I. Process Find the first derivative and plug the -value of the coordinate of the relative min/ma & solve for f () Find the left- and right-hand side derivatives using Or lim c f ( ) f ( c) c Eamples: For each: determine any relative etrema and then determine the derivative at each relative etremum. 1. f( ) = on the interval (-1, ).. f( ) = f( ) = f( ) = + 4

82 Critical Values A critical value is a number c in the domain of f for which f () c = 0or f () c does not eist. *Relative Etrema Occur Only at Critical Numbers * One result of this theorem is that, on a closed interval, absolute etrema must occur at local etrema or at the endpoints. Long story short: Find the derivative, set = 0, solve for, plug back into original function. Eamples: 5. f( ) Find the critical value(s) = f( ) = Method for Finding Absolute Etrema on a Closed Interval [a,b] (1) Find the critical numbers of f in (a,b). () Evaluate f at each critical number in (a,b). (3) Evaluate f(a) and f(b). (4) Compare: the least of these y-values is the minimum; the greatest is the maimum. 7. Find the maimum and minimum values of + 5 f( ) = on the interval [0, 5] Find the maimum and minimum values of 3 f( ) 1 = on the interval [0, 4] Find the maimum and minimum values of g ( ) = on the interval [-1, 1].

83 83 Calculus for Business Lesson- First Derivative Test Name: Date: Objective: Learn about Increasing and Decreasing Functions & the First Derivative Test for Etrema Graphically Numerically Sign of the Derivative f increasing When s go up, y s go up. f ( ) > 0 f decreasing When s go up, y s go down f ( ) < 0 First Derivative Test If f changes from to + at c, f has a relative minimum at (c, f(c)). If f changes from + to at c, f has a relative maimum at (c, f(c)). If f doesn t change sign at c, then f(c) is not a relative etremum for f. Where c is a critical value. Process: Step 1 Find all critical numbers of f in the given interval; break the interval into smaller intervals using these critical values, points of discontinuity, and the endpoints. Step Create a sign chart, by picking an -value in each interval and finding the sign of f there. Step 3 For each sub-interval state whether f is increasing or decreasing there. Eample 1. Tell where f 3 ( ) = + is increasing, decreasing, and identify any relative etrema Tell where f( ) = is increasing, decreasing, and identify any relative etrema. 3. Use the graph of f to tell where f is increasing, decreasing, and identify any relative etrema.

84 84 Calculus for Business Lesson- Concavity, Inflection, nd Derivative Test Name: Date: Objective: Concavity- To use derivatives to determine concavity and points of inflection Function has a hill (concave down) or a valley (concave up) Points of Inflection- Location at which a function goes from being concave down to concave up (or vice versa) Concave Up Concave Down Inflection Second Derivative Test for Concavity Process: 1. Find the 1 st derivative. Set = 0 and solve 3. Find the nd derivative 4. Set = 0 and solve (result could be a point of inflection) 5. Create sign chart Guided Eample 1 1 f + 3 f '( ) = + 3 ( ) = 0 = + = 0, 1 f "( ) = = + 1 = 1/ Intervals (, 1) (-1, -1/) (-1/, 0) ( 0, ) Test Values - -3/4-1/4 Sign of f () Sign of f () Conclusion Increasing Concave down Increasing Concave down Increasing Concave up Increasing Concave up Since the function went from concave down to concave up at = -1/, that must be the inflection point.

85 85 Determine relative etrema, concavity, points of inflection g ( ) = +.. f( ) = + 3 Let f be a function with f () c = 0and the second derivative f eists on an open interval containing c. Then if f (c)>0, then f has a relative minimum at (c, f(c)), if f (c)<0, then f has a relative maimum at (c, f(c)), if f (c)=0, then test fails; must use the 1st Derivative Test for Etrema (could be ma, min, or neither) Find all the relative etrema 3 3. f( ) = f ( ) = + 1.

