mei casio tasks for as puremei.org.uk/files/ict/mei-casio-tasks-as-core.pdf · mei casio tasks for...

MEI Casio Tasks for AS Pure

TB v2.2 26/06/2018 © MEI

Task 1: Coordinate Geometry – Intersection of a line and a curve

1. Add a new Graphs screen: p5

2. Add a quadratic function as Y1, e.g. Y1=x²–4x+1 :

fs-4f+1l

3. Add a line as Y2, e.g. Y2=x–3 : f-3l

4. Plot the curves: u

5. Find the points of intersection of the line and the curve: yy

Use the cursor keys to navigate between the points of intersection: $!

Questions

What is the relationship between the x-coordinates of the points of intersection and the equations of the line and curve?

Hint: 0452 xx

Does this work for other curves and lines?

Problem (Try the problem with pen and paper first then check it on your calculator) Find exact values of the coordinates of the points of intersection of the following:

y = x2

and y = 2 x + 3 y = x2

– x and y = 2 – x

y = x2

– 2x + 2 and y = 2x + 1

Further Tasks

What happens if the coefficient of 2x is not 1?

Find a line and a curve that intersect at the points (5, 16) and (-2,2).

Can you find an example of a line and a curve that would have: o Exactly 1 point of intersection? o No points of intersection?

Investigate the number of points of intersection of two curves.


TB v2.2 26/06/2018 © MEI

Task 2: Coordinate Geometry – Perpendicular lines


2. Switch the derivative on: LpNNNNNNqd

3. Add the function Y1 = 2x + 1 and plot it: 2f+1lu

4. Display a perpendicular line (a normal): Sketch > Norm re

Use !/$ to move the position of the point and the perpendicular line.

Press l to display the equation of the normal.

Question

What is the relationship between the gradients of the lines?.

What happens when you change the equation of the first line? You should try negative and non-integer values.

Problem (Try the problem with pen and paper first then check it on your calculator)

Find the equation of the line that is perpendicular to the line through (5,1) and (1,3) and

passes through the point (3,4).

Further Tasks

Investigate the relationship between the equations of perpendicular lines when they

are written in the form ax + by = c.

Given two points A and B how would you find the equation of the perpendicular bisector?


TB v2.2 26/06/2018 © MEI

Task 3: Differentiation – Exploring the gradient on a curve


2. Switch the derivative on: LpNNNNNNqd

3. Add the function Y1 = x² and draw it: fslu

4. Add the tangent at the point: Sketch > Tangent rw

Use !/$ to move the position of the point on the curve.

Question

How is the gradient of the tangent to the curve at a point related to the point?

Verify your comments by trying some other curves of the form baxy 2. You might find it

useful to examine a table of values of the gradient: p7u. (Compare X and Y’1).

To add/edit curves toggle the Graphs/Text screens with u.

Problem What is the relationship between a point on the curve and the gradient of the tangent to the curve at that point for:

y = x3

y = x4

y = x5

y = xn

Further Tasks Investigate the relationship between a point on the curve and the gradient of the tangent to the curve at that point for:

y = ax2

or y = ax3

y = ax2

+ bx + c or y = ax3

+ bx2

+ cx + d


TB v2.2 26/06/2018 © MEI

Task 4: Differentiation – Introduction to Stationary Points


2. Add a quadratic function as Y1, e.g. Y1=x²–4x+1 :

fs-4f+1l

3. Add the derivative of Y1 as Y2, e.g. Y2=2x–4 : 2f-4l


Questions

How is the graph of the quadratic function related to its derivative?

What is happening on the graph of the derivative when the quadratic function is at a minimum?

Are there any quadratic functions that have a maximum?

Problem For the following curves plot the graphs and their derivatives. Use the derivative graph to find where the curve has a local maximum or minimum:

y = x2

+ 4x + 1

y = 4 – 6x – x2

y = x3 – 3x

y = x

3 – 3x

2 + 3x

What do you notice about the derivative graph at a local minimum of maximum?

