TI-84 Graphing Calculator Guide designed to … Graphing Calculator Guide designed to accompany. ......

TI-84 Graphing Calculator Guide designed to accompany Calculus Concepts: An Informal Approach to the Mathematics of Change 5e Preliminaries 1 0.1 Setup and Calculations 1 0.1.0 Turning the Calculator ON or OFF 1 0.1.1 Settings 1 0.1.1a Mode 1 0.1.1b Statistical Setup 1 0.1.1c Graphing Window Format 2 0.1.2 Calculating 2 0.1.3 Accessing 2ND and ALPHA Functions 3 0.1.4 Using ENTRY and ANS to Reuse Expressions 4 0.1.5 Storing and Recalling Values 4 Chapter 1 Ingredients of Change: Functions and Limits 5 1.1 Functions: Four Representations 5 1.1.1 Entering a Function into Y1 5 1.1.2 Evaluating a Function that is in Y1 5 1.1.2a Changing the number of decimal places reported by the calculator 6 1.1.2b Re-evaluating a previous entry 6 1.1.2c Editing a previous entry 6 1.1.3 Using Equation Solver 6 1.1.4 Graphing a Function 7 1.2 Function Behavior and End Behavior Limits 9 1.2.1 Investigating End Behavior 9 1.2.1a Graphically investigating end behavior 9 1.2.1b Numerically investigating end behavior 9 1.3 Limits and Continuity 10 1.3.1 Numerically Investigating Function Behavior 10 1.3.2 Graphing a Piecewise Defined Function 10 1.4 Linear Functions and Models 11 1.4.1 Entering Data 11 1.4.2 Graphing a Scatter Plot 12 1.4.2a Setting up STAT PLOT 12 1.4.2b Viewing a scatter plot 12 1.4.3 Calculating First Differences 12 1.4.4 Fitting a Linear Model to Data 14 1.4.5 Aligning Data 15 1.4.6 Using Unrounded Models in Y1 16 1.4.7 Restoring a Missing Column on the List Screen 16 1.5 Exponential Functions and Models 17 1.5.1 Calculating Percentage Changes 17 1.5.2 Calculating Percentage Changes for Data 18 1.5.3 Fitting an Exponential Model to Data 18

1.6 Models in Finance 19 1.6.1 Evaluating a Loan with Simple Interest 19 1.6.2 Evaluating a Loan with Compound Interest 19 1.6.3 Calculating Doubling Time 20 1.7 Constructed Functions 22 1.7.1 Constructing a Function Using Multiplication 22 1.7.2 Constructing a Function Using Composition 23 1.7.3 Approximating the Model of an Inverse Function 24 1.8 Logarithmic Functions and Models 25 1.8.1 Fitting a Logarithmic Model 25 1.8.2 Fitting the Inverse of a Log Model 25 1.9 Quadratic Functions and Models 27 1.9.1 Fitting a Quadratic Model 27 1.9.2 Calculating Second Differences 28 1.10 Logistic Functions and Models 29 1.10.1 Fitting a Logistic Model 29 1.11 Cubic Functions and Models 30 1.11.1 Fitting a Cubic Model 30 1.12 Cyclic Functions and Models 31 1.12.1 Graphing a Sine Function 31 1.12.2 Fitting a Sine Model 31 Chapter 2 Describing Change: Rates 33 2.1 Measures of Change over an Interval 33 2.1.1 Calculating Change 33 2.1.2 Calculating Average Change 33 2.1.3 Calculating Percentage Change 34 2.1.4 Graphing a Secant Line 34 2.4 Rates of Change – Numerical Limits and Nonexistence 35 2.4.2 Numerically Estimating Rate of Change 35 2.4.4 Graphing the Tangent Line 36 Chapter 3 Determining Change: Derivatives 37 3.1 Simple Rate-of-Change Formulas 37 3.1.1 Writing the derivative of a regression function 37 3.1.1a Accessing coefficients from a regression function 37 3.1.1b Calculating with coefficients from the regression function 38 3.1.1c Using coefficients from the regression function in Y= 38 3.1.2 Evaluating a function in Y= using coefficients from a regression 38 3.4 Rates of Change for Composite Functions 39 3.4.1 Calculations involving the Chain Rule 39 3.4.1a Using Y= with the Chain Rule 39 3.4.1b Graphing a function involving the Chain Rule 39 Chapter 4 Analyzing Change: Applications of Derivatives 40 4.0 nDeriv — TI’s Numerical Derivative Routine 40 4.0.1 Using nDeriv to Calculate and Graph 40 4.0.1a Calculating with nDeriv 40 4.0.1b Graphing with nDeriv 41

4.0.1c Calculating with nDeriv in Y= 41 4.1 Linearization and Estimates 42 4.1.1 Graphing and Calculating with Linearizations 42 4.1.1a Graphing a linearization 42 4.1.1b Calculating with a linearization 42 4.2 Relative Extreme Points 43 4.2.1 Solving for Zeros of the Derivative 43 4.4 Inflection Points and Second Derivatives 44 4.4.1 Locating an Inflection Point 44 Chapter 5 Accumulating Change: Limits of Sums and the Definite Integral 45 5.2 Limits of Sums and the Definite Integral 45 5.2.1 Using NUMINTL to Estimate the Accumulated Change 45 5.2.1a NUMINTL with Right-Rectangles 45 5.2.1b NUMINTL with Left-Rectangles 46 5.2.1c NUMINTL with Midpoint-Rectangles 47 5.6 The Definite Integral - Algebraically 48 5.6.1 fnInt — TI’s numerical integration routine 48 5.6.2 Evaluating a Piecewise-Defined Integral 49 5.7 Differences of Accumulated Change 50 5.7.1 Calculating Area of Between Two Intersecting Curves 50 Chapter 6 Analyzing Accumulated Change: Integrals in Action 51 6.1 Perpetual Accumulation and Improper Integrals 51 6.1.1 Numerically Evaluating an Improper Integral 51 6.3 Calculus in Economics: Demand and Elasticity 52 6.3.1 Locating the Point of Unit Elasticity 52 Chapter 7 Ingredients of Multivariable Change: Models, Graphs, Rates 53 7.1 Multivariable Functions and Contour Graphs 53 7.1.0 Evaluating a Multivariable Function 53 7.1.3 Solving a Multivariable Function for One Input Variable 53 7.2 Cross-Sectional Models and Rates of Change 55 7.2.1 Fitting a Cross-Sectional Equation 55 7.3 Partial Rates of Change 56 7.3.1 Evaluating a Partial Derivative at a Point 56 7.4 Compensating for Change 57 7.4.1 Evaluating the Derivative at a Point 57 7.4.2 Evaluating the Slope of a Tangent Line at a Point 57 Chapter 8 Analyzing Multivariable Change: Optimization 58 8.2 Multivariable Optimization 58 8.2.1 Solving a system of two linear equations 58 8.3 Optimization under Constraints 59 8.3.1 Solving a system of three linear equations 59 8.3.2 Identifying the type of constrained point 60

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Preliminaries 0.1 Setup and Calculations 0.1.0 Turning the Calculator ON or OFF

• Use ON at the bottom left of the key pad to turn the calculator on. If the calculator is not on the home screen when turned on, 2ND MODE

• To turn the calculator OFF, use

[QUIT] returns to the home screen.

2ND ON. The 2ND

function key is the blue key near the top left of the key pad.

0.1.1 Settings

This guide assumes that the calculator has been setup as follows: 0.1.1a Mode

• Use the cursor and settings should be as in the screenshot.

ENTER

• QUIT (

keys to change any settings that are incorrect. The cursor keys, , are located near the top right of the keypad.

2ND MODE

) to exit the MODE screen.