86 86 Calculus for Business Lesson- Optimization Name: Date: Optimization Problems Applied Minimum and Maimum Problems Method for Solving: Step 1 Define variables needed to describe the quantity to be optimized. Step Write a primary equation for the quantity to be optimized. Step 3 Use other info given in the problem to write a secondary equation that relates the variables used in the primary equation. Solve this equation to get one variable in terms of the other. Use this equality to rewrite the primary equation in terms of one variable. Step 4 Find a feasible domain for the function you wrote in Step 3, i.e., upper and lower bounds on the variable that make sense for your problem. Step 5 Find absolute or relative ma or min of your function on the feasible domain. Step 6 Write your answer in an English sentence. Eample 1 Sr. Karlien wants to create a rectangular garden. She has available 100 feet of fencing to make the border. What are the dimensions of the plot with the largest area that can be enclosed with this fence? Eample Find positive numbers whose product is 19 and whose sum is a minimum. Eample 3 A manufacturer wants to design an open bo having a square base and a surface area of 108 square inches. What dimensions will produce a bo of maimum volume? Eample 4 A bo with a square base and open top must have a volume of 3,000 cubic cm. Find the dimensions of the bo that minimize the amount of material used

87 87 Mied Problem Set- Optimization 1. A bo with an open top and a square base is to have a surface area of 108 square inches. Determine the dimensions that will maimize the volume of the bo.. Two posts, one 1 feet high and the other 8 feet high, are 30 feet apart. They are anchored by two wires running from the top of each pole to a single stake in the ground at a point between the two posts. Where should the stake be placed so that the minimum amount of wire is used? 3. A bus stop shelter has two square sides, a back, and a roof. The volume is 56 cubic feet. What dimension will allow for the least amount of material to be used? 4. Scrumptious Soup Company makes a soup can with a volume of 50 cm3. What dimensions will allow for the minimum amount of metal to produce the can? 5. A jumbo-size can of baked beans has a volume of 600 cm3. What dimensions will allow for the minimum amount of metal to produce the can? 6. The surface area of a can of chunked chicken requires 60 square inches of material. What dimensions allow for maimum volume? 7. A large can of tuna requires a surface area of 100 square inches. What dimensions provide the maimum volume? 8. A wire of length 1 inches can be bent into a circle, a square, or cut to make both a circle and a square. How much wire should be used for the circle if the total area enclosed by the figure(s) is to be a minimum? A maimum? circle square 9. A window consisting of a rectangle topped by a semicircle is to have an outer perimeter P. Find the radius of the semicircle if the area of the window is to be a maimum.

88 A rectangular field as shown is to be bounded by a fence. Find the dimensions of the field with maimum area that can be enclosed with 1000 feet of fencing. You can assume that fencing is not needed along the river and building. 11. A company manufactures cylindrical barrels to store nuclear waste. The top and bottom of the barrels are to be made with material that costs $10 per square foot and the rest is made with material that costs $8 per square foot. If each barrel is to hold 5 cubic feet, find the dimensions of the barrel that will minimize the total cost. 1. The operating cost of a truck is cents per mile when the truck travels miles per hour. If the driver earns $6 per hour, what is the most economical speed to operate the truck on a 400 mile turnpike? Due to construction, the truck can only travel between 35 and 60 miles per hour. 13. A furniture business rents chairs for conferences. A contract is drawn to rent and deliver up to 400 chairs for a particular meeting. The eact number would be determined by the customer later. The price will be $90 per chair up to 300 chairs. If the order goes above 300 chairs, the price would be reduced by $0.5 per chair for every additional chair ordered above 300. This reduced price would be applied to the entire order. Determine the largest and smallest revenues this business can make under this contract. 14. The speed of traffic through the Lincoln Tunnel depends on the density of the traffic. Let S be the speed in miles per hour and D be the density in vehicles per mile. The relationship between S and D is D approimately S = 4 for D 100. Find the density that will maimize the hourly flow A commercial cattle company currently allows 0 steer per acre of grazing land. On average a steer weighs 000 pounds at the market. Estimates by the Department of Agriculture indicate that the average weight per steer will be reduced by 50 pounds for each additional steer added per acre of grazing land. How many steer per acre should be allowed in order to optimize the total market weight of the cattle?

