Calculator Guide: TI-84 Plus CE

[RuiAce]

Using the TI-84 Plus CE in QCE Mathematics

As per the votes, I believe the TI-84 Plus CE likely one of the more popular calculators used by QCE students. It's the most superior option from the TI-84 series. It's not as powerful as the TI-Nspire series, but I'm convinced it has more than enough functionality to be a heavily useful calculator nonetheless. It is not worth buying another calculator if you already have this one.

The following is a guide to the operations I believe are useful to QCE Maths Methods and QCE Specialist Maths (all units). It's hoped that every cornerstone of the syllabus (that can be trivialised with a calculator) will be addressed. If there is functionality requested in the two mathematics syllabuses not addressed, please leave a comment so that I know what to add on. (Or if there's a more efficient way of doing something!) Please feel welcome to ask about additional functionality outside of the syllabuses, but I can't promise that I can cover everything confidently.

As a foreword, I recommend to not use the calculator for EVERYTHING in the course. The calculator is meant to help you, not do things for you (and sometimes, do it slower). You should always make a judgement on if the calculator is appropriate to the problem at hand.)

In this guide, buttons on the calculator will be denoted in bold font. When multiple symbols appear in bold font, it's assumed that you must press the keys in that order. (Spaces are used to separate one button from the next, even when it's obvious enough to not be required.)

It may be worth mentioning that the TI-84 Plus CE does come in various colours. This is obviously for aesthetic, and does not impact the calculator's functionality.

Common basic functionality

Getting started
The on button is located at the bottom right of the calculator. As expected, this switches the calculator on. The first time you use it, you might be greeted by a screen like this.

It's usually harmless to not show this screen again. (After all, you can always just reopen this guide if you want to see it again.) Press 1 to toggle this, or press 2 if you'd rather see it again later just in case.

You will then be taken to a blank screen as the calculator awaits further instructions.

The calculator needs to be recharged every once in a while. There is a battery level indicator in the top-right corner of the calculator. Note that the orange border shows up only when the calculator is being recharged.

You should find one or two charging cables when unpacking the box the calculator came with. If you ever lose it and need to repurchase it, I believe that the calculator takes a USB Type-B. The charging port is along the right face of the calculator. Plug the USB Type-B end into the calculator, and the other end into an appropriate device. (If the other end is a USB Type-A, i.e. the one found on computers, you can usually just plug it into the computer for recharging purposes.)

(It is likely that your calculator came with a lid. These work the same way as for scientific calculators.)

Shift keys
The TI-84 Plus CE has two 'shift' keys, similar to many scientific calculators. They are also used by first pressing the shift key, followed by a relevant other button on the calculator. If you pressed it by accident, you can always press it again to get out of the shift key mode.

The blue 2nd key toggles the blue operation/symbol above relevant keys. For example, pressing 2nd⏵sin will let you access the sin^-1 operation above the sin button.

The green alpha key toggles the green operation/symbol above relevant keys. For example, pressing alpha⏵sin will let you access the E symbol above the sin button.

The basic operations
Unlike scientific calculations, computations are submitted through the enter key in the bottom right corner. This replaces the classic equal sign on scientific calculators. Because the enter key is so fundamental, and obviously used to submit an operation, at times it will be omitted from computations. It's expected that you can press this button yourself at the very end.

The four basic operations are accessed in a similar way to scientific calculators, through the +, −, × and ÷ keys on the right. The graphics calculator also obeys the order of operations. (Note that the multiplication symbol shows up as an asterisk, and the division symbol shows up as a forward slash.)

For example, to compute \( 9 - 3\times 5\), I type 9 − 3 × 5 and obtain:

The bracket keys ( and ) are located above the numbers 8 and 9. For example, to compute \( (9-3)\times 5\), I type ( 9 − 3 ) × 5 and obtain:

Fractions are stored in one of the menus. This particular menu is preset to be in f1, which can be accessed through alpha y=. (The y= button is the white button at the top left of the calculator.)

You can press 1 to access a classic \( \frac{a}{b} \) fraction. If you press 2, you obtain a mixed numeral. (Recall that these are used to type expressions like \( 9 \frac34\right).)

After choosing a fraction, the direction keys (see: navigation) are used to jump between the numerator and denominator (and the integer for mixed numerals). So for example, alpha y= 1   9 ▼ 1 2 gives:

Notice how the answer is automatically converted to a simplified fraction wherever possible. However, if it's impossible to leave it as a fraction, it'll automatically be converted to a decimal form.