0.1.1b Statistical Setup • Specify the statistical setup using

STAT

• 5 selects 5: SetUpEditor

• Type 2ND 1 , 2ND 2 , 2ND 3 , 2ND 4 , 2ND 5 , 2NDCommas are necessary. The comma key is located directly above 7 on the keypad.

6

• ENTER

executes the setup command and returns “Done” when the calculator has completed the task.

2 5e Calculus Concepts; LaTorre, et al. TI-84 Calculator Guide



• STAT ENTER shows the complete set of lists.

• QUIT (2ND MODE) exits the list editor and returns to the home screen.

0.1.1c Graphing Window Format

• Specify the graphing window setup using FORMAT (2ND ZOOM). The ZOOM key is located directly under the screen. Use the cursor and ENTER keys to change any settings that are incorrect.

• QUIT (2ND MODE) exits the graphing window format screen and returns to the home screen.

0.1.2 Calculating

When evaluating expressions using a calculator, the order of operations is important. *It may be necessary to use parentheses that do not appear in handwritten expressions. * Negative signs are not the same as minus signs on a TI calculator. The negative sign (–) key is along the bottom row of the keypad. The minus key is third from the bottom in the right column of the keypad. Practice Expressions:

• 7 32( 5)− −− −

• 4 15 21 4

− − −−

• 108 7689 24

−−−

• 2

2

(2 4)(5 3)+−

Preliminaries 3 TI-84 Calculator Guide



• 2 2

2 2

10 58 3 ( 2)

−+ − −

• 21( 106) 35( 40)2(51 58)

− − −−

0.1.3 Accessing 2ND and ALPHA Functions

Most of the keys on the TI-84 have multiple uses. A key’s main function is indicated on the key itself. Above the key are indicated the two other functions in blue and green type. • Blue type indicates a function accessed using the blue 2ND key near the top left of the

keypad before typing the desired key For example, to access the symbol for use in calculations type 2ND xe .

• Green type indicates a function accessed using the green ALPHA key directly beneath the 2ND key.

For example, to access the letter V, type ALPHA 6.




0.1.4 Using ENTRY and ANS to Reuse Expressions

The TI-84 can recall previous expressions it has evaluated. The TI-84 stores the last 25 expressions that were evaluated on the home screen, even after the screen has been cleared. • To recall a previously evaluated expression, use ENTRY (2ND ENTER).

Repeatedly typing ENTRY recalls expressions from most recent entered to earliest entered.

• After recalling an expression to the home screen, the cursor keys as well as delete, DEL, and insert, INS (2ND DEL) can be used to modify the expression.

• ENTER is used to re-evaluate the expression. 0.1.5 Storing and Recalling Values

The TI-84 can store a numerical value for later use in an expression or calculation. Each of the alpha keys can be used to access a storage location. • The STO key stores a number for later use. With a number or expression on the

home screen, type STO A (ALPHA MATH) to store the number into storage location A. Any of the letters can be used for storage. It is best to avoid X because the calculator uses this position to store calculated input values.

For example, to store the real number equivalent of 32 5+

(to 32 decimal-place

accuracy) in T...

• Type the expression 32 5+

on

the home screen • Type STO T (ALPHA 4) • ENTER evaluates the

expression and stores the result in T.

• The recall the number stored in an alpha location, type the letter (either alone or in an

expression). For example, to use the number that has been stored in T while evaluating another

expression... • Type the expression 24 5T−

(Use ALPHA 4 to type T.) • ENTER evaluates the expression.



Chapter 1 Ingredients of Change: Functions and Limits 1.1 Functions—Four Representations 1.1.1-1.1.2 Entering and evaluating a function (CC5e p. 4) Given ( ) 3 2tg t e t= − , evaluate (4)g .

1.1.1 To enter a function into Y1 in the equation editor... • Y= (located directly under the calculator

screen) opens the equation editor

• Enter the right side of the equation into Y1 using 2ND LN [e^] for e and X,T,θ,n for x X,T,θ,n is used to represent the input variable in a function. The other letters

(obtained using ALPHA ) are storage units that hold a specific number.

• 2ND MODE [QUIT] returns to the home screen

1.1.2 To evaluate the output value of the function entered in Y1... • VARS accesses the Variables

screen

• Highlight Y-VARS using (right arrow) selects the VARS Y-VARS menu

• 1 (or ENTER) selects 1: Function

• 1 (or ENTER) selects 1: Y1 ( keystroke summary) VARS Y-VARS Function Y1

• ( 4 ) indicates the input value to be evaluated

• ENTER evaluates g at t = 4 The output is (4) 155.794g ≈




1.1.2a To set the number of decimal places report by the calculator... • MODE accesses the mode screen

Mode should usually be set as in the near right screenshot.

• To set rounding to three decimal places, highlight 3 on the decimal options line... (down arrow) to the line beginning with FLOAT then (left arrow, four times)

• 2ND MODE [QUIT] returns to the home screen

1.1.2b To re-evaluate a previous entry... • 2ND ENTER [ENTRY] copies a

previous command (that has not yet been deleted)

• ENTER re-evaluates the function

1.1.2c To edit a previous entry... • 2ND ENTER [ENTRY] copies

the previous command

• Use to move the cursor over the 4 then change 4 to 9

• ENTER re-evaluates the function The output is (9) 24,291.252g ≈

1.1.3 Using Equation Solver (CC5e p.6) For ( ) 3 2tg t e t= − , solve ( ) 300g t = .

• MATH 0 (0: Solver) accesses a solver dialog screen similar to the near right screenshot

• Select the top line using (up arrow) to access the EQUATION SOLVER screen

Chapter 1 7 TI-84 Calculator Guide



• Complete the equation to be solved so that it reads 0 = Y1 – 30 then ENTER (or ) to return to the solver dialog screen

• Position the cursor on the X = line (shown as X = 0 in the far right screenshot)

• ALPHA ENTER [SOLVE] returns a solution to the equation

When ( ) 300g t = , 4.636t ≈

For an equation with only one solution, any value in the domain of the function will return the solution. For an equation with multiple solutions, the solution closer to the starting value is returned.

1.1.4 Graphing a function (CC5e p. 5, Example 1) Graph ( ) 80(1.013 )tp t = million people where t is the number of years since the end of 1900, 0 100t≤ ≤ .

• Enter the function into Y1... • Y = opens the equation editor

• With the cursor located to the

right of the = on the Y1= line, CLEAR any content from Y1

• Enter the formula for ( )p t into

4Y1

To set the graph’s input interval... • WINDOW (located directly under

the calculator screen) • Set Xmin to –10 by typing (-) 10 • Set Xmax to 10 by typing 10




To obtain the graphing window to the right above ZOOM ZoomOut ENTER twice then ZOOM ZoomFit (to adjust the output interval)

To set the graph’s output interval... • ZOOM (located directly under the

calculator screen) • Select 0: ZoomFit either

• by scrolling down using until 0: is highlighted then using ENTER

or • by typing 0

( keystroke summary) ZOOM 0 (ZoomFit)

To change the graphing window, either

• Use WINDOW and ZoomFit as before

or • Use one of the first three

options in the ZOOM menu To return to the graphing screen from any other screen... • GRAPH (located directly under the calculator screen)




1.2 Function Behavior and End Behavior Limits 1.2.1 Investigating end behavior (CC5e p. 16) Graph a function before numerically estimating its end behavior 1.2.1a Graphically investigating end behavior

Graph ( )1

xf xx

=+

on the interval 10 10x− ≤ ≤ .

• Enter the function into Y1 on the

Y = screen

• Set the WINDOW parameters... • Xmin = –10 • Xmax = 10

• ZOOM 0 (ZoomFit) to automatically set the output interval and graphs the function

1.2.1b Numerically investigating end behavior

Given ( )1

xf xx

=+

, numerically estimate lim ( )x

f x→∞

.