Further Tasks

Find different cubics that have 0, 1 and 2 points where the tangent to the curve is horizontal (a stationary point).

Is it possible to find a quadratic that doesn’t have a stationary point?


TB v2.2 26/06/2018 © MEI

Task 5: Integration – Area under a curve


2. Add the function Y1 = x² and draw it: fslu

3. Find the area under the curve using the Integral function: G-Solv > ∫dx > ∫dx: yueq

4. Set the lower limit to 0 and find the area under the curve for different values of the upper limit, e.g. to find the area between 0 and 1 press: 0l1l

Questions

What is the relationship between the area and upper limit?

What is the relationship if f(x) is changed to a different power of x?


Find the area under y = x3 between x = 0 and x = 3.

Further Tasks

Investigate the area under y = x2 between x = a and x = b.

Investigate the areas under functions that are the sums of powers of x:

e.g. y = x³ + 3x² + 4x +1


TB v2.2 26/06/2018 © MEI

Task 6: Functions – Transformations


2. Add a function, e.g. f(x) = x², Y1=x²: fslu

3. Add the function f(x+a ) + b, Y2=(x+A)²+B:

Njf+afks+agl

4. Plot the curves using modify: y

Questions

What transformation maps f(x) onto f(x+a)+b?

Does this work if other functions are entered for f(x)?


Show that (x+2)3 + 3 = x³ + 6x² + 12x + 11.

Hence sketch the graph of y = x³ + 6x² + 12x + 11.

Further Tasks

Show that f(x) = x4 – 8x³ + 24x² – 32x + 13 can be written in the form (x+a)

4 + b and

hence find the coordinates of the minimum point on the graph of y = f(x).

Set Y2 so that it plots f(cx). What transformations map f(x) onto f(cx)?


TB v2.2 26/06/2018 © MEI

Task 7: Equations of Circles

1. Go into Conic Graphs mode: p9

2. Select the (X–H)²+(Y–K)²=R² form of a circle: NNNNl

3. Use MODIFY to investigate circles in this form: q

4. Select the AX²+AY²+BX+CY+D=0 form of a circle: dddNl

5. Set A=1 and then use MODIFY to investigate circles in this form: q

Questions

For circles of the form (x – h)² + (y – k)² = r² what is the radius and the position of the

centre of the circle?

For circles of the form x² + y² + bx + cy + d = 0 what is the radius and the position of the centre of the circle?

Problem (Try the question with pen and paper first then check it on your calculator) Find the radius and the centre of the circle x² + y² – 4x + 2y – 4 = 0. Find the exact values of

the coordinates of the points of intersection with the y-axis.

Further Tasks

Investigate circles that pass through the origin. Find a condition that the coefficients must satisfy.

Investigate circles of the form x² + y² + bx + cy + d = 0 that do not intersect either the x

or y axes.

What can you deduce about the centre of a circle that has the x and y axes as

tangents.


TB v2.2 26/06/2018 © MEI

Task 8: The Factor Theorem

1. Go into Table mode: p7

2. Add Y1 = x³ – 2x² – x + 2 : f^3$-2fs-f+2l

3. Use SET to set the table to Start: –5, End: 5, Step: 1: yn5ld

4. Display the table: u

5. Go into Graph mode and plot the graph of this function: p5u

Questions

How do this table and graph confirm that x³ – 2x² – x + 2 = (x + 1)(x – 1)(x – 2)?

What is the factorised form of the following cubics: y = x³ + 4x² + x – 6 y = x³ – 4x² – 11x + 30

y = x³ – x² – 8x + 12 y = x³ – 7x² + 36

Problem (Try the question with pen and paper first then check it on your calculator) Show that (x – 2) is a factor of f(x) = x³ + 4x² – 3x – 18. Hence find all the factors of f(x).

Further Tasks

Find examples of cubics that only have one real root.

Investigate using the factor theorem for polynomials of other degrees, e.g. quadratics or quartics.

Investigate the polynomial solver: pafw.