Also note that similar to most scientific calculators, the numerator can be typed first. Here, if I type 1 2 alpha y= 1, then 9], I obtain:

Unlike scientific calculators, negative numbers must be accessed through the negative number symbol (-). This symbol is below the number 3. If you use the minus sign, the calculator will display a syntax error (which you can quit by clicking 1, or fix by clicking 2). The output from typing (-) 3 is shown on the second line here.

Powers are accessed through the caret symbol ＾ above the division symbol. If I want to compute \( 3^3\), I type 3＾3 to obtain:

Note that squaring (taking powers of 2) and reciprocating (taking powers of -1) are considered common operations, and hence they have shortcut buttons on the calculator. These are found along the left-hand column. After becoming proficient at the calculator, you should use these buttons for squares and reciprocals, rather than manually typing powers of 2 and negative 1.

The period (i.e. the dot) . under the 2 is used for decimal points. Decimal points are approached the same way as for scientific calculators.

Navigation and editing
The four arrow keys ▲▼◀ ▶ are on a small circular palette towards the top right of the calculator. They're used in general for scrolling purposes. When getting started, it helps to know that these can be used to access previous operations in the history.

Unlike for scientific calculators, after typing a series of equations you'll find they're listed one immediately below the next. Here are just some random computations done one after another.

Suppose I realise I made a mistake and want to change the second equation to \( 9^2 - 8^2 - 7\). Now for a short example like this, perhaps retyping it is enough. But editing is the more efficient procedure once you start dealing with extremely long-winded expressions.

I can press the up key four times in a row, i.e. ▲▲▲▲, to access the said equation. You'll notice that the equation requested becomes highlighted.

(If you press the up button too many times, you can go back down with the down button ▼.)

I now press the enter key. Notice that by doing so, the equation I've requested has now been pre-filled for me for my next operation.

I can now use the left arrow key ◀ to scroll back to where the 3 on \(9^3\) is. A flashing black box appears over the 3 on my device (but not on my screenshot). (Again, if I press the left arrow key too many times, I can use the right arrow key ▶ to get back.)

Now I simply type 2 to replace it and press enter, and obtain my new answer.

Notes:
- Rather than pressing the direction keys repeatedly, I could've just held the key down for a longer duration. Like scientific calculators and keyboards, the graphics calculator will then scroll in the direction of the key I've pressed. This is faster for longer equations.
- The up/down arrow keys can also be used to access previously obtained answers, instead of the expressions submitted.
- Notice how by hovering my cursor over the 3, I could replace it with a 2. It's important to understand that the TI-84 Plus CE is defaulted to replace mode.

To delete a symbol instead of replacing it, hover over the symbol the same way as above (i.e. with the arrow keys). Then, there is a del button towards the top along the middle column. Click this button to delete the symbol you've currently hovered over.

Pressing 2nd del enters the insert mode (ins). When in insert mode, your cursor turns into a blinking underscore.

Now when you type, rather than overwriting what was already there, you insert immediately BEFORE where the underscore is placed. If you make a mistake with this it's usually not hard to correct it; it's just annoying.

If for some reason you decided you need to eliminate the entire equation you just typed, the clear button above the caret (i.e. powers) wipes the whole thing away. The button is quite self-explanatory - simply click clear and the equation disappears.

The clear button works like that only when something's already typed. If you have nothing already typed, the clear button wipes clean your screen. (That is, you're sent back to the blank screen.) Don't worry though - the calculator has quite a fair bit of memory. You can still use the up arrow keys to access anything in the history, should you need it!

Memory keys
After a messy computation (for example, \( \frac{1.319^3\times 34}{2.5\pi} - 88\sin (65^\circ)\)) you may want to store the number somewhere for use later. The memory keys can be used here. They're quite similar to the ones on scientific calculators.

The most important memory key is perhaps the ans key. This always stores the value in the computation immediately before your current one. The ans key can be manually accessed through 2nd (-) (it is above the negative symbol key).

The ans key also shows up by itself sometimes. This can be useful when you want to split up computations onto two lines (for say, ease of reading). Consider my current screen.

Suppose now I press × 2 . 7 5. I obtain the following output - notice how 'Ans' is automatically displayed on my screen even though I didn't press it!