• Make sure that the function is still entered as Y1 and that MODE is set to FLOAT.

• Use Y1 to evaluate f at increasing values of x. This example starts with 1x =

and increments by multiples of 10.

lim ( ) 1x

f x→∞

≈

• Use Y1 to evaluate f at decreasing

values of x. This example starts with 2x = −

and increments by multiples of 10.

lim ( ) 1x

f x→−∞

≈




1.3 Limits and Continuity 1.3.1 Numerically investigating function behavior (CC5e p. 24)

Investigate the behavior of 2 16( )

4tr tt−

=−

as 4t → .

• Enter the function into Y1

• Use Y1 to evaluate r at close input values to the left of 4.

4lim ( ) 8t

r t−→

≈

• Use Y1 to evaluate r at close input values to the right of 4.

4lim ( ) 8t

r t+→

≈

1.3.2 Graphing a piecewise defined function (CC5e p. 29)

Graph 2 2 when 4

( )3 2 when 4

x xd x

x x + <

= − + ≥

• Enter the piecewise defined function in Y1 as 2(X 2)(X 4) ( 3X 2)(X 4)+ < + − + ≥ The statements (X< 4) and (X≥ 4) are test statements. A test statement returns the value 1 if the statement is true and 0 if the statement is false. The symbols < and ≥ are options on the 2ND MATH [TEST] screen.

• To graph the function... • WINDOW

• Xmin = –2 • Xmax = 6

• ZOOM 0 (ZoomFit)




1.4 Linear Functions and Models 1.4.1 Entering data (CC5e p. 37) Enter the data given in the table into lists L1 and L2.

• STAT returns options for editing data in lists. The EDIT option at the top of the screen should be highlighted.

• 1 (or ENTER) chooses 1: Edit

enabling data entry/editing ( keystroke summary) STAT Edit If there is data in L1 or L2, use to highlight the list name then CLEAR ENTER to clear the list. Warning: using DEL when the list name is highlighted will delete the storage area for the list itself. In order to reinstate a deleted list, see 1.4.7

• To enter values into the lists, use the number keys. To move around in the lists, use the arrow keys. • 5 enters 5 as the first input in L1

• ENTER (or ) moves the

cursor down

• moves the cursor to L2 where output values may be entered

• To add an entry to the middle of a list, highlight the position in which the entry

belongs and use 2ND DEL [INS] to create a blank entry and move the remainder of the list down.

• To delete an entry within a list, highlight the entry (not the list name) and use DEL to remove the entry and move the remainder of the list up.

• To replace an entry within a list, highlight the entry and use the number keys to enter the replacement.

Years 5 6 7 8 9 10

Companies (%) 50 47 44 41 38 35




1.4.2 Graphing a scatter plot of the data (CC5e p.37) Graph a scatter plot of the data entered into lists L1 and L2 in 1.4.1

1.4.2a To set STAT PLOT • 2ND Y= [STAT PLOT] opens a list of

plots. Generally Plot1 will be used for scatter plots.

• 1 (or ENTER) accesses the options menu for 1: Plot1. Set parameters to...

• Plot1 is On • Type: scatter plot • Xlist: L1 • Ylist: L2 • Mark: (large square is easier to see)

1.4.2b Viewing a scatter plot • ZOOM 9 (9: ZoomStat) graphs

the scatter plot with window settings that show all of the data points. ( keystroke summary) ZOOM 9 (ZoomStat)

To turn off equations that may also graph...

• Y= If an equation is turned on (= is highlighted) and the equation will graph. An equations graph can be turned off by placing the cursor on the = and using ENTER. Turning off the equations graph does not delete the equation from Y= .

1.4.3 Calculating first differences (CC5e p. 37) Calculate the first differences of the data entered in 1.4.1

First differences are the incremental changes in successive output values. If first differences are close, a linear function should be considered as a model for the data.

• STAT Edit to bring up the lists




To calculate first difference using the lists, L3 and L4 will be used as intermediate storage and the first differences will be calculated in L5. (The ith first difference iD is calculated from the ith and (i + 1)st output values as

1i i iD y y+= − . Lists L5, L4, and L3 are used to store the components of this equation.)

To copy from L2 into L3 and L4 ... • Highlight L3

• 2ND 2 [L2] completes the

equation at the bottom of near right screen L3 = L2

• ENTER copies the L2 data to L3

• Similarly, highlight L4, then 2ND 2 [L2] ENTER copies the L2 data to L4

To prepare lists L2 and L3 to be used in the first difference calculation... • Highlight the last entry in L3 and

delete using DEL

• Highlight the first entry in L4 and delete using DEL

To calculate the first differences and store them in L5... • Highlight L5

• 2ND 4 – 2ND 3

completes the equation at the bottom of the near right screen L5 = L4 – L3

• ENTER returns the first differences in L5 All of the first differences are equal to –3




1.4.4 Fitting a linear model to data (CC5e p. 39) The table shows the retail sales of electricity for years between 2003 and 2008. Find a linear model for the data.

To enter the data... • STAT Edit and enter data in L1

and L2

To view a scatter plot of the data... • ZOOM 9 (ZoomStat) to view the

scatter plot

To prepare Y= to receive the model equation... • Y= to verify that Y1 is clear (the equation can be stored in any

of the ten function locations) and the Plot1 is turned on • 2ND MODE [QUIT] returns to

the Home Screen To request a linear regression... • STAT to access the CALC

options

• 4 selects 4: LinReg(ax+b) the linear regression

( keystroke summary) STAT CALC LinReg

To choose a storage position for the regression equation... • VARS Y-VARS Function Y1

selects Y1 as the storage position for the equation

Year Retail Sales quadrillion kWh

2003 1.20 2004 1.23 2005 1.27 2006 1.30 2007 1.33 2008 1.35




• ENTER The equation form as well as the

values for the parameters appear on the Home Screen and are pasted into Y1 (see Y=)

• GRAPH shows the equation graph along with the scatter plot

To view the graph without the scatter plot... • Y= • to Plot1 (Plot1 will appear to

be flashing) • ENTER to turn Plot1 off • GRAPH (or choose options using STAT PLOT)

1( ) 0.031 60.604r t t≈ − quadrillion kWh gives the retail sales of electricity to

commercial customers where t is the year, data from 2003 through 2008. 1.4.5 Aligning data (CC5e p. 40) Input data is often aligned (renumbered) to reduce the magnitude of the coefficients. In this example, the data used in 1.4.4 is aligned and remodeled.

• STAT Edit to edit L1 • Highlight L1 and complete the L1

equation L1= L1 – 2000 • ENTER

To re-run the regression... • STAT CALC 4 (LinReg) • VARS Y-VARS Function Y2 • ENTER

to find a linear model and paste it in Y2

2 ( ) 0.031 1.110r t t= − quadrillion kWh gives the retail sales of electricity to commercial customers where t is the number of years since 2000, data from 2003 through 2008.




1.4.6 Pasting all reported digits of a model into Y1 (CC5e p. 41) see also 1.4.4 Although models are generally reported to three decimal places, it important to use all of

the digits found by the calculator when working with the model to reduce the possibility of round-off error.

To paste a model with all digits found by the calculator into the Y= list:

• STAT CALC 4 (LinReg) (4 selects a linear model. Options 5, 6, 7, 8, 9, 0, A, B, or C select other model types.)

• VARS Y-VARS Function Y1 selects Y1 as the equation (Other options are Y2, Y3, … Y0)

• ENTER runs the regression

• Y= shows the equation as pasted

into Y1 1.4.7 Trouble-shooting: Restoring a missing column on the List Screen If delete DEL, rather than CLEAR, is selected to remove data from one of the lists, the

entire list will be deleted.

If the set of lists is incomplete re-enter all lists...

