© Teachers Teaching with Technology (Scotland)

©Teachers Teaching with Technology (Scotland)

Teachers Teaching with Technology

T3 Scotland

Calculus

Introduction to DifferentiationSketching the Derivative

Introduction to Intregration

CALCULUS

AimTo demonstrate how the TI-83 can be used to introduce differentiation and integration

Introduction to differentiationTangents to the curve. (Mathematics 1 (H))Objectives

Mathematical objectivesBy the end of this session you should be able to

• understand the concept of gradient function• find by graphical methods the gradient function• establish the link between gradient function and differentiation

Calculator objectivesBy the end of this session you should be able to

• use the [2nd][DRAW 5:Tangent( ] menu to plot tangents

Introduction to differentiationSketching derived functions. (Mathematics 1 (H))Objectives

Mathematical objectivesBy the end of this session you should be able to

• understand the concept of gradient function• find by graphical methods the gradient function• establish the link between gradient function and differentiation• establish the link between the shape of a graph and its gradient function

Calculator objectivesBy the end of this session you should be able to

• use the [MATH 8:nDeriv( ] function to plot differentiated function• use of [VARS] menu• switch On / Off graphs in the [Y =] menu

Introduction to IntegrationArea under a curve. (Mathematics 2 (H))Objectives

Mathematical objectivesBy the end of this session you should be able to

• find the area under a curve • establish the relationship between the function and the area obtained

Calculator objectivesBy the end of this session you should be able to

• Use of the [2nd][CALC 7:∫ f(x)dx] to obtain area under a curve.

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� T3 Scotland Calculus Page 2 of 9

To plot the gradient functions on the TI-83 use[2nd][DRAW 5:Tangent( ]

The TI-83 will display the graph and a curser will appear(not necessarily in the position shown).We want to plot the tangent line to the curve at x = 1, so [1][ENTER].

The TI-83 draws the tangent to the curve at x = 1.The equation of this tangent is displayed (bottom left).From this equation the gradient of the tangent line can be obtained, in this case, 2.This process is repeated for the x values shown in the table over leaf.

INTRODUCTION TO DIFFERENTIATIONTangents to curves

Calculator skills sheetBefore commencing this unit the STAT PLOTS on your TI must be switched off. To ensure that this is the case follow this procedure.

Press 2nd Y=STAT PLOT

A screen like this is displayed.You want to switch off all the STAT PLOTS so

choose option 4 and ENTER .This screen is displayed.

Example

You are now ready to attempt the first example.

It is suggested that for this example you set the following [WINDOW] range.

On the [Y=] screen enter the function Y1= x2.[GRAPH] this function.

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8. Make a statement about what you think the gradient function of y = axn might be:

____________________________________________________________________________

____________________________________________________________________________

INTRODUCTION TO DIFFERENTIATION

Gradient of TangentsInvestigation

4. Conjecture: For y = x6 the gradient function is ____________

5. Conjecture: For y = x7 the gradient function is ____________

6. Conjecture: For y = x8 the gradient function is ____________

7. Conjecture: For y = x9 the gradient function is ____________

1. y = x2

Conjecture: For y = x2 the gradient function is ____________

Equation of tangentGradient of tangent

x 1 2 3 4 5

2. y = x3

Conjecture: For y = x3 the gradient function is ____________

Equation of tangentGradient of tangent

x 1 3 4 5

� �3 � �3 � �3 � �3 � �3

3. y = x4

Conjecture: For y = x5 the gradient function is ____________

Equation of tangentGradient of tangent

x 1 2 3 4 5

=4× =4× =4× =4×=4×

2

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Graphing the derivative

Calculator Skills Sheet

Given a function it is possible to set up the TI to plot the given function and its derivative automatically.

On the [Y=] screen and move the cursor to Y2 = as shown.

We now want to call up the TI function called nDeriv to this location.

To achieve this press [MATH].

The TI will display this screen. Choose [8:nDeriv( ]and [ENTER]

The TI “pastes” nDeriv( function into Y2 =.

We must now tell the TI which function to differentiate and for which variable, at which value.To obtain Y1 press [VARS], chooseY-VARS menu, then 1:FUNCTION then 1:Y1 and press [ENTER].The screen displays are shown below.

