Methods CAS Calculator Summary Sheet PDF

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A CAS calculator summary sheet for preparing for the end of year Mathematical Methods Examination Number 2...

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Changing Document Settings: To change the number of decimal places, change from radians to degrees (or vice versa), change calculation mode, etc: press doc   7  2 Defining Functions:

Basic functions: use f(x) := Hybrid functions: use Define

   The := symbol can be found by pressing ctrl   Either type in ‘Define’ or press menu  1 1

  ,     button and select  ,  to enter the functions Afterwards press the and their respective domains. Calculus: 

d  shift + for  

or menu  4  3

Note 1: If the function inside the integral is a general function containing k and x , x multiplied by

k must be written as k x x , not kx .

   Note 2: the modulus sign can be found under the  button. Always use the modulus sign when using your CAS to find an area.

d   shift – for d

or menu  4  1

To find the value of derivative at a given x value: menu  4  2

 d  solve   f ( x )  0, x   dx  . ‘solve’ can be typed in or found under Finding stationary points: menu  3  1. Solving Equations: Solving for a given variable: type in solve(equation, variable you want to solve for). To find all solutions within a given domain – something that is quite useful when solving trigonometric equations – type in solve(equation, variable you want to solve for)| a x b where a and b are the lower and upper bounds of the domain respectively. The | and  signs can be found by first pressing ctrl and then pressing the = sign. Finding the inverse function: Swap x and y around in the original equation. Then use solve(resulting equation, y) to find the inverse function. Make sure to replace y with f-1(x) when writing your final answer. Note: When solving inequalities, it is usually easiest to replace the inequality sign with an = sign when typing the equation into the solve function on the calculator. You can then use the solutions to figure out the range of x values that satisfy the inequality.

Solving a system of equations: go to menu  3  7 or you can type in solve(equation 1 ‘and’ equation 2,x). Finding maxima and minima: The easiest way to find exact values for max./min. values in a given domain is by using fmax(f(x), x, lower bound of domain, upper bound of domain) or fmin(f(x), x, lower bound of domain, upper bound of domain). If you’ve defined f(x), it is particularly easy to find the coordinates of the max. or min. as you can simply substitute the x value into f(x) to find the corresponding y value.

Probability and Statistics: n

Cr

can be found by typing in ncr(n,r).

solve

  f (x )dx 0.5,m  where L is the lower m

L Finding the median, m, of a continuous function: bound of the domain. The lower and upper quartiles can be found using 0.25 and 0.75, respectively, instead of 0.5.

Distributions: can be found under menu  6  5  2. Normal Cdf Input a range of X values to find the corresponding probability. If the upper or lower range is  , the  symbol can be found under the   button. 3. Inverse Normal Input the area to the left of a given X value to find the corresponding X value. If the area you are given is to the right of the X value, you need to do 1 – that area in order to find the area to the left of that X value. Remember that  = 0 and  = 1 corresponds to the standard normal distribution, Z, and can be used to find the corresponding Z score.

A. Binomial Pdf This gives the probability for a given outcome. Leave the ‘X Value’ cell blank in order to get the entire distribution starting at X =0. B. Binomial Cdf This gives the probability for a range of outcomes. Remember that the range is inclusive so make sure that the range you are given is inclusive as well. Confidence Intervals: For the Pˆ distribution: menu  6  6  5 The confidence interval will be given by (CLower, CUpper) Graphing Screen Besides simply graphing functions, the graphing screen can be used to find decimal approximations for intersection points, intercepts, max./min., etc. These can be found under menu  6. If the function does not specify a certain number of decimal places for your answer, this means you need to provide an exact value, which must be found in the calculator screen using one of the

functions that has been listed above. If more decimal places are required, go to menu  8 to change the number of digits that are displayed. To change the window, go to menu  4  1 and change the dimensions/scale of the axes. This works a lot better than simply using the zoom functions. When sketching a graph, ensure that the window on your screen matches the axes on the exam. This can be done by changing the x and y axes as well as the scaling on the axes. Less Frequently Used Calculator Functions To find the equation of a tangent line at a given x value: menu  4  9 tangentLine(expression, x, x value) To find the equation of a normal line at a given x value: menu  4  A normalLine(expression, x, x value) factor(expression)

menu  3  2

expand(expression)

menu  3  3

To create a matrix in order to perform matrix operations: menu  7  1. Appendix: Binomial distributions where you need to determine the number of trials, ‘n’: The easier case: You know how many outcomes satisfy the criteria If you know how many outcomes satisfy the criteria, you can create an equation and solve it using solve(equation,n), where n is the unknown in the equation. This will be the case for questions where the question uses the words “at least one” since the outcomes that satisfy the criteria will be given by 1 – Pr(X = 0). Similarly, if the question uses the words “at least two,” the outcomes that satisfy the criteria will be given by 1 – Pr(X = 0) – Pr(X = 1). Note 1: Even when the question says that the probability of the event has to be “at least” a given value, you should replace the  sign with an = sign when typing the equation into the solve function on the calculator. Note 2: Remember that n must be a whole number so the value of n that your calculator gives must be rounded. If the question says that the probability must be “at least” a given value, which is usually the case, you must always round n up. Never round n down in these situations. The harder case: You don’t know how many outcomes satisfy the criteria If you don’t know how many outcomes satisfy the criteria, you can either use a guess and check method with the binomCdf(n, p, lower X value, upper X value) function or else you can use a spreadsheet. Note that the n and upper X values should be the same number. Using a spreadsheet: For questions that require you to find Pr( X  x ) k , first type the value of x into cell A1. Increase this number by 1 each time you move down a cell in column A. If you need to do this a large number of times, it is faster to type =A1 + 1 in cell A2 and then move the mouse towards the bottom right of cell A2 until it becomes a + sign. After this happens, hold down the mouse button for 2 seconds. You can now drag the equation down column A as far as you want to. Lastly, click on the last cell you want to be filled in and all of the selected cells will be filled in.

Next, go to cell B1. Type ‘=’ and then press menu  4  2  B, which will bring up the Binomial Cdf distribution. In the ‘n’ and ‘upper bound’ fields, type in a1. Fill in the ‘p’ field with the probability of success and fill in the ‘lower bound’ field with the value of x that you were given. Press enter until cell B2 is highlighted. To drag the formula down, use the same method as described above: go to the bottom right of cell B1 until the mouse becomes a + sign. Then hold the mouse button down for 2 seconds and drag the formula down as far as you want to. Lastly, click on the last cell you want to be filled in and all of the selected cells will be filled in. Now it’s simply a matter of scrolling down column B until you find a probability that exceeds k . Write down the value in column A that corresponds to this probability. This is the n value you are looking for....