• STAT (EDIT) 5 (SetUpEditor)

• 2ND 1 , 2ND 2 , 2ND 3 , 2ND 4 , 2ND 5 , 2ND 6 (commas are necessary)

• ENTER

• STAT (EDIT) 1 (Edit) shows the complete set of lists




1.5 Exponential Functions and Models 1.5.1 Calculating percentage changes (CC5e, p. 48)

Find the percentage change from June 2006 to July 2006

• 100×(6.024–5.365)/5.365 • ENTER


• 100×(24.212–21.562)/ 21.562 • ENTER


• 100×(97.315–86.663)/ 86.663 • ENTER

1.5.2-1.5.3 Exponential data (CC5e, p. 50) The table shows northern cod population by decade. Find an exponential model for the data.

• STAT Edit and enter data in L1 and L2

• With STAT PLOT 1 On, ZOOM 9 (ZoomStat) to view the scatter plot

Decade (since 1963)

Population (billion)

0 1.72 1 0.63 2 0.24 3 0.085 4 0.032




1.5.2 Calculating percentage changes for data Calculate the percentage differences of the data from CC5e p. 50.

• Copy the output data from L2 into

L3 and L4

• Delete 1.72 in L3 and 0.32 in L4

• Highlight L5 and complete the equation as L5 = 100×( L4 – L3)/ L3

• ENTER 1.5.3 Fitting an exponential model to data Fit an exponential model to the data from CC5e p. 50.

• STAT CALC 0 selects 0: ExpReg 0: ExpReg is found using STAT CALC and either the up or down arrow keys to scroll through the menu options. ( keystroke summary) STAT CALC LinReg

• VARS Y-VARS Function Y1 selects Y1 as the storage position for the equation

• ENTER runs the regression

• GRAPH to view the function graphed with the scatter plot




1.6 Models in Finance 1.6.1 Evaluating a loan with simple interest (CC5e pp. 57- 58) Calculate the future value at time t of the amount a college student will owe on a $1000 loan from her aunt with 4% interest charged per year on the face value (present value) of the loan.

A loan (or investment) with simple interest changes value at the start of each successive year.

• Y= accesses the near right screen

shot

• 1000 (1 + .04X) enters the right side of the present value equation into Y1

• 2ND WINDOW [TBLSET] accesses the Table Setup; use the options as shown in the near right screen shot

• 2ND GRAPH [TABLE] accesses the Table

• 0 1 2 3 4 5 use input values incrementing by 1 to see the amount owed for each or the next five years

1.6.2 Evaluating a loan with compound interest (CC5e, p. 59) Calculate the future value at time t of the amount a college student will owe on a $1000 loan from a bank with 4% interest compounded monthly.

• Y= accesses the near right screen shot

• 1000 (1 + .04 /12) ^ X enters the right side of the future value equation intoY1 Because the loan is compounded monthly, the only appropriate input values are

multiples of 112

.




• 2ND WINDOW [TBLSET accesses the Table Setup; (use the options as shown in the near right screen shot)

• 2ND GRAPH [TABLE] accesses the Table

• 0 1/12 1 2 3 4 5 (use input values for the start, after one month, one year, two years, …five years.

The amount owed will reach $1221 after five years of monthly compounding.

1.6.3 Calculating doubling time (CC5e, p. 63) Calculate the doubling time for an investment with an APR of 6.9% compounded quarterly

40.0692 1

4

x

P P = +

can be rewritten as 40.0692 1 1

4

x = +

• Y= to enter the RHS of the

equation into Y1

• 1 (1 + .069 /4) ^ (4X) enters the right side of the doubling time equation for Investment A into Y1

• MATH 0 (Solver) accesses the Equation Solver

• Move the cursor to the top line and complete the equation to be solved as Y1 – 2

• ENTER accesses the screen shot

• Highlight the right side of the X= equation. Type 1 to change the starting guess to 1




• ALPHA ENTER [SOLVE] returns approximately 10.132

The doubling time for Investment A is 10.25 years because interest is compounded quarterly.




1.7 Constructed Functions 1.7.1 Evaluating a function constructed using multiplication (CC5e, p. 67)

( )( ) 287.411 0.266xD x = million bottles gives the sales of 12-ounce bottles of sparkling water when the price is x dollars per bottle. Calculate the revenue from the sales of sparkling water when the price of one 12-ounce bottle is $2.50.

• Y=

• 287.411 (0.266^X) enters the right side of the sales equation

• moves the cursor to Y2

• X×Y1 completes the equation in Y2

• 2ND QUIT returns to the Home Screen

• Y2 (2.50) ENTER

At a price of $2.50 per 12-ounce bottle of sparkling water, the revenue is $26.221 million.




1.7.2 Evaluating a function constructed using function composition (CC5e, p. 69) The level of contamination in a certain lake is modeled as ( )f p p= parts per million when the population of the surrounding community is p people. The population of the surrounding community is modeled as 2( ) 400 2500p t t= + people where t is the number of years since 2000.

• Y1 accesses the screen on the near right

• 400 X2+ 2500 enters the right side of the equation ( )f p into Y1


2ND X2 ( Y1 ) ENTER uses the output of p as the input of f

Find the level of pollution of the lake in 2007.

• 2ND QUIT returns to the Home Screen

• Y2(7) ENTER evaluates Y2(Y1(7))

In 2007, the level of pollution in the lake was 148.661 parts per million. Alternate method: use two steps to evaluate a function constructed by composition.

• Y1 accesses the screen on the near right

• 400 X2+ 2500 enters the right side of the equation ( )f p into Y1


• 2ND X2 ( X ) ENTER




Y1 is the inside function for the composition

• 2ND QUIT returns to the Home Screen • Y1(7) ENTER evaluates the

population in 2007

• Y2(2ND (-)) ENTER evaluates the pollution at a population of 22,100

1.7.3 Approximating the model of an inverse function (CC5e, p. 70)

Given a function 1( ) 133

p d d= + , find a model that approximates an inverse of the function.

Use the function p to generate at least three points with values for d in L2 and corresponding values for ( )p d in L1. Then fit an appropriate model to the data.

• Type RHS of 1( ) 133

p d d= + into Y1

• STAT 1 opens the list editor

• Enter at least three values for d in L2 0, 66, and 132 are good choices because they are divisible by 33.

• Highlight L1 and type VARS Y-VARS 1 1 ( 2ND 2 ) to complete the equation L1 = Y1(L2)

• ENTER returns the values 1, 3, and 5 in L1. The inverse of a linear function is also a linear function.

• STAT CALC 4 VARS Y-VARS 1 2 ENTER returns the equation

( ) 33 33d p p= − feet models the depth below the surface when the atmospheric pressure is p atm.




1.8 Logarithmic Functions and Models 1.8.1 Fitting a logarithmic model (CC5e, p. 78) The table shows altitude given air pressure. Find a logarithmic model for the data.

• STAT 1 brings up the lists

• Enter the data in L1 and L2

• ZOOM 9 returns the scatter plot

• To generate the log regression STAT9 VARS

ENTER ENTER ENTER (and stores the equation in Y1) • GRAPH

( ) 76.174 21.331lnA p p≈ − thousand feet when p inches of mercury is the air pressure, data from 0.33 13.76p≤ ≤

1.8.2 Fitting the inverse of a log model (CC5e, p. 80) Find a model to approximate the inverse of the log model found in 1.8.1.

• STAT 1 and enter at least three

values for p into L2.

• Highlight L1 and type VARS Y-VARS 1 1 ( 2ND 2 ) to complete the equation L1 = Y1(L2)

• ENTER returns the values for ( )A p in L1.

Pressure (Hg”)

Altitude (thousand feet)

0.33 100 0.82 80 2.14 60 5.56 40 13.76 20




The inverse function for a log function is an exponential function.

• STAT CALC 0 VARS Y-VARS 1 2 ENTER returns the equation and pastes it into Y2.