The other memory keys are useful when you know you're not going to use ans, i.e. the result from the line immediately above for the next computation. There are 27 other memory keys for numbers on the TI-84 Plus CE - one for each of the letters A-Z, and also θ. These take up most of the 'alpha' green keys on the calculator.

The expression is first typed into the screen. To store its value into a memory key, the sto-> key above the on key is used. This displays an arrow on the screen. The memory key (a.k.a. variable) comes after this arrow. So for example, 2.413 + 9.1354 sto-> alpha 4 will print:

(This is because alpha 4 leads to the memory key T being displayed.)
Now, suppose I wish to compute \(2T - 3\). I can simply press 2 alpha 4 − 3 and obtain my answer!

(Worthwhile remark: for multiplication, the times symbol does not need to be explicitly typed for memory keys. The calculator knows to interpret \(2T\) as \(2\times T\). This is why I often refer to them as variables as well. The ans key also obeys this rule.)

If you accidentally pressed enter without storing, don't worry. The answer can also be stored into a memory key.

Brightness
The calculator's brightness is increased through 2nd ▲ and decreased through 2nd ▼. (Hopefully you see the sun icon that's commonly associated with brightness!) These are pretty self explanatory; set your brightness as you feel appropriate. And preferably before the exam too, not during.

A note on menus and the catalog
The keys f1, f2, f3 and f4 have preset menus. These will be used throughout the rest of the guide.

There are also many more menus in the TI-84 Plus CE. Common ones include math, vars, distr, matrix, stat and many more. Again, these will be explored later on if required.

These menus are all navigated through the direction arrows, as in the navigation section. They can be exited through the quit button, accessible by 2nd mode.

Each of the functions in the menu can be accessed by scrolling down to it, and then pressing enter. Alternatively, they can be accessed by pressing the number or letter assigned to it. For example, in this menu, I can access the "logBASE(" function by simply pressing 5, or by scrolling over it and then pressing enter.

It's up to you to remember how each of the (important) functions work. But when you open one of these menus not through the f-keys, i.e. for example math, some commands in the menu have a small catalog documentation. These are accessed by hovering over the command, and then pressing the + key. The catalog doesn't tell you what the input arguments are; it just tells you what needs to be put in. It's a good thing to keep at the back of your mind if you forget what goes where in a calculator function.

Modes
The mode button is next to the 2nd button on the calculator. When you click on it, you'll be displayed a bunch of settings you may wish to change on the calculator.

- MathPrint/Classic - You can try setting your calculator onto classic mode. It basically displays equations through a more older, outdated format. (For example, powers aren't displayed as powers, but rather the caret symbol itself is displayed.) Personally, I recommend just leaving on MathPrint.

- Normal/Sci/Eng - The sci option can be triggered to display output in scientific notation. The normal mode just displays the number as is. (Eng is a special display mode used in engineering.)

- Float/0/1/2/3/4/5/6/7/8/9 - Switching out for a digit rounds your final answer to that many decimal places. For example, switching to 0 rounds to the nearest whole number, whilst switching to 5 rounds to 5 decimal places. Leaving it as float is usually recommended - it attempts to display as many digits as the calculator can display.

- Radian/Degree - A self explanatory toggle. Always double check what your question gives you to know if you're working in radians or degrees!

- Function/Parametric/Polar/Seq - A setting to choose what type of function you wish to graph. Usually, you leave it as function for \(y=f(x)\) curve. Parametric plots are mentioned in specialist.

- Sequential/Simul - When graphing multiple curves, the calculator usually graphs one completely before moving onto the next. This is done in the sequential mode. When you switch to simul, you're telling the calculator to graph both of them simultaneously. It's up to you which you prefer out of the two.

- Real/\(a+bi\)/\(re^{\theta i}\) - Only important for specialist students, who require complex numbers. Usually Real is enough, since \(i\) still works in real mode. (But computations like \(\sqrt{-1}\) do not, so very occasionally it is preferred to switch to \(a+bi\).)

- Full/Horizontal/Graph-Table - By default on full, when graphing, the curve takes up the entire screen. The other two modes are split-screens. Horizontal shows the equations of the curves simultaneously. Graph-table displays a table of values alongside the plot. This falls down to preference I suppose.

- Fraction type - Usually you should keep this as n/d. Mixed numerals are a bit pointless in my honest opinion.

- Answers: Typically you should leave this as auto. When your answers include fractions or decimals, it tries to stick with fractions/decimals depending on what you had in the input. Switching it to dec forces all answers to be given as decimals, which is sometimes unnecessary.