Finally add [,] [x] [,] [x] [ )] to complete the sequence shown.

We are now in a position to plot derived functions!

T3 Scotland Calculus Page 5 of 9

Graphing the derivative

ExamplePlot the derivative when y = x2

Solution On the [Y=] screen, enter x2 in the Y1 position as shown.Change the graph style of Y1 to a thick line by placing the cursor on the \ and pressing [ENTER].

Set the [WINDOW] range as shown

The TI should display this.The graph of Y1= x2 is displayed as a thick lineThe graph of the derivative of Y1, i.e. Y2, is displayed as a thin line.

Why is the derived function a straight line ?____________________________________________________________________________________________________________________________________________________________________

Why are the y values of the derived function negative when x < 0 ?______________________________________________________________________________________________________________________________________________________

Now try plotting the derived function when y = x3

You may need to change the [WINDOW] range to the following.

The TI should display these graphs

Why is the derived function a quadratic shape ?____________________________________________________________________________________________________________________

Why is it always positive ?____________________________________________________________________________________________________________________

T3 Scotland Calculus Page 6 of 9

Plot the following along with their derived functions.

Sketch the results on these axis.

Use this [WINDOW]range.

1. 2.

3. 4. + 3

y x y x

y x y x

= =

= − =

2 12

3

2 2

2 2

Comment on your findings.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Graph these functions on the TI-83.Sketch the result onto these axis.Sketch the derivative without using the TI.

5. 6.

7. - 2 8. + 3

y x y x

y x y x

= =

= =

13

12

2

2 3

2 3

Check your answers by switching on the derived function again.

Switch off the derived function by highlighting the equals sign and pressing [ENTER to get the configuration shown.Notice that the = sign on the Y2 line is not highlighted.

To enter y = ½x2 , press [(][1][÷][2][)][x][x2]

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Match the graphs shown with their derivatives

9.

10.

11.

12.

13. E.

D.

C.

B.

A.

The TI now shades in the area under the curve (between the curve and the x-axis), between 0 and 1.It also displays the area 0.3.

To see how to convert this answer into a fraction see Calculator Hint Sheet 7 (Section 2)

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Example

You are now ready to attempt the first example.

It is suggested that for this example you set the following [WINDOW] range.

On the [Y=] screen enter the function Y1= x2.[GRAPH] this function.

To find the area under a curve use the function[2nd][CALC 7:∫f(x)dx]

By repeating this procedure for the values of the upper and lower limits shown on the tables over leaf, should result with the display shown here.

The display changes to a request for a lower limit.In this example the lower limit is zero, so [0][ENTER].

The display changes to a request for an upper limit.In this example the upper limit is one, so [1][ENTER].

INTRODUCTION TO INTEGRATIONArea under a curve

Calculator skills sheetBefore commencing this unit the STAT PLOTS on your TI must be switched off. To ensure that this is the case follow this procedure.

Press 2nd Y=STAT PLOT

A screen like this is displayed.You want to switch off all the STAT PLOTS so

choose option 4 and ENTER .This screen is displayed.

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AREA UNDER A CURVE

Investigationx1 = Lower limit

x2 = Upper limit

1. y = x

Conjecture:For y = x area between y = 0 and y = a is ____________________

Area

x1 --- x2 0 --- 1 0 --- 2 0 --- 3 0 --- 4 0 --- 5

2. y = x2

Conjecture:For y = x2 area between y = 0 and y = a is ____________________

Area

x1 --- x2 0 --- 1 0 --- 2 0 --- 3 0 --- 4 0 --- 5

3. y = x3

Conjecture:For y = x3 area between y = 0 and y = a is ____________________

Area

x1 --- x2 0 --- 1 0 --- 2 0 --- 3 0 --- 4 0 --- 5

4. y = x4

Conjecture:For y = x4 area between y = 0 and y = a is ____________________

Area

x1 --- x2 0 --- 1 0 --- 2 0 --- 3 0 --- 4 0 --- 5

÷2 = ÷2 = ÷2 = ÷2 = ÷2 =

÷3 = ÷3 = ÷3 = ÷3 = ÷3 =

÷4 = ÷4 = ÷4 = ÷4 = ÷4 =