( ) 35.552(0.954 )Ap A ≈ inches of mercury gives the air pressure at A thousand feet above sea level, data from 0.33 13.76p≤ ≤




1.9 Quadratic Functions and Models 1.9.1 Fitting a quadratic model (CC5e, p. 88) The table shows the occurrence of premature, under-5.5 lb babies as a percentage of births to mothers classified by prenatal weight gain. Find an exponential model for the data.


• Enter the data into L1 and L2

• ZOOM 9 returns the scatter plot

• STAT5 VARS1 1 ENTER

• GRAPH returns the scatter plot with the function

2( ) 0.029 2.286 79.685P g g g≈ − + gives the percent of babies that weigh less than 5.5

pounds who were born before 37 weeks to mothers who gained g pounds, data from18 43g≤ ≤

Weight Gain

(pounds)

Low birth-weigth, premature babies

(percent) 18 48.2 23 42.5 28 38.6 33 36.5 38 35.4 43 35.7




1.9.2 Calculating second differences (CC5e, p. 91) Calculate second difference for the percentage of U.S. Internet users who shop online.


• Enter the output data into L2 and copy it into L3 and L4.

• Delete the last entry from L3 and the first entry from L4.

• Highlight L5 and enter 2ND 4 – 2ND 3 to complete the equation L5 = L4 – L3

• ENTER returns first differences in L5

• Copy L5 to L3 and L4.

• Delete the last entry from L3 and the first entry from L4.

• Highlight L5 and enter 2ND 4 – 2ND 3 to complete the equation L5 = L4 – L3

• ENTER returns the second differences in L5 The second difference are –0.3, –0.3, –0.2, and –0.2

Year Online Shopper (percent)

2008 84.2 2009 86.0 2010 87.5 2011 88.7 2012 89.7 2013 90.5




1.10 Logistic Functions and Models 1.10.1 Fitting a logistic model (CC5e, p. 96) Find a logistic model for the number of NBA basketball players who are taller than a given height as presented in Table 1.44 on page 96 of CC5e.

• STAT ENTER brings up the lists

• Enter the data in L1 and L2

• With STAT PLOT 1 On, ZOOM 9 to view the scatter plot

If the data is modeled without aligning, the results are as follows: Note that the value of a is difficult to report because of the leading zeros.

• STAT ALPHA B VARS 1 1

• ENTER • ZOOM 9 graphs the scatter plot

and function

To align the input values:

• STAT 1 • Highlight L1 and complete the

equation as L1= L1 – 68 • ENTER returns the aligned values

to L1

• Y= CLEAR clears the previous equation from Y1

• STAT ALPHA B VARS Y-VARS 1 1 ENTER finds the aligned logistic equation and stores it in Y1

• ZOOM 9 graphs the scatter plot and function

0.462

485.896( )1 0.007 hp h

e=

+ NBA players are taller than 68h + inches, data from

0 20h≤ ≤




1.11 Cubic Functions and Models 1.11.1 Fitting a cubic model to data (CC5e 105) Find a cubic model for the data. (This data is from Table 1.47 on page 105 of CC5e with input data aligned to minimize magnitude of coefficients.)

• ENTER the aligned data into L1 and L2 of the STAT lists

• With STAT PLOT 1 On,

ZOOM 9 to view the scatter plot

• STAT 6 VARS 1 ENTER selects the cubic regression

• ENTER finds and stores the regression in Y1

• ZOOM 9 returns the scatter plot and graph

Years since 2000

Employees (thousand people)

1 9473 2 9233 3 9187 4 9226 5 9409 6 9580 7 9577




1.12 Cyclic Functions and Models

1.12.1 Graphing a sine function (CC5e p112) Graph the function ( ) 6sin(1.57 4.71) 8g x x= − + .

To enter the function for graphing... • Y= 6 SIN ( 1.57 X - 4.71 ) + 8 To graph ( )g x over 8 8x− ≤ ≤ ... • WINDOW

• XMIN = (-)8 and XMIN = 8

• ZOOM 0

1.12.2 Fitting a sine model to data (CC5e p115) Find a sine model to fit the data in Table 1.49 on page 114 of CC5e with the input data aligned so that Dec ’08 is 0, Jan ’09 is 1, etc.

• Enter 0 through 12 into L1 of the STAT list and enter the output data into L2.

• ZOOM 9 shows the scatter plot

• STAT ALPHA C selects the sine regression SinReg

• ENTER

• VARS 1 1




• ENTER

• ZOOM 9

( )( ) 0.652sin 0.517 2.450 10.618p t t≈ − + cents/kWh gives the retail prices for electricity in the commercial sector where 0t = at the end of December, 2008, data from 0 12t≤ ≤ .



2 Describing Change: Rates 2.1 Measures of Change over an Interval 2.1.1 - 2.1.2 Calculating change and average rate of change (CC5e, p. 134) A model for temperature is 2( ) 0.8 10 49 Ff t t t= − + + where t is the number of hours since 7 A.M.

• Enter RHS of ( )f t into Y1 2.1.1 To calculate change from 11:00 A.M. to 4:30 P.M...

• On the home screen, enter the formula

for calculating change: f (9.5) – f (4) using Y1 instead of f. VARS 1 1 (9.5) – VARS 1 1 (4)

Because change is used to calculate average change (and percentage change when applicable), storing the result in an alpha location can be useful.

• STO N (ALPHA LOG•

) ENTER

Temperature decreased by 4.4°F between 11:00 A.M. to 4:30 P.M. 2.1.2 To calculate average change from 11:00 A.M. to 4:30 P.M...

• On the home screen, enter the formula for

calculating average change: ( (9.5) (4)) / (9.5 4)f f− − either using Y1 instead of f or if change was calculated and stored in N, enter the formulas as N/(9.5-4)

• ENTER

Temperature decreased by an average of 0.8°F per hour between 11:00 A.M. to 4:30 P.M.

34 5e Calculus Concepts; LaTorre, et al. TI-84 Guide



2.1.3 To calculate percentage change...

In the context of temperature, percentage change has no valid interpretation. However, the calculation is shown here as an example of how to calculate percentage change quickly when it is applicable.

• On the home screen, enter the formula for

calculating average change: ( (9.5) (4)) / (9.5) 100f f f− ⋅ using either Y1 instead of f or if change was calculated and stored in N, enter the formulas as N/Y1(9.5)*100

• ENTER

2.1.4 Graphing a Secant Line (CC5e p. 134) For the function 2( ) 0.8 10 49 Ff t t t= − + + draw a secant line from the point at 4t = to the point at 9.5t = .

• With f entered in Y1, graph using the following WINDOW Xmin = 0

settings:

Xmax = 10 Ymin = 50 Ymax = 90

GRAPH

• On the home screen, use DRAW (2ND PRGM

) to open the drawing menu

• 2 returns Line( on the home screen.

• , Y1 ( 4 ) , 9.5 , Y1 ( 9.5

) ) completes the command line

• ENTER

(4, f(4)) to (9.5, f(9.5))

returns to the graphing window and draws the line from

Chapter 2 35 TI-84 Guide



2.4 Rates of Change – Numerical Limits and Nonexistence

2.4.1 refers to a section in the Excel 2007 Guide 2.4.2-2.4.3 Numerically estimating a rate of change (CC5e p. 161) Given ( ) 32(1.12 )tF t = , numerically estimate '(3.5)F correct to three decimal places.