- Stat diagnostics: Not useful for methods/specialist, but I like to keep this on.

- Stat wizards: Keep this on. For stats commands, it gives you guidance as to what variables go where.

- Set clock: Self explanatory. Ideally just do this in your free time, and then never touch it again.

- Language: If you want a different language, that's your choice.

The mathematical methods toolbox

(Mathematical) Functions
Functions are basically handled through the graphing interface on the TI-84 Plus CE. Obviously their graphs need to be dealt with this way. But interestingly enough, evaluation of expressions can be done through functions as well.

To enter the graphs interface, press the y= button on the top left of the calculator. This takes you to a series of expressions for \(y\) that initially looks like this.

All functions on the TI-84 Plus CE must be declared in terms of \(x\). One way of doing this is just to use the memory key X, accessed through alpha sto->. A more convenient method is to use the x,τ,θ,n button next to the alpha button. When the graphing mode is set to 'function' (as opposed to, say, parametric) this button always prints an X for you!

Each of the buttons Y1 to Y0 can be used to store a function. (You need to scroll down past the Y9 to see the Y0.) This effectively means that we can store 10 functions at a time. To store a function, simply write the expression in terms of X, in the function you wish to store it in. For example, here I store \(f_5(x) = 2x^2 - 4\):
2    x,τ,θ,n    x²    −    4

Suppose now we wish to evaluate \(f_5(1.25)\). We can press 2nd mode to quit the graphing interface, and return to the home screen.

Stored within f4 is the YVAR menu. This is accessed through alpha trace.

It can also be accessed through the vars menu, next to the clear button. Scroll right to Y-VARS, and select 1. Function. The same menu as from f4 will appear.

The syntax is then very similar to function syntax. I basically just need to type Y5(1.25)! Therefore, I press 5 to access Y5 from the menu, followed by ( 1 . 2 5 ). This gives the following output. (I also entered it manually to check that my function was working properly.)

Common mathematical functions
Many common mathematical functions are already built into the calculator. You should be able to locate buttons for the following functions:
- √: Square roots, accessed through 2nd  x²
- ln: The natural logarithm (i.e. base \(e\) logarithm)
- log: The base-10 logarithm. Always be careful when you're using this one that you don't actually need base \(e\)!
- sin, cos, tan: The trigonometric functions, all compatible with both radians and degrees.
- e^x: The exponential function, accessed through 2nd  ln.
- 10^x: The base-10 exponential, accessed through 2nd  log. You might not need this one much, if at all.
- sin^-1, cos^-1, tan^-1: The inverse trigonometric functions, all compatible with both radians and degrees. Accessed through 2nd   [TrigFunction] .

The two constants \(\pi\) and \(e\) can also be found on the calculator. π is accessed through 2nd ＾, and e is accessed through 2nd ÷. These buttons are one above/below the other.

The math menu contains various other mathematical functions you can use. These are also in the f2 menu, accessed through alpha window. Noteworthy mentions here include:
- abs: The absolute value; option 1 in the menu.
- logBASE(: Logarithm of any base: option 5 in the menu.
- ^x√: The \(x\)-th root of a number: option 6 in the menu. Allows you to compute cube roots, fourth roots, etc.. Alternatively just use powers of fractions.
(Others will be mentioned later.)

All of these common maths functions can be used in your function declarations for the Y-VARs as well.

Graphing toolkit
First, declare the function(s) you wish to graph as above.

At any time once you've done doing this, you can press the graph button at the top right corner of the calculator. The TI-84 Plus CE traces out the curve for you as you go. Below is the curve of \(f_5(x) = 2x^2 - 4\) from earlier.

Note that if you accidentally press the graph button, it can be time consuming waiting for the graph to be fully sketched. If you wish to abort the sketch, press the on key. This stops the curve sketching, and you can now quit back to where you were earlier.

The graph is coloured green because Y5 is currently associated with the colour green. You may wish to note that if you scroll all the way to the left, you can hover over that colour. If you press ENTER, you'll be allowed to change the colour if you wish. You can also change the line style. These are optional aesthetics - it's unlikely you'd need them in the exam.

You can zoom into the graph ordinary, or by specifying the boundaries. If you press the window button on the calculator, you're presented with this palette.