89 16. Catching Rainwater. A 115-ft 3 open-top rectangular tank with a square base ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an ecavation charge proportional to the product y. If the total cost is c = 5( + 4y) + 10y, what values of and y will minimize it? 17. Designing a Poster. You are designing a rectangular poster to contain 50 in of printing with a 4-in. margin at the top and bottom and a -in. margin at each side. What overall dimensions will minimize the amount of paper used? 18. Vertical Motion. The height of an object moving vertically is given by s = -16t + 96t + 11, with s in ft and t in sec. Find (a) the object's velocity when t = 0, (b) its maimum height and when it occurs, and (c) its velocity when s = Finding an Angle. Two sides of a triangle have lengths a and b, and the angle between them isθ. What value of θ will maimize the triangle's area? (Hint: A = ½ ab sin θ.) 0. Designing a Can. What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 1000 cm 3? 1. Designing a Can. You are designing a 1000-cm 3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure r units on a side. The total amount of aluminum used up by the can will therefore be A = 8r + 7 rh the ratio of h to r for the most economical can is?. Designing a Bo with Lid A piece of cardboard measures 10- by 15-in. Two equal squares are removed from the comers of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular bo with lid. (a) Write a formula V() for the volume of the bo. (b) Find the domain of V for the problem situation and graph V over this domain. (c) Use a graphical method to find the maimum volume and the value of that gives it. (d) Confirm your result in (c) analytically. 89

90 3. Designing a Suitcase. A 4- by 36-in. sheet of cardboard is folded in half to form a 4- by 18-in. rectangle as shown in the figure. Then four congruent squares of side length are cut from the corners of the folded rectangle. The sheet is unfolded, and the si tabs are folded up to form a bo with sides and a lid. (a) Write a formula V() for the volume of the bo. (b) Find the domain of V for the problem situation and graph V over this domain. (c) Use a graphical method to find the maimum volume and the value of that gives it. (d) Confirm your result in (c) analytically. (e) Find a value of that yields a volume of 110 in Quickest Route. Jane is mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline from the point nearest the boat. She can row mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time? 5. Inscribing Rectangles. A rectangle is to be inscribed under the arch of the curve y = 4cos(0.5) from = -π to = π. What are the dimensions of the rectangle with largest area, and what is the largest area?

91 91 Calculus for Business Lesson- Business Applications Name: Date: Objective- To learn about maimizing yield and revenue while minimizing costs of inventory control. Cobb-Douglas productivity models Recall: Revenue (R())- amount received in sale of units produced. Maimizing Revenue involves determining the price at which a quantity should be sold in order to obtain maimum revenue. Profit (P())- revenue minus cost: R() C(); where R is revenue and C is cost Maimizing Profit involves determining the price at which a quantity should be sold in order to obtain maimum profit. Eample 1: Rosie s Discount Mart sells paperback books. At price $p, Rosie can sell q( p) = 5 p + 55p + 60 books. What price would give Rosie the greatest revenue? Eample : Mark s restaurant can produce one chicken sandwich for $. The sandwiches sell for $5 each and at this price, his customers buy 100 sandwiches each month. Because of rising costs from suppliers, the restaurant is planning on raising the price of the sandwich. Based on the results of the previous price increases, Mark estimates that he will sell 10 fewer sandwiches each month for every $1 he increases the price. At what price should the sandwiches be sold to maimize Mark s profit? What is the maimum profit?

92 9 Maimizing yield involves optimization situations in which an increase in one variable causes a decrease in another related variable. Eample 3: Taylor s Orchard has always planted 40 trees per acre, with a yield of 300 apples per tree. For each additional tree planted per acre, the yield drops by 5 apples per tree. How many trees should be planted per acre for maimum yield? Inventory Control: deciding the most appropriate time to produce quantity of an item. Possibilities: 1. Produce all of a given item at the beginning of a year. Advantages: All items are on hand for immediate sale. Cheaper to produce at bulk rate. Disadvantage: Need to store unsold items. Produce items throughout the year as needed. Advantage: No cost to store items Disadvantage: Will cost more to produce in smaller quantities Eample: MAC Boats anticipates a demand for 1,000 fishing boats over the net year. The start-up costs of each production run are $5000, and it costs the company $40 to store each boat during the year. How many boats should be made during each production run to minimize total costs? How many production runs should there be?