• (1.12^X)

•

enters the right side of the equation into Y1 (Y1 (X) - Y1 (3.5) ) / (X -

enters the formula for the slope of the secant in Y2

3.5)

• 2ND

table options WINDOW returns the

• Indpnt: ASK and Depend AUTO

•

should be highlighted 2ND GRAPH returns the table, use DEL column

to clear any values in the X

• .49

(3.49)F (in the X column) ENTER

returns in Y1 and the slope of the secant line in Y2

• 3.499 ENTER 3.4999 ENTER 3.49999 ENTER 3.499999 ENTERshown to the far right

completes the table as

• To see more decimal places for

the slope of the secant line, use the cursor to highlight values in the Y2 column

• Clear the X column (by using DEL

to remove each entry); then insert close values from the right

36 5e Calculus Concepts; LaTorre, et al. TI-84 Guide



2.4.4 Graphing the Tangent Line (CC5e p. 161) Given ( ) 32(1.12 )tF t = , graph the tangent line at 3t = .

• With the function F stored in Y1, set WINDOW Xmin = 0

settings to:

Xmax = 9 Ymin = 30 Ymax = 90

• From the graphing window, DRAW (2ND PRGM

) opens the drawing menu

• 5 (5: Tangent( ) returns to the graphing window with the cursor on function Y1

• 3 chooses the input value 3t = for the point of tangency

• ENTER

draws the tangent line

To clear figures drawn using DRAW... open DRAW and choose 1 (1: ClrDraw)



3 Determining Change: Derivatives 3.1 Simple Rate-of-Change Formulas 3.1.1 Writing the derivative of a regression function (CC5e p. 197) Write a model g and the rate-of-change model 'g for the data in Table 3.1 on page 197 of CC5e. Fit the regression model.

• Enter the data into STAT list.

• Fit a regression equation to the data and store in Y1. In this example, a quadratic

regression QuadReg is used. QuadReg returns an expression of the form 2ax bx c+ + and stores the coefficients a, b, and c individually.

3.1.1a Accessing coefficients from a regression model When the calculator fits a regression equation to data, it stores the coefficients from the equation as Statistics Variables until the next regression run. These coefficients can be accessed from the home screen or pasted into other expressions as follows... •

opens the variables menu

• 5 selects the 5: Statistics menu (The right screen shot shows the Statistics menu as it appears when it is opened.)

• EQ () opens the regression

equations menu

• 2 selects the coefficient a (other coefficients, b through e, can be selected using options 3 through 6)

• ENTER

returns the value of the coefficient

Coefficients can also be used in other expression as in 3.1.1b and 3.1.1c.




3.1.1b Calculating with coefficients from the regression function Use the coefficients from the regression model to calculate the coefficients for the rate-of-change model... • On the home screen, type

2 a (VARS 5 EQ 2) to calculate 2a.

ENTER

• Type b (VARS 5 EQ 3) ENTERto return the value of b.

The rate of change formula for the data is '( ) 9.107 0.625g t t≈ − .

3.1.1c Using coefficients from the regression function in Y= Use the coefficients from the regression model to write the rate-of-change model in Y2... • In the Y=

• Type the derivative formula using the coefficients stored by the TI-84:

list, place the cursor beside Y2=

Y2= 2 a X + b To access the coefficient a, type VARSTo access the variable X, type

5 (5: Statistics) EQ 2 (2: a)

X,T,θ,n To access the coefficient b, type VARS

5 (5: Statistics) EQ 3 (3: b)

3.1.2 Evaluating a function in Y= using coefficients from a regression model For the model g and its derivative 'g from 3.1.1, evaluate '(5)g .

• On the home screen, type Y2 ( 5 ) ENTER to evaluate '(5)g when 'g is stored in Y2=

In 1998, the maintenance cost for a vehicle driven for 15,000 miles was increasing by

about $44.91 per year. ($44.92 if the rounded model is used for the calculation)




3.4 Rates of Change of Composite Functions 3.4.1 Calculations involving the Chain Rule

Given function 1( ) 24 52t ff

= + , ( ) 1 0.04 gf g e= + , and ( ) 0.6 0.2g h h= + where t, f, and g can be

composed, graph the composite function t f g and the derivative of the composite function over 0 9h≤ ≤ . 3.4.1a Using Y= with the Chain Rule Using Y1 through Y4 helps clarify expressions in a composite function, especially when the functions are nested.

• Enter the functions into Y2, Y3, and Y4 and the composite form of the functions into Y1...

Y1 = Y2(Y3(Y4(X))) Y2 = 24(1/X) + 52 Y3 = 1 + 0.04e^X Y4 = 0.6X+0.2 In each function use X,T,θ,n as the input variable.

• Enter the derivative formulas dtdf

, dfdg

, and dgdh

into Y6, Y7, and Y8 respectively and

the chain rule derivative formula dt dt df dgdh df dg dh

= ⋅ ⋅ into Y5...

Y5 = Y6(Y3(Y4(X)))*Y7(Y4(X))*Y8(X) Y6 = –24X^(-2) Y7 = 0.04e^X Y8 = 0.6 In each function use X,T,θ,n as the input variable.

3.4.1b Graphing a function involving the Chain Rule

• In Y= turn graphing off for function Y2, Y3, and Y4 as well as Y6, Y7, and Y8 by placing the cursor on each equal sign and typing ENTER

. Only Y1 and Y5 will have highlighted = signs indicating that they will graph.

• Set WINDOW Xmin = 0

settings to

Xmax = 9

ZOOM 0



Chapter 4 Analyzing Change: Applications of Derivatives 4.0 nDeriv — TI’s numerical derivative routine 4.0.1 Using nDeriv to calculate and graph For a function ( ) 32(1.12 )xf x = , calculate '(7)f and graph 'f using nDeriv.

nDeriv is TI’s numerical derivative routine. It returns a numerical value for the

derivative of a function given a specific input value. It can also be used to graph a slope-graph for a function.

The format used for nDeriv is nDeriv( function , input variable , input value )

4.0.1a Using nDeriv to calculate a derivative value Calculate '(7)f . • Enter the RHS of the equation for f

in Y1.

• Return to the home screen. nDeriv is accessed as follows... •

opens the MATH menu. (The MATH menu appears as in the screen shot on the left.)

• 8 (8: nDeriv( ) returns nDeriv( to the home screen

• Type the parameters Y1, X, 7 VARS (commas are necessary)

Y-VARS 1 1 , X , 7 )

• ENTER

runs the command and returns a numerical answer.

keystrokes MATH 8 VARS Y-VARS 1 1 , X , 7 )

ENTER

'(7) 8.107f ≈




4.0.1b Using nDeriv to graph a slope graph Graph 'f over 0 10x≤ ≤ .

• Enter the RHS of the equation for f

in Y1.

• Place the cursor to the right of Y2=

• Paste nDeriv into Y2 as

MATH 8 VARS Y-VARS 1 1 , X , X ) (Use parameters Y1, X, X )

ENTER

To graph the slope graph... • Highlight = next to Y1 and ENTER

to turn off graphing of Y1

• Set WINDOW Xmin = 0

to

Xmax = 10

• ZOOM

0 (ZoomFit) to graph

nDeriv graphs more slowly than other function graphs because the calculator has to numerically estimate the pixel position for each pixel input value.

4.0.1c Using nDeriv in Y2 to calculate a derivative value

Calculate '(7)f .

With f in Y1 and nDeriv in Y2... • On the home screen type

Y2 ( 7 )

• ENTER

to evaluate

'(7) 8.107f ≈




4.1 Linearization and Estimates 4.1.1 Graphing and Calculating with Linearizations (CC5e p. 252) The linearization of f at t c= is ( ) ( ) '( ) ( )Lf t f c f c t c= + ⋅ − .

4.1.1a Graphing a linearization

Graph the tangent line that represents the linearization of3 2( ) 12.92 185.45 729.35 10,038.57f t t t t= − + − + at 7t = .

Even though the linearization is a tangent line and can be graphed using the

DRAW Tangent option, this option does not lend itself to calculating using the linearization.

• Set up Y=

using Y1= for the function f ,

using Y2= for the slope '(7)f , and using Y3= for the linearization Lf .