The only options you should ever need to change are:
- Xmin and Ymin: The least value for the \(x\) and \(y\) components respectively.
- Xmax and Ymax: The greatest value for the \(x\) and \(y\) components respectively.
- Xscl and Yscl: Frequency of the tick marks on the \(x\) and \(y\) axes. Initially they are set to 1, so the \(x\) and \(y\) axes will print tick marks at 0, 1, 2, 3, 4, ... and 0, -1, -2, -3, -4... (Sometimes, changing Xscl to a multiple of \(\pi\) is useful for graphing with trigonometric functions.)
Let the other parameters change themselves accordingly.

You really shouldn't need the zoom menu on the calculator at all in the exam. You can play around it with you wish. But what's useful to note is that option 6, i.e. ZStandard will reset the graph zoom to default parameters. The defaults boundaries are [-10,10] for both axes, with tick marks at each 1 unit.

Operations on graphs
After you've sketched your graph(s), there are some little operations that your calculator lets you do. The trace button is the first button of interest here. It basically lets you move along points on the graph. Use the left and right arrow keys to do this. Use the up and down arrow keys to switch between graphs.

The table button, accessed through 2nd graph, generates a table of values for the graph that you've sketched.

More operations are found in the calc menu. Access this through 2nd trace.

- value: Simply evaluates the expression at a given \(x\) value
- zero: Attempts to find an \(x\)-intercept on the graph. You need to specify a lower bound, an upper bound, and an initial guess for the intercept. These can be done using the left/right arrow keys.
- minimum: Attempts to find a minimum on the graph. Works as above.
- maximum: Attempts to find a maximum on the graph. Works as above.
- intersect: Attempts to find a point of intersection of two curves on the graph. Works as above.
- dy/dx: Computes the derivative at a specific point on the curve. Works similar to trace.
- \(\int f(x)dx\): Computes the (signed) area under the curve, from a lower bound to an upper bound.

Note that only one zero, min, max and intersection can ever be found. This is why the calculator expects boundaries and/or initial guesses. Your choice for the boundaries/initial guess should be reasonable. The following is a demonstration for one of the points of intersection with Y1 = 2X-1 and Y2 = X^2-X.

(The point of intersection is at approximately \(x=2.618\).)

Graphing with translations and dilations
There are two common methods used for transformation graphing. Method 1 is slower, but works on all kinds of functions. Method 2 is faster, but only for a select few functions.

Consider our earlier curve \(f_5(x) = 2x^2 - 4\). Suppose we now wish to define \(f_3(x) = 2f_5(x-4)\). This can be done by simply setting Y3 equal to 2 Y5 ( x - 4). Recall that x is typed as x,τ,θ,n, and Y5 is accessed through the Y-VARS menu.

If I now attempt to graph, I get the translated and dilated curve alongside the original curve.

The other method uses one of the apps. Click the apps button, then the up button (this takes you to the bottom). Select Transfrm - this takes you into transformation graphing.

Now go back to  y=. Note that the QUIT-APP option above can be accessed by first hitting the up arrow, then scrolling right over to it. Apart from that, you'll notice that Y1 and Y2 now look a bit different. If you scroll all the way to the left, and press Y1, you'll be shown the "transformation graphing" interface.

You can scroll down to ->Y1:, and then browse left and right through the transformations that this app supports. Here I chose a simple cosine curve. Then, after I graph it, I'm now displayed a fancier interface.

The up and down arrow keys now allow me to specify parameter values for A, B, C and D. Scroll so that the shaded equal sign is the parameter you wish to change. (In my case, I would be changing the parameter D.) Then, simply type another number to change the value of that parameter!

The numeric solver
This is very handy when some equations would otherwise take forever to solve, or are otherwise impossible to solve. Press the math menu, then scroll up once (this takes you to the bottom). The Numeric Solver is the very last option.

Type in the expression on the LHS and on the RHS for the equation you wish to solve. For example. suppose I wish to solve \(e^x = 1+\ln (x+1)\).
e^x    x,τ,θ,n    enter    1    +    ln    x,τ,θ,n    +    1    )    enter.

After you do this, you're taken to a new interface. Here, you specify an initial guess value for X. For this example, I'll just randomly sub in X=2 for this. (Usually, if possible, you use a graph to help choose this initial guess.)
Then, you can also specify bounds on the variable. Here I just don't bother, and leave it as its preset valuse.

I then go back up and press SOLVE, by using the "graph" button here. (Note: For some reason, this button doesn't show up when you're editing the bounds.)