TB v2.2 26/06/2018 © MEI

Task 9: Differentiation – Gradient graphs


2. Add a cubic function as Y1, e.g. Y1=x³ – x² – 2x+2 :

f^3$-fs-2f+2l

3. Add the derivative, d

d

y

x , as Y2:

dY2= (Y1) |

dx x

x

iwqq1$fl


Question for discussion

How is the shape of the gradient graph related to the gradient of the tangent to the curve?

Verify your comments by trying some other functions for Y1.

Problem Change your original function so the gradient function has one of the following graphs:

Extension Task

Find the point on the function 3 26 9 1y x x x where the tangent has its maximum

downwards slope. Investigate the point with maximum downward slope for other cubic functions.

The derivative is in Option > Calc


TB v2.2 26/06/2018 © MEI

Task 10: Reduction to Linear Form Investigation into Moore’s Law Moore's law is the observation that the number of transistors in an integrated circuit doubles approximately every two years. The table below gives the maximum number of transistors per integrated circuit for a computer produced in that year:

Year 1972 1978 1985 1989 1993 1999

Transistors per integrated circuit

2,500 29,000 275,000 1,180,000 3,100,000 24,000,000

The number of transistors, N, may be modelled by a function of the formtN kb , where t is

the number of years after 1970 and k and b are constants.

i) Show that in terms of this model the graph of 10log N against t should be a straight

line and state its gradient and y-intercept in terms of k and b.

ii) Use the data to find the equation for N in terms of t.

iii) In 2015 a computer was produced with 7,100,000,000 transistors per integrated circuit. Is this value consistent with your model?

Using the CG-20 Data can be entered in the Statistics mode: MENU > 2. To calculate the log of all the values in a list enter the formula in the List name (the List command is in the OPTN menu). Use GRAPH to plot a scatter graph of with Xlist: List1 and YList: List3. To find the regression line select CALC > X > ax+b


TB v2.2 26/06/2018 © MEI

Task 11: Solutions of Trigonometric Equations (Degrees)

1. Select Graphs mode: p5

2. Check the angle type is set to degrees: SHIFT > SET UP and scroll down to Angle.

3. Enter the graph Y1=sin x : hfl

4. Enter the graph Y2=0.5 : 0.5l

5. Set the View-Window to TRIG: Lewd

6. Draw the graphs: u

7. Use G-Solve to find the points of intersection: Lyy

You can use the cursor (!/$) to move between the points of intersection. Try finding

the points of intersection for other values of Y2 (e.g. Y2 = 0.75 or Y2 = –0.3). Questions

What symmetries are there in the positions of the points of intersection?

How can you use these symmetries to find the other solutions based on the value of sin

-1x given by your calculator? (This is known as the “principal value”.)

Problem (Try the question just using the sin-1 function first then check it using the graph) Solve the equation: sin x = 0.2 (–360° ≤ x ≤ 720°)

Further Tasks

Investigate the symmetries of the solutions to cos x = k and tan x = k.

Investigate the symmetries of the solutions to sin 2x = k.

Try using the SolveN function in your calculator.


TB v2.2 26/06/2018 © MEI

Task 12: Gradients of tangents to the exponential function y=ex


2. In SET UP set Derivative: On: LpNNNNNNqd

3. Draw the graph y = e

x. Y1=ex:

LGflu

4. Add the tangent at the point (Sketch > Tangent): rw

Use !/$ to move the position of the point on the curve.

Questions

What is the relationship between the point and the gradient of the tangent on y = ex ?

How does this relationship change for the graphs of y = e2x

, y = e3x

, … ?

Problem (Check your answer by plotting the graph and the tangent on your calculator)

Find the equation of the tangent to the curve y = e2x at the point 1x .

Further Tasks

Find the tangent to y = ex that passes through the origin.

Find the gradient of the tangent to y = 3x when 0x .


TB v2.2 26/06/2018 © MEI

Task 13: Quadratic Inequalities


2. Add the curve Y1=(x–A)(x–B) :

jf-afkjf-agkl

3. Plot the curves using modify: y

Find the range of values of x for which the curve lies below the y-axis and is the solution to

the inequality

(x – a)(x – b) < 0.