• Turn off graphing for Y2.

• Set WINDOW

Xmin = 0 to

Xmax = 10 • ZOOM

0 (ZoomFit)

4.1.1b Calculating with a linearization

Calculate (7.5)Lf using a linearization of 3 2( ) 12.92 185.45 729.35 10,038.57f t t t t= − + − + at 7t = .

• Set up Y=

as in 4.1.1a

• On the home screen, type Y3 ( 7.5

)

• ENTER

to evaluate

(7.5) 9572.465Lf ≈




4.2 Relative Extreme Points 4.2.1 Solving for zeros of the derivative (CC5e p. 259) Given 3 2( ) 0.395 6.67 30.3 3661p x x x x= − + + , graph '( )p x .

• Enter the right side of the equation

into Y1

• Enter the derivative function into Y2

• Turn off graphing of Y1

• Set the window using Xmin = 0 and Xmax =

WINDOW 10

• 'p 0 returns the graph of

There are two places where 'p crosses the horizontal axis

• MATH

EQUATION SOLVER O accesses the

• Set the eqn: 0 = Y2

the derivative is found in Y2 since

• ENTER

the far right returns the screen to

iThe two zeros for ' 0p = occur on the interval between 0x = and 10x = . A

guess closer to 0 will return the input location of the relative maximum and a guess closer to 10 will return the location of the relative minimum.

• With the cursor on the 1,

ALPHA ENTERinput location

returns an

One solution of ' 0p = occurs at 3.156x ≈




4.4 Inflection Points and Second Derivatives

4.4.1 Locating an inflection point (CC5e p. 276)

Given 0.969

83( )1 5.94 tp t

e−=+

, find the location of the inflection point.

• Enter the function p in Y1

• Enter nDeriv (Y1, X, X)

in Y2

• In WINDOWXmax =

, set Xmin = 0 and

8

• ZOOMgraphs of

0 (ZoomFit) returns the p and 'p

• TRACEto the near right (Notice that the

4 returns the screen

title indicates Y1 is selected) • Use to select the derivative

function Y2

• Use (repeatedly) to move the cursor to the left of the maximum point on the derivative graph

• Use (repeatedly) to move the ENTER

cursor to the right of the maximum

ENTER

• ENTER returns the maximum value of the derivative function

• Use to move the cursor to the function and obtain the output value at the inflection point The inflection point of p has coordinates ( )1.847, 41.5 .



Chapter 5 Accumulating Change: Limits of Sums and the Definite Integral 5.2 Limit of Sums and the Definite Integral 5.2.1 - 5.2.2 Using the NUMINTL Program to Estimate the Accumulated Change (CC5e p. 331)

Given 1.7(0.8 ) when 0 20( )

10.21 3ln when 20 30

t tm t

t t ≤ ≤

= − + < ≤

Estimate signed areas using rectangles. A download for NUMINTGL is available at

://www.ticalc.org/pub/83plus/basic/math/calculus/. 5.2.1a Calculate the sum of the areas of the ten right-rectangles over 0 to 20

• Y=

Enter the first piece of the piecewise function into Y1

• PRGM

and select NUMINTGL

• ENTER

to enter the program

• 1 ENTER• 1

to continue ENTER

to draw rectangles

• 0 ENTER

•

to enter the left endpoint 20 ENTER

to enter the right endpoint

• 2 ENTER to select right rectangles 10 ENTER

to use ten rectangles

http://www.ticalc.org/pub/83plus/basic/math/calculus/�




•

•

to step through the program from the graph to the sum ENTER

to step through the program

The right-rectangle area approximation using ten rectangles is 5.975.

• 3 to quit the program

5.2.1b Calculate the sum of the areas of the five right-rectangles over 20 to 30

• Y=

Enter the second piece of the piecewise function into Y1

• and select NUMINTGL

• ENTER

to enter the program

• 1 ENTER• 1

to continue ENTER

to draw rectangles

• 20 ENTER

•

to enter the left endpoint 30 ENTER

to enter the right endpoint

• 2 ENTER to select right

rectangles 5 ENTER

to use five rectangles

•

to step through the program from the graph to the sum




• ENTER

to step through the program

The right-rectangle signed area approximation using five rectangles is -4.536.

• 3 to quit the program 5.2.1c Calculate the sum of the areas of left-rectangles NUMINTL can be used to calculate using left-rectangles similarly to the calculations in 5.2.1a and 5.2.1b except that left-rectangles (1) should be choosen. 5.2.2 Calculating the sum of the areas of midpoint rectangles NUMINTL can be used to calculate using left-rectangles similarly to the calculations in 5.2.1a and 5.2.1b except that midpoint-rectangles (4) should be choosen. 5.2.3 Using NUMINTL to estimate a limit of sums Repeated use of NUMINTL can be used to generate the right column in a limit of sums table like Table 5.9 5.5.1 refers to a section in the Excel 2007 Guide




5.6 The Definite Integral - Algebraically 5.6.1 fnInt — TI’s numerical integration routine (CC5e p. 379) Calculate the total area of the regions between the curve of 2( ) 6g x x x= + − and the x-axis over

5 5x− ≤ ≤ as shown in the second QE on page 379.

• Enter f in Y1

• Return to the home screen

• MATH

9 (9: fnInt)

• Complete the expression fnInt(Y1, X, -5, -3) ENTER

• ENTER

copies the previous line down.

• Edit the line to read fnInt(Y1, X, -3,

• 2)

ENTER

• Repeat the previous three steps to

find the area of the third region

• Sum the absolute values of three signed areas

• Sum the signed areas together to

find 5

5

( )f x dx−∫




5.6.2 Evaluating a piecewise definite integral (CC5e p. 380) Evaluate the definite integral over 1.9 12.0m− < < of the function 'p in Example 2.

• Enter the three pieces of 'p in Y1,

Y2 and Y3.

• Return to the home screen • MATH• Complete the expression

fnInt(

9 (9: fnInt)

Y1, X, -1.9, .7) ENTER

• 2ND ENTER• Edit the line to read

fnInt(

Y2, X, .7, •

3.6) ENTER

• • Edit the line to read

fnInt(

ENTER

Y3, X, 3.6,

10.1)

• Sum the signed areas




5.7 Differences of Accumulated Change 5.7.1 Calculating Area of Between two intersecting curves (CC5e p. 388) Calculate the total area of the regions between the two curves in the QE on page 388.

• Enter the right side of the equations for f and g into Y1 and Y2.

To graphically estimate the intersection of the two functions...

• WINDOW• Xmin=0 and Xmax=

• 10

ZOOM

0 [ZoomFit] returns the graph

• TRACE

5 [intersect]

• Place the cursor on one of the functions and

ENTER

• Place the cursor on the other function and

ENTER

• Use the to move the cursor to the intersection point and

ENTER

• 2ND MODE

[QUIT] to return to the home screen

To calculate the total area using fnInt... • MATH• Complete the expression as

fnInt(

9 returns fnInt

Y2-Y1, X, 0, 4.651) + MATH 9 Y1-Y2, X, 4.651,

10)



Chapter 6 Analyzing Accumulated Change: Integrals in Action 6.1 Perpetual Accumulation and Improper Integrals 6.1.1 Numerically evaluating an improper integral with no upper bound numerically (CC5e pp. 418)

Numerically estimate 0.06

2

4.3 xe dx∞

−∫

• Enter function in Y1 • Use fnInt in Y2

MATH 9 VARS 1 1 , X , 2 ,

X)

Use Tables to evaluate the definite integral from 2 to N for increasing values of N • WINDOW

(TBLSET)

• Useand ENTER to select Ask and ENTER

to select Auto

• 2ND GRAPH

(TABLE)

• In the X column enter 50 100 200 400 800 1600

The output values are returned in Y1 and the value of the definite integral

0.06

24.3

x xe dx−∫ for increasingly larger values

of x are in Y2.

