CAS – Specialist Maths

1. General i. Solve from the algebra menu, | symbol can be used to restrict domain. Simultaneous equations can also be solved. ii. Use Define (from Actions Menu) if a function is going to be used a number of times. iii. sto key (ctrl var) can be used to store a value in a pronumeral iv. the stored value can be substituted into the defined function v. Complete the square from Algebra menu can be used to get the general form of a circle.

vi-viii Factorise over Q and R Putting in ‘,x’ changes the factorisation field to R rather than Q. Select Menu then Algebra.

expand As well as expanding products it can be used to write in partial fraction form (i. and ii.)

In the Algebra menu, under B: Trigonometry are texpand and tcollect which can manipulate trigonometric expressions (iii. and iv.)

fMax and fMin (Calculus menu) Used appropriately these functions can give the x coordinate of the absolute maximum or absolute minimum. i. The absolute maximum of the function is shown to be unrestricted as x approaches -∞. ii. On the domain 0 < x < 4 the absolute max occurs at x = 2. iii. On the domain -3 < x < 4 the absolute max occurs at x = -3. The absolute max value can be found by substitution.

Graphing As well as function, which must be of the form f(x) = …, in the Graph Entry/Edit menu there is Equation which includes many common template forms such as

2 2

2 2

( ) ( ) 1x h y ka b− −

+ = for an

Ellipse.

2. Vectors a. basics

In the Matrix & Vector menu there are some Vector options (if required) that find the Unit vector, the dot product and the magnitude (under Norms option).

The vector 3i 4 j+

can be entered as [3 4].

b. scalar and vector resolutes For the vectors 2i 3 j 6ka = − + +

and i 2 j 2kb = − +

.

Define them then:

i. scalar resolute of a

in the direction of .b

ii. vector resolute of a

in the direction of .b

iii. vector resolute of a

perpendicular to .b

c. linear dependence E.g. Find the value of y if the vectors i j 3ka = − +

, 2i 2 j 3kb = − +

and 4i j 3kc y= + +

are linearly dependent.

3. Circular Functions a. functions The range of circular functions, reciprocal circular functions and inverse circular functions are in the trig menu and can be used to evaluate, solve, graph, differentiate and integrate.

b. domain Domain function can be typed in or found in the catalogue. For an increasing or decreasing function, the range can be found by substituting the endpoints

4. Complex Numbers There are two menus related to Complex numbers. One under Number and one under Algebra.

In the Number Menu there is an option called Complex Number Tools with options that cover those shown.

Note: cis ir re θθ =

In the Algebra Menu there is an option called Complex which allows you to factorise and solve over the set of complex numbers.

5. Differentiation

i. Differentiate a function in terms of x Shortcut key: Shift – is derivative ii. Evaluate the derivative at a point (e.g. x = 1). iii. Find the second derivative

iv. Implicit differentiation (Calculus menu)

E.g. Find 2 if 2 10dy x y ydx

+ =

Rational Functions PropFrac (found in Fraction Tools under the Number menu) can be used to write a fraction as a proper fraction. This may be helpful in determining some asymptotes of a

graph such as the line xy32

= in the

example shown.

6. Anti-differentiation and applications a. basics i. Find an antiderivative Shortcut key: Shift + is integral ii. Find a definite integral. These can be used in area, volume and rate problems also. Recall when differentiating or integrating trigonometric functions (e.g. sin x) you should (generally) be in radian mode. iii. An integral can be used in solve. On a graph page under menu analyse graph, integral or bounded area can be used to find areas.

b. arc length Menu 4: Calculus B: Arc Length E.g. Arc length for 2( ) 2f x x= − between x = 1 and x = 4.

7. Differential Equations a. Solving

The Differential equation solver can be found in the Calculus menu.

E.g. 1 Solve the differential

equation 2dy xdx

= . This generates

the general solution E.g. 2 Solve the differential

equation 2 2 3dy xdx

− = , given y = 4

when x = 0.

E.g. 3 Solve 2dy ydx

= given y =

100 when x = 0.

E.g. 4 In solving the differential

equation 22 4 0dx xdt

− − = , where x

= 2 at t = 0, you need to solve the answer presented from deSolve for x to obtain a rule for x in terms of t,

b. Euler’s method

E.g. Given 9dy xdx

= − and y = 0

when x = 10, use Euler’s method with a step size of 0.1 to approximate y when x = 10.2. To four decimal places approximation is 0.2049

c. Direction/Slope Fields

To sketch the direction (slope) field

for dy x ydx

= + , from the Graph

Entry/Edit section choose ‘Diff Eq’ and then enter the rule for the differential equation. Note: when referencing y it needs to match the start of the rule i.e. y1 in this one. An initial condition can also be entered to generate a specific solution.

8. Vector functions To sketch the path of the vector function r( ) cos( )i sin( ) jt t t= +

, from the Graph Entry/Edit menu choose Parametric and then enter the rules for x(t) and y(t). The domain can also be set, e.g.

02

t π≤ ≤ .

On the calculator page you can then refer to x1(t) and y1(t) to find say arc length between t = 0 and t = π

9. Probability and Statistics Use of normal and binomial distributions and inverse normal is assumed.

a. p-value in one and two tail tests for hypothesis testing Menu 6: Statistics 7: Stat tests 1: z test Input method is stats.

b. Confidence intervals Menu 6: Statistics 6: Confidence intervals 1: z interval Input method is stats.

10. Other Advice Make sure you know your settings and can adjust them; radian/degree, float and graph window settings.

Make sure you can troubleshoot common calculator problems: include multiplication sign for instance in ax, adjust window settings if you cannot see the graph, in function mode on graph screen rule must be in terms of x

You could consider using a new problem (doc Insert Problem) for each extended response question so that if you want to go back to it later you don’t have to scroll through all your other questions workings. The difference between a new Problem and a new Page is that Problems allow you to reuse variables without causing conflicts.

11. Some sample complex number problems Below are two sample complex number problems that can be solved with CAS.

(a) Given 3 4u i= + , find 6 2

uu−

in Cartesian form.

(b) Find the Cartesian equations for { : 1 }z z z i− = + and { : 4 }.z z i z i+ = − +

(c) Find the Cartesian equations for 3( 1)4

Arg z π+ = .

Items to have setup in your calculator going into the exam.

• Make sure both calculator and graph settings are on float and best to be in radian mode. • Have euler set up on a calculator page. • Have problem pages set up for each extended response question.

Vectors

Scalar resolute a in direction of b

Vector resolute (parallel component) a in direction of b

Perpendicular component.

Define vectors.

Call functions – accessible via var button.

Check that sum of vector components gives original vector a.

Angle between vectors a and b.

Complex Numbers

Could define your own cis notation.

Vector Functions

Can define r(t) as a vector with i and j components.

v(t) as the derivative.

Speed as the magnitude of velocity i.e. the norm of the velocity vector.

Could use fMin and fMax on speed rule.

Arc length for parametric equations is the integral of the magnitude of the velocity vector. It can be defined as the integral of the norm of the velocity vector.