93 93 Cobb-Douglas Productivity Model a The productivity of a plant or factory is given by: K a 1 q = y where q is the number of units produced, is the number of employees, and y is the operating budget or capital. K and a are constants that are determined by each individual factory or plant, with 0 < a < 1. Eample: Brian s Beach Shop manufactures surfboards. Daily operating costs are $80 per employee and $5 per machine. The number of surfboards produced each day is given by q = 4.5 y, where is the number of employees and y is the number of machines. If Brian wants to produce 90 surfboards each day at minimum cost, how many employees and how many machines should he use? Point of Diminishing Returns the point at which the rate of growth of the profit function begins to decline. The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on 1 3 advertising is P ( s) = s + 6s Find the amount of money the company should spend on advertising 10 in order to yield the maimum profit. Find the point of diminishing returns.

94 94 Mied Problem Set- Business Applications 1. Marge is planning a casino bus trip. If 100 people sign up, the cost is $300 per person. For each additional person above 100, the cost per person is reduced by $ per person. To maimize Marge's revenue, how many people should go on the trip? What is the cost per person?. A charter dinner cruise boat holds 50 people. The company will charter the boat for 35 or more people. If 40 people are on board, the cost per person is $150. For each additional person, the cost per person is reduced by $3. How large a group should be on the cruise to maimize the revenue? What is the cost per person? 3. A peach orchard has an average yield of 90 bushels per tree if there are 0 trees per acre. For each additional tree per acre, the yield decreases by 3 bushels per tree. How many trees should the orchard plant per acre to maimize the yield? What is the total yield? 4. An orange grove plants 5 trees per acre and gets a yield of 116 bushels of oranges per tree. For each additional tree planted per acre, the yield decreases by 4 bushels per tree. How many trees should be planted per acre to maimize the yield? What is the total yield? 5. Matt's Top 40 rents movies. If the rental fee is $4 each, Matt knows he can rent 100 movies per week. For each additional $1 increase in the rental fee, Matt loses 10 rentals per week. What rental fee should Matt charge for a movie to maimize his revenue? If each movie costs Matt $, what should his rental price be to maimize profit? 6. Barb's Babysitting charges $8 per hour and, at that rate, averages 0 jobs each week. For each additional $1 charge per hour, the number of jobs per week declines by two. What should Barb charge per hour to maimize revenue? If Barb spends $ per job on supplies, what should she charge per hour to maimize her profits? 7. When Jerry's Jalopies charges $0 to do an oil change, there are 80 customers per month. For each additional $1 charge, the number of customers per month drops by four. If it costs Jerry $5 per customer for the supplies, what should he charge for an oil change to maimize profits? How many customers will there be each month? 8. Missy's Tutoring charges $35 per hour for a tutoring session and has 60 clients each week. For each additional $1 charge, there are two fewer clients each week. It costs $1 per client for supplies. What should Missy charge per hour to maimize profits? How many clients will there be each week? 9. A bicycle plant assembles 000 bicycles per month. Each production run costs $100, and it costs $0 to store a bicycle for a month. How many production runs should the plant use to minimize inventory costs? How many bicycles are assembled in each production run? 10. A soda bottling company bottles 0,000 cases of lime soda each year. Each production run costs $1400, plus a storage cost of $18 per case. How many production runs should the company use to minimize inventory costs? How many cases are bottled in each production run? 11. A tetbook publisher estimates that the demand for a new calculus book will be 6000 copies. Each book costs $1 to print, and set-up costs are $1800 for each printing run. Storage costs $3 per book per year. How many books should be printed per printing run and how many printings should there be to minimize inventory costs? 1. A car dealer epects to sell 500 new convertibles this year. Each convertible costs the dealer $16,000 plus a fied $5000 delivery fee per order. It costs $500 to store each car for a year. How many orders should be made and how many cars should there be in each order to minimize inventory costs?