0.06

2lim 4.3 63.563

x x

xe dx−

→−∞≈∫




6.3 Calculus in Economics – Demand and Elasticity 6.3.1 Locating the point of unit elasticity (CC5e p. 446) Given the demand function in Example 3, locate the point of unit elasticity.

• Enter ( )D p in Y1 • Enter '( )D p in Y2 • To enter η into Y3...

• MATH

• Complete the expression for

1 (selects absolute value)

η X,T,θ,n X(times) VARS (Y-VARS) 1 (Function) 2 (Y2) / VARS

(Y-VARS) 1 (Function) 1 (Y1) )

• To solve η = 1,

• MATH•

0 (Solver) VARS

(Y-VARS) 1 (Function) 3 (Y3) - 1

•

returns the screen where a reasonable guess for X must be entered (consider the context)

• With the cursor on the X= row

ALPHA ENTER

to solve the equation

13.757p ≈



Chapter 7 Ingredients of Multivariable Change: Models, Graphs, Rates 7.1 Multivariable Functions and Contour Graphs 7.1.0 Evaluating a multivariable function (CC5e p. 521)

Evaluate the function ( , , , ) 1ntrF P r n t P

n = +

at (10,000, 0.05, 4, 17).

• Enter F in Y1

APLHA 8 ( 1 + ALPHA X / ALPHA LOG ) ^ ( ALPHA LOG APLHA 4

)

• MATHeqn: must contain Y1

0 (0: Solver) The equation

•

should be listed

returns the screen beginning Y1 =… all variables

• Enter the values of P, R, N and T

• MODE

to return to the homescreen

• VARS

1 1

• ENTER

Note: There is not a way to clear the stored values on the calculator. If you wish to see the current value of any variable, type the variable name on the homescreen using the ALPHA key and hit ENTER. When dealing with single variable functions, the functions were stored in Y1 as a function of x. With multivariable functions, the solver is used to store multiple variable values and to use Y1 on the homescreen to evaluate the function.

7.1.1 – 7.1.2 refer to sections in the Excel 2007 Guide




7.1.3 Solving a multivariable function to find the value of one of the input variables (Example 4 CC5e pp. 529-530) Solve for the value of t that corresponds to v = 5 on the (10.45 10 )(33 ) 2000v v t+ − − = contour curve.

• Enter H in Y1

• • On the EQUATION SOLVER

SCREEN,

0 (0: Solver)

complete eqn:0= Y1 –

2000

• Store the value of 5V =

• With the cursor in the T = row, ALPHA ENTERof T so that

returns the value (5, ) 2000H T =




7.2 Cross-Sectional Models and Rates of Changes 7.2.1 Fitting a cross-sectional equation to data set (CC5e p. 543)

• STAT

1 Enter data into L1 and L2

• Y=

Confirm that Plot1 is turned on

•

9 returns the scatter plot

• MODE

returns to the home screen

• STAT 5 (QuadReg) VARS 1 1 ENTERequation in Y1

returns the quadratic

• GRAPH

the scatter plot shows the equation over




7.3 Partial Rates of Change 7.3.1 Evaluating a partial derivative at a point (CC5e p. 553) For the function ( , ) (1.0614 )tF P t P= , calculate (7500, 10)PF .

• Enter F in Y1

• MATH

0 (0: Solver)

• Confirm that Y1 is referenced in the equation 0 =

• Enter the values of P and T

• 2ND MODE

returns to the home screen

•

8 (8: nDeriv)

• Complete the expression nDeriv(Y1, P, 7500)

• ENTER




7.4 Compensating for Change 7.4.1 Evaluating the derivative at a point (CC5e p. 563)

Calculate the derivative of 1

0.4

0.6

3.960.3

yx

=

at 10x = .

• Enter y in Y1

• Returns to the home screen • MATH

Complete the expression. 8 (8: nDeriv)

• ENTER

returns the derivative

7.4.2 Evaluating the slope of a tangent line at a point (CC5e p. 566)

• Enter B in Y1

• 0 (0: Solver)

• Confirm that Y1 is referenced in the equation 0 =

• Enter the values of W and H

• Return to the home screen

• Enter h

w

BB−

using MATH

(8: nDeriv)

8

• ENTER tangent

returns the slope of the

( )67,129

3.851 pounds/inchdwdh

≈



Chapter 8 Analyzing Multivariable Change: Optimization 8.2 Multivariable Optimization 8.2.1 Solving a system of two linear equations (CC5e p. 590) Step 1: Arrange the terms so that variables and constants appear in the same order in each

equation and the right side of the equation = 0

0.6 1.8 0.03 00.2 1.2 0.03 0

x

y

f x yf y x= − + == − + =

0.6 0.03 1.80.03 0.2 1.2

x yx y+ =+ =

Step 2: Set up the system of matrices [ ][ ] [ ]A X B= where

[ ] [ ] [ ]0.6 0.03 1.8, , and

0.03 0.2 1.2x

A X By

= = =

Step 3: Solve: [ ] [ ] [ ]1X A B−=

• 2ND X-1 (MATRIX) (Edit) ENTER

([A])

• Enter the dimensions of the matrix 2 (rows) 2 (columns)

• .6 ENTER .03 ENTER .03

ENTER .2 ENTER

• 2ND X-1 (MATRIX) (Edit) ([B]) ENTER

• 2 (rows) 1 (columns)

• 1.8 1.2

• 2ND MODE [QUIT]

• 2ND X-1 1 ([A]) X-1 2ND X-1 2 ([B])

• ENTER returns the solution

2.720, 5.592x y= = (2.7,5.6) 0.7f ≈




8.3 Optimization Under Constraint 8.3.1 Solving a system of three linear equations (CC5e p. 602)

Step 1: The system of equations 1 (1)2 (1.5)1.5 8.5

yxx y

+ = + = + =

λλ is rewritten as

0 1 1 11 0 1.5 2

1.5 0 8.5

x yx y

x y

+ − λ = − + − λ = − + + λ =

Step 2: The matrices[ ][ ] [ ] are written asA X B= 0 1 1 11 0 1.5 21 1.5 0 8.5

xyλ

− − = −

Step 3: The system is solved as [ ] [ ] [ ]1X A B−=

• 2ND X-1 (MATRIX) (Edit) ENTER ([A])

• Enter the dimensions of the matrix 3 (rows) 3 (columns)

• Enter the values

• 2ND X-1 (MATRIX) (Edit) ([B]) ENTER

• 3 (rows) 1 (column)

• Enter the values

• 2ND MODE [QUIT]

• 2ND X-1 1 ([A]) X-1 2ND X-1 2 ([B])

• ENTER




8.3.2 Identifying the type of constrained point (CC5e p604)

• In Y1, enter .3^X Y^.4 • In Y2, enter 18X +Y -200

• Go to the EQUATION SOLVER

(MATH 0) verify that Y1 is the equation

• ENTER ( or )changes screens so that the values X = 6.666 and Y = 80 are stored in the calculator

• Return to the home screen, • Y1 ENTER

(6.666,80) 5.4038f ≈ is a

critical point

• Go back to the EQUATION SOLVER, use to return to the page shown to the near right and change the equation to 0 = Y2

• ENTER ( or ) changes screens so that the close value for y: Y = 79.5 can be stored in the calculator

• ENTER solves the constraint equation for x, given that y = 79.5




• Return to the Homescreen, Y1 ENTER returns the close value, (79.5) 5.403769693f ≈

• Repeat the previous 4 bulleted steps to evaluate (80.5) 5.403769889f ≈ The constrained optimal point, ( ), (6.666,80) 5.4038f x y f= ≈ is verified to be a

constrained maximum because close points on either side of the optimal point have lower output values.

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