The numeric solver is sometimes subject to what's called floating point errors, and it looks like this has happened here. The solution is actually at X=0, but instead it gave me a number that's just extremely close to 0, namely \(2.8592\times 10^{-7}\). At times like these, usually we just conclude that surely enough, the answer should in fact be 0.

Note: Many equations in the syllabus can be solved without the numeric solver program. But I've also bumped into cases where I felt it was just necessary. Always make a judgement about when to use this!!!

Polynomial roots finder
There is an app that helps find all approximate roots of polynomial expressions. In apps, choose PlySmlt2 (option 9). Then choose 1 for POLYNOMIAL ROOT FINDER.

The options you should set are:
- order: Degree of the polynomial.
- Real/a+bi/re^(θi): Form that the roots should be given in. (If you know you're gonna have non-real roots, use a+bi.)
(I recommend leaving the rest alone.

After you set these, you can now attempt to solve for the roots of a polynomial. For example, suppose I want the roots of \(x^3-x^2+2x-2=0\). I enter in each of the coefficients (scrolling to the right to access each box). Then I click graph to access the SOLVE function, which gives me all of the roots of the polynomial.

The binomial coefficient (Combinations)
The combination is one of the functions in the f2 menu. Recall that this is found by pressing alpha window.

Select option 8, i.e. the nCr option. Then input your values for \(n\) and \(r\). (Use the right arrow key to move from the upper index to the lower. The following example computes \( \binom{9}{4} = 126 \) for me.

Scientific notation
The EE symbol is used to represent scientific notation on the TI-84 Plus CE. It is accessed through 2nd  ,. Note that the comma is above 7 on the calculator.

A number in scientific notation \(a\times 10^b\) is typed as a EE b. So for example, to compute
\[(9.109\times 10^{-31}) + (1.673\times 10^{-27})\] I type
9 . 1 0 9 2nd , (-) 3 1 + 1 . 6 7 3 2nd , (-) 2 7

The answer is, unsurprisingly, \(1.6739109\times 10^{-27}\).

Numeric derivatives
The graphics calculator cannot differentiate algebraically for you. However, it can compute the value of the derivative at a specific point for you. This can be very time saving if you have to deal with the product or quotient rule, since typing in the substituted expression usually takes longer time.

The numeric derivative calculator is found in the f2 menu as option 3 When selected, it looks like this.

The first box contains the variable you wish to differentiate with respect to. Typically, I always use X (even if the question uses something else like t), just because the x,τ,θ,n button is very accessible. Then, the second box contains the expression to be differentiated. This can be one of the Y-VARS if you've stored an expression there already, or it can be an expression you typed in. The third box contains the value of \(x\) for which the derivative must be evaluated at.

For example, I wish to find the (approximated) value for the derivative of \(x e^x \sin (x)\) at \(x = \pi\). You should try replicating this yourself.

Tangent lines
This is an interesting little extra feature supported by the TI-84 Plus CE. Whilst I do not advise using this to find the equation of the tangent, it's a useful check tool to keep at the back of your head.

Tangent lines can be added through the draw menu. This is accessed through 2nd prgm. It is option 5 of the menu.

In the resulting interface, you can either scroll left or right to choose the point you want the tangent to be at. Alternatively, I prefer just entering the value in manually. Here, I've sketched the curve \(y=\sin x\), and suppose I wish to find the tangent line at \(x=\pi\). Then I just type π in and obtain:

.

You can verify algebraically that the tangent line is \(y = -2x+2\pi\). Note that the calculator has run into some floating point error here, since it didn't display the exact value of 2 for the gradient. This is one of many reasons why I do not recommend using it to find the equation of the tangent for you. But for something bizarre like \(y=xe^x \ln(x)\sin(x)\tan(x)\) at \(x=\pi\), it might be the better alternative.

Locating maxima and minima
In the calc menu (accessed via 2nd trace), the options 3. minimum and 4. maximum can be used to help find local extrema as well. But you have to be careful.

Here I have assigned Y1 to correspond to the function \(x(x-2)(x+2)\). Suppose I wish to find the local maximum on the graph. I need to specify a left and right bound, as well as a guess, which I will do with the arrow keys here.

The calculator tells me that the maximum is at approximately \(x=-1.154701\). Indeed, the maximum is actually at \(x=-\frac2{\sqrt3}\).