95 95 Calculus for Business Lesson- Implicit Differentiation Name: Date: So far we have been working with functions in eplicit form (equations solved for y in terms of ). Now we will learn how to work with implicit forms of equations (equations not solved for y or not easily solved for y). Eplicit Form of y = function of ; y written eplicitly in terms of 1 y = Implicit Form of y = function of ; an equation that relates y to but where y cannot necessarily be isolated y = 1 Implicit Differentiation Step 1 Differentiate both sides of the equation with respect to. Step Collect all terms which contain dy on one side of the equation and everything else on the other side. d Step 3 Factor dy out of all the terms on the one side. d Step 4 Solve for dy Note: dy d d will be in terms of and y. Eample 1 Graph + y = 16 and find dy d implicitly. Find all points where the graph has a horizontal tangent. Using implicit form of dy d : dy Using eplicit form of d : Find all points where the graph has a vertical tangent. Using implicit form of dy d : Using eplicit form of dy d :

96 96 Eample a. Find the first derivative of b. Find f () y+ y= by implicit differentiation. Eample 3 y a. Graph the hyperbola: = b. Find an equation of the tangent line to the graph of the hyperbola y = 1 at the point (3,-). 6 8 Eample 4 Find an equation of the tangent line to the astroid 3 y = at the point (8,1).

97 97 Calculus for Business Lesson- Related Rates Name: Date: Objectives: Identify a mathematical relationship between quantities that are each changing. Use one or more rates to determine another rate. Process: Step 1: Draw a diagram. Step : Determine which quantities and rates are given, and which to be found. Step 3: Identify the primary function to use. (Often this is a formula from geometry.) Step 4: Differentiate with respect to the independent variable Step 5: Write a related rates equation Step 6: Substitute known quantities and solve for desired rate. [NOTE: Do not substitute known quantities before this last step!] Type 1: Eplicit Function of One Variable Eamples 1. Air is being blown into a sphere at the rate of 6 cubic inches per minute. How fast is the radius changing when the radius of the sphere is inches?. The edge of a cube is increasing at a rate of inches per minute. At the instant the edge is 3 inches, how fast is the volume increasing?

98 3. A point moves along the curve ( ) (a) y = 3 such that its -coordinate is increasing at 4 units per second. At the moment = 1, how fast is the y-coordinate changing? Interpret your answer based on the shape of the graph and the location of the point. 98 (b) At the moment = 1, how fast is the point s distance from the origin changing? Type - Implicit Function of One Variable 4. A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? Type 3: Functions of Two Variables rates given 5. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of cm /min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm? Type 4: Functions of Two Variables 1 rate given--secondary equation needed 6. A water tank has the shape of an inverted circular cone with base radius m and height 4 m. If water is being pumped into the tank at a rate of m 3 /min, find the rate at which the water level is rising when the water is 3 m deep.

99 99 Mied Problem Set- Related Rates 1. A conical tank is being filled with water. The tank has height 4 ft and radius 3 ft. If water is being pumped in at a constant rate of cubic inches per minute, find the rate at which the height of the cone changes when the height is 6 inches. Note the difference in units.. A searchlight is positioned 10 meters from a sidewalk. A person is walking along the sidewalk at a constant speed of meters per second. The searchlight rotates so that it shines on the person. Find the rate at which the searchlight rotates when the person is 5 meters from the searchlight. 3. A person 5 feet tall is walking toward an18 foot pole. A light is positioned at the top of the pole. Find the rate at which the length of the person s shadow is changing when the person is 30 feet from the pole and walking at a constant speed of 6 feet per second. 4. The length of a rectangle increases by 3 feet per minute while the width decreases by feet per minute. When the length is 15 feet and the width is 40 feet, what is the rate at which the following changes: a. area b. perimeter c. diagonal