Questions for discussion

If the product of two numbers is negative what does this tell you about the numbers?

Will you always be able to find x-values for which a quadratic is negative?

What would the solution to (x – a)(x – b) > 0 look like?

Problem (Try the problem with pen and paper first then check it on your software)

Sketch the graph of y = 2x² – x – 6 and hence solve the inequality 2x² – x – 6 ≥ 0.

Further Tasks

Find the range of values for k such that x² – 4x + 3 = kx has two distinct roots.

Investigate y > mx + c and y > ax² + bx + c graphically.


TB v2.2 26/06/2018 © MEI

Teacher guidance Notes on using the Modify function It is useful for students to be familiar with this mode first. When in Modify mode the parameters and the step size can be changed with the cursor keys or values can be directly typed in. When in Modify mode the cursor keys are used to change the parameters and cannot be used to move the axes. All moving of the axes and zooming is disabled in Modify mode. To move the axes or zoom press EXIT to come out of Modify mode. The axes can then be set to the appropriate values. To re-enter modify mode press EXIT again to return to the list of functions then F5 to go back into Modify mode. Task 1: Coordinate Geometry – Intersection of a line and a curve This task can be used to introduce the intersection of a line and a curve. Some students might find it helpful to plot the function obtained by subtracting the linear function from the quadratic and observing its roots. Problem solutions: y = x

2 and y = 2 x + 3 (–1, 1) and (3, 9)

y = x2 – x and y = 2 – x

(–2, 4) and (1, 1)

y = x2 – 2x + 2 and y = 2x + 1 (–√3+2, –2√3+5) and (√3+2, 2√3+5)

The curve 42 xxy and the line 62 xy intersect at (5, 16) and (-2, 2).

Task 2: Coordinate Geometry – Perpendicular lines For the calculator to display the equation of the perpendicular line the Derivative function needs to be switched on. The equations of the perpendicular lines will have the coefficients written in decimal form and it might be helpful to discuss with students why writing these as fractions is preferable in some circumstances. It might be helpful to get the students to create a table like the one below, so that they can record their results in a systematic way.

Gradient of Line Gradient of Perpendicular Line

Problem solution: The perpendicular line has equation y = 2x – 2.


TB v2.2 26/06/2018 © MEI

Setting the View-window to initial Leq keeps the increments when moving points to

0.1. Task 3: Differentiation – Exploring the gradient on a curve The aim of this task is for students to investigate (or verify if they have already met it) the rule for differentiating polynomials. It can be used as an introduction to the topic or to consolidate what they have already learnt.

Students should be encouraged to use the table view (p7u) as well as the

graphically view so they have both a numerical and graphical appreciation of the rules. It might be helpful to get the students to create a table like the one below, so that they can record their results in a systematic way.

x Gradient d𝑦

d𝑥

Task 4: Differentiation – Introduction to Stationary Points This task highlights the link between finding stationary points and maximum/minimum points on curves. It might be helpful to get the students to create a table like the one below, so that they can record their results in a systematic way.

Upper Limit Area

The G-Solv menu (y) has options for finding maximum/minimum points which some

students will find useful. Problem solutions: y = x

2 + 4x + 1

min: (–2, –3) y = 4 – 6x – x

2 max: (–3, 13)

y = x3 – 3x

min: (1, –2) max: (–1, 2)

y = x3 – 3x

2 + 3x no maxima or minima

The final example can be used to discuss stationary points that are points of inflection and this can lead into the first of the extensions tasks.


TB v2.2 26/06/2018 © MEI

Task 5: Integration – Area under a curve The aim of this task is for students to investigate (or verify if they have already met it) the rule for integrating/finding the area under polynomials. It can be used as an introduction to the topic or to consolidate what they’ve already learnt. Problem solution: The area is 20.25. The first of the further tasks is an opportunity for students to investigate:

0 0

f( )d f( )d f( )d

b b a

a

x x x x x x .