Now, this only worked because I picked my values for the bounds and my guess carefully. If I screw up any of the bounds, and/or the guess, sometimes it'll work and other times it won't! (For the interested, this is a consequence of how approximation algorithms work. Calculators don't usually find these values by algebra, but rather algorithms that rely only on numeric computation.)

Note also that the bounds should restrict the local maximum to also be the global maximum. For example, in this example, I rig things so that it converges to the global maximum on the specified interval. But the global maximum is NOT the local maximum here!

(If I moved my guess elsewhere, I could've gotten it to find the local maximum for me instead. This is why sometimes it's hard to say which "maximum" the calculator will find; i.e. whether it's the local maximum, or a global maximum.)

The same principles apply for the minimum.

Plotting the derivative
It is possible to also plot the derivative curve. The trick is that when choosing a value for X, all we do is just let that value be X again! Sounds like a bit of a hack, but it works.

Here, I plot \(y = x(x-2)(x+2)\) along with \( \frac{dy}{dx}\).

The trade-off is this this tends to be ridiculously slow. (But I mean, what were you expecting?! The calculator has to compute a TON of derivatives for you to generate this plot!)

Numeric integrals
The numeric integral is also found in the f2 menu. It is option 4. Much like with derivatives, it can only evaluate integrals that have boundaries on them.

It is also used pretty much the same way. The integration boundaries are specified, followed by a Y-VAR or an expression to integrate, followed by the variable of integration. The below computation evaluates \( \int_1^{\ln 2} xe^x\,dx \).

Areas under curves with the aid of a graph
If you've already sketched a curve, option 7: ∫f(x)dx in the calc (2nd trace) menu computes the signed area under the curve for you. All you need to do is specify the lower and upper limits.

For example, for \(y=x(x-2)(x+2)\), suppose I wish to find \( \int_1^2 y\,dx\). I will manually enter the values of 1 and 2 here:

The signed area under the curve is \(-2.25\) according to the calculator.

Okay, so what if I want an unsigned area then? That is, what if I just want the magnitude of a bunch of areas?

For the calculator, it turns out that the shortcut formula we can use here is \(A = \int_a^b |f(x)|\,dx\). That is, we wrap what's inside the integral in absolute values. So for this example, I can just go back to computing the integral of |Y1| over those boundaries:

It's interesting to note that when I put the absolute values inside the integral, I no longer need to split the integral apart either! For example, suppose I want the magnitudes of the areas from \(-2\) to \(2\). I just simply do this!

Areas between curves
For the area between curves problems, we basically use this formula now:
\[ A = \int_a^b |f_1(x) - f_2(x)|\,dx. \]
For example, suppose I require the areas between \(y = x(x-2)(x+2)\) and \(y=2x\) in the first quadrant. To do this, I need to start by finding the points of intersections of the curves. It's not hard to derive algebraically that the points of intersection I require are \(x=0\) and \(x=\sqrt{6}\). Alternatively, you'll find that the point of intersection is automatically stored in the variable X. You can keep it there, but I personally like to move it into another variable, say A.

From here, you can now just compute the integral.

Calculus and the numeric solver
One of the remarkable things to note is that calculus expressions can be used in the numeric equation solver as well. When doing the methods paper, I found myself faced with the problem
\[ \int_0^h \ln(5x+e) - 1\,dx = \frac{89.59155183}{2}.\]
In hindsight, this was perhaps not the intended method of solving the question. But it was something that could be dealt with on the numeric equation solver - check it out! (It gave me the value \(h\approx 17.232\).)

Numeric second derivative
The trick here is to compute the derivative OF a derivative. It follows the same trick with actually graphing the derivative function.

The following example computes \( \frac{d^2}{dx^2} \left( \frac{x+e^x}{1+\sin (x)} \right) \) at \(x=0\), which is something I imagine you would not like to do by hand. It found very quickly for me that the value is \(-1\)! Note that it ran into floating point errors here, once again.

The binomial distribution
Note that I am assuming you have not turned off stat wizard in the mode options. If you did, please turn it back on!

Functions related to probability distributions can be found in the distr menu. This is accessed through 2nd vars. Note that the vars key is located under the four direction symbols. You need to scroll further down to find the functions that are related to the binomial distribution.

binompdf( is used to determine the value of \(P(X=x)\), where \(X\) is a random variable following a binomial distribution. The parameters you must input are:
- trials: The \(n\) parameter in the binomial distribution, corresponding to the number of times the Bernoulli trial is repeated.
- p: The \(p\) parameter in the binomial distribution, corresponding to the probability of a single success in the Bernoulli trial.
- x value: The value of \(x\) required in \(P(X=x)\).