100 The volume of a tree is given by V = C h where C is the circumference of the tree in meters at 1π ground level and h is the height of the tree in meters. Both C and h are functions of time t in years. dv a. Find a formula for. What does it represent in practical terms? dt b. Suppose the circumference grows at a rate of 0. meters/year and the height grows at a rate of 4 meters/year. How fast is the volume of the tree growing when the circumference is 5 meters and the height is meters? 6. a. When the radius of a spherical balloon is 10 cm, how fast is the volume of the balloon changing with respect to change in its radius? b. If the radius of the balloon is increasing by 0.5 cm/sec, at what rate is the air being blown into the balloon when the radius is 6 cm? c. When the volume of the balloon is 50 cubic cm, at what rate is the radius of the balloon changing? 7. When hyperventilating, a person breathes in and out very rapidly. A spirogram is a machine that draws a graph of the volume of air in a person s lungs as a function of time. During hyperventilation, the person s spirogram trace might be represented by V = cos( 00πt ) where V is the volume of air in liters in the lungs at time t minutes. a. Sketch a graph of one period of this function. b. What is the rate of flow of air in liters/minute? Sketch a graph of this function. c. Mark the following on each of the graphs above. i) the interval when the person is breathing in ii) the interval when the person is breathing out iii) the time when the rate of flow of air is a maimum when the person is breathing in

101 101 Calculus for Business Lesson- Antiderivatives and Indefinite Integrals Name: Date: Anti-derivative = Anti-differentiation = Working backwards Eample: is an antiderivative of The general form of the antiderivative is given as F() + C, where C is any constant, called the constant of integration. Any antiderivatives of f will differ only by a constant. Notation: Let y = + C. Then dy d =. This is called a differential equation, an equation that involves, y, and a derivative of y. We can re-write this equation as dy = d. This is called the differential form of the equation. In general, Let y = F( ) and F ( ) = f( ). Then dy d = f() dy = f() d is a differential equation and is the differential form of the equation. We write this using integral notation: y = f ( ) d = F( ) + C is the indefinite integral sign; it means the antiderivative of f with respect to. f() is the integrand, the function to be antidifferentiated. = the variable of integration C = the constant of integration Rules of Integration 0 d = n d = k d = kf ( ) d =

102 10 Steps for Integrating: (1) Re-write the original integral into a form in the table. () Integrate (take the antiderivative). (3) Simplify. Eamples 1 1. d = d = d = 4. ( t 1 ) 4 dt = 10

103 103 Calculus for Business Lesson- Antiderivatives and Initial Conditions and Particular Solutions Name: Date: Objective- To learn how to find anti-derivatives involving particular values and initial conditions. dy The general solution to the differential equation 3 1 d = is = = ( 3 1) dy (3 1) d y d If we are given the value of y for one value of this is called an initial condition then we can get a particular solution; that is, we can find a particular constant C and write a specific antiderivative as our answer. Eamples 1. Given F( ) = ( 3 1) d and F () = 4. Find the particular solution. F() =. Find g() if = and g(0) = -1. g ( ) 6 3. A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 ft. Using the fact that the acceleration due to gravity is 3 ft/sec², (a) find the position function giving the height s as a function of time t. (b) When does the ball hit the ground? 103

104 4. The rate of growth dp/dt of a population of bacteria is proportional to the square root of t, where P is the dp population size and t is the time in days ( 0 t 10). That is, k t dt =. The initial size of the population is 500. After 1 day the population has grown to 600. Estimate the population after 7 days The Grand Canyon is 1800 m deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of the time t in seconds. How long will it take the rock to hit the canyon floor? 104

105 105 Calculus for Business Lesson- The Fundamental Theorem of Calculus & MVT Name: Date: This is the important theorem that links the branches of calculus--differential and integral. Fundamental Theorem of Calculus If the function f is continuous on [a, b] and F is an antiderivative of f on the interval [a, b], then b a f ( ) d = F( b) F( a) This says that taking a definite integral, the limit of a Riemann sum, the area under a curve, is the same as taking an antiderivative. Look Ma, No long sum! b b Notation: f ( ) d = F( ) ] = F( b) F( a) a a Eample 1 Use the Fundamental Theorem of Calculus to evaluate the following definite integrals: a. 7 3 dt b. ( ) d t dt d. 3 c. ( t 9 ) d 105

106 106 1 Eample Find the area of the region bounded by the graphs of y =, the -ais, =1 and =. Eample 3 Find the area under the graph of y = + sin from 0 to π. 0 π Eample 4 Evaluate 0 1d 106