Task 6: Functions – Transformations If students have met trigonometric functions then these work well for this task. If you are going to try a variety of functions then it is good to use Y2=Y1(x+A)+B For the problem students should expand the function using either a binomial expansion or by multiplying out the brackets.

The graph of the function is the graph of 3f( )x x translated by

2

3

.

Be careful with second further task (horizontal stretches) – they can look like vertical stretches for many functions but this is an excellent discussion point. Students should also take care with the scales on the axes here as these can cause

confusion. f( ) sinx x or 3f( )x x x are good functions to use for this.

Task 7: Equations of Circles This task uses the Modify mode – see the notes above. Students who have not done much investigative work before might need some support structuring their approach: suggest that they change one value at a time and then record what is happening for each. Students should link the form x² + y² + bx + cy + d = 0 to the completed square form of a quadratic. Problem solution: Centre (2, –1), radius 3. For a circle to pass through the origin D=0. If the x and y-axes are tangents, then the centre must lie on either the line xy or the line

xy .


TB v2.2 26/06/2018 © MEI

Task 8: The Factor Theorem This task is intended to reinforce the link between the numerical values of roots, algebraic factors and points of intersection with the x-axis. In discussions students should be encouraged to explain how both the table and the graph indicate what the factors are. It might be useful for some students to practise expanding products of three factors before attempting this task. Students will also need to be shown, or to develop, strategies for dividing by a factor such as equating coefficients, long division or division by the box method. Questions: y = x³ + 4x² + x – 6 : y = (x – 1)(x + 2)( x + 3)

y = x³ – 4x² – 11x + 30 y = (x – 5)(x – 2)(x + 3)

y = x³ – x² – 8x + 12 y = (x – 2)² (x + 3)

y = x³ – 7x² + 36 y = (x – 6)(x – 3)(x + 2) The third question can be used to demonstrate an example of a cubic with a repeated root. Problem solution: x³ + 4x² – 3x – 18 = (x – 2)(x + 3)² Task 9: Differentiation – Exploring the gradient on a curve The aim of this task is for students to investigate (or verify if they have already met it) the shape of derivative functions. They should be encouraged to discuss why the derivatives have the shape they do in terms of the gradient of the tangent to the curve at different points. It can be used as an introduction to the topic or to consolidate what they have already learnt. Problem solution (possible solutions):

𝑦 = 𝑥3 − 𝑥 𝑦 = 𝑥3 + 𝑥2 + 𝑥 𝑦 = 𝑥4 − 𝑥2

Task 10: Reduction to Linear Form

i) Gradient = b10log Intercept = k10log

ii) tN 397.15.4645

iii) Model gives a value of 5,625,176,604. Whilst this is different to the value for the computer it is not an unreasonable estimate, considering the extrapolation from the data given in the question.

Task 11: Solutions of Trigonometric Equations (Degrees) This task encourages students to think about the symmetries of the trigonometric graphs and use these in finding solutions to equations. V-Window: Trig is useful to set the axes to appropriate sizes. The scale on the x-axis is dependent on whether the calculator is in degrees or radian mode. Problem solution: x = –348.46°, –191.54°, 11.54°, 168.46°, 371.54°, 528.46°.


TB v2.2 26/06/2018 © MEI

Task 12: Gradients of tangents to the exponential function y=ex

The aim of this task is for students to be able to find the gradients and equations of tangents to exponential functions. Once students have observed that the derivative is the same as the y-coordinate they should

explore other curves of the form y = eax .

Problem solution:

14.778 7.389y x

The second of the further tasks requires students to rewrite 3xy as (ln3)e xy .

Task 13: Quadratic Inequalities This task focusses on solving quadratic inequalities by sketching graphs. Students should be encouraged to relate the roots of the quadratic with the possible values of the factors to determine whether the product is positive or negative. Substituting in some values can help confirm the solution is valid. It is important to highlight where the solution can be written as a single inequality and where it should be written as two separate inequalities.

Problem solution: 3

2x or x ≥ 2

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