After you plug in these values, scroll down and click paste. The command that will compute \(P(X=x)\) is automatically filled in for you, so you only need to press enter to get your probability.

The following example computes \(P(X=14)\), where \(X\sim Bin(20, 0.7)\).

binomcdf( is used to determine the value of \(P(X\leq x)\) instead. The parameters are the same, but this example now computes \(P(X\leq 14)\) for me. (Recall that for the CDF, the \(\leq\) symbol is used. Not the \(<\) symbol, i.e. not the strict inequality!)

Note that to compute \(P(a\leq X\leq b)\), you must compute \(P(X\leq b) - P(X< a)\), i.e. \(P(X\leq b) - P(X\leq a-1)\). That is, you must take the difference of two binomcdf( expressions separately.

invBinom( is a function that essentially works with upper quantiles. Or at least, upper quantiles in the discrete random variable setting. Here, suppose that we're given a value of \(\alpha\). We wish to find the smallest value of \(x\), such that \( P(X\leq x) \geq \alpha\).
The parameters you must input are:
- area: The value of \(k\) in the inequality \(P(X\leq x) \geq \alpha \).
- trials: The \(n\) parameter in the binomial distribution as above.
- p: The \(p\) parameter in the binomial distribution as above.

For example, suppose \(X\sim Bin(20, 0.7)\) and I wish to find the largest value of \(x\) such that \(P(X\leq x) \geq 0.3\).

The calculator tells me \(x=13\). Therefore, I expect \(P(X\leq 12)\) to be less than 0.3, and \(P(X\leq 13)\) to be greater than 0.3. Indeed, I can check this is true!

(Note: This is actually working with lower quantiles in the discrete random variable setting. I would hope that you don't need to do this for the exam! I just include it for sake of completeness.)

The normal distribution
Note that I am assuming you have not turned off stat wizard in the mode options. If you did, please turn it back on!

The functions related to the normal distribution should be the first that you see in the distr menu. (Accessed via 2nd vars.)

I personally believe it should be unlikely that you require normalpdf( very often. This literally computes the value of \(f(x)\) at a given value \(x\), where
\[ f(x) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}. \]
That is, it literally just subs a value into the probability density function. The probability density function itself is not that interesting for us. You can play around with it - the syntax is pretty much the same as for normalcdf(.

However normalcdf( will be of far greater use to you, since it computes \( \int_a^b f(x) \,dx\), i.e. \(P(a\leq X \leq b)\). Note that for the normal distribution, you must specify both the lower and upper bounds \(a\) and \(b\). The parameters you need to provide are:
- lower: The lower value \(a\). It's pre-set to \(-10^{99}\), i.e. a very large number. That way, if you just want to compute \(P(X\leq b)\), you can skip over writing a value for lower.
- upper: The upper value \(b\). Note that if you want to compute \(P(X\geq a)\), you can set \(b\) to be a very large number.
- μ: The \(\mu\) parameter in the normal distribution, corresponding to the mean of the random variable.
- σ: The \(\sigma\) parameter in the normal distribution, corresponding to the standard deviation of the random variable. (NOT the variance!)

(Note that μ and σ are default set to 0 and 1 respectively. This corresponds to the standard normal distribution. It lets you compute \(Z\) probabilities more quickly.)

For example, to compute \(P(Z\leq 0)\), where \(Z\sim N(0,1)\), I simply set my upper value to be 0 and obtain an answer that might appear a bit unsurprising. (Don't forget that floating point errors are a thing!)

For example, to compute \(P(0\leq X \leq 15)\), where \(X\sim N(5, 16)\), I note that the standard deviation is 4. This gives:

invNorm( is used to help find quantiles of a normal distribution. Recall that:
- The upper \(\alpha\)-quantile of a continuous random variable \(X\) is the value of \(t_\alpha\), such that \(P(X > t_\alpha) = \alpha \).
- The lower \(\alpha\)-quantile of a continuous random variable \(X\) is the value of \(t_\alpha\), such that \(P(X \leq t_\alpha) = \alpha\).
(Recall that the strictness of the inequality does not matter for continuous distributions.)

The parameters you need to provide are:
- area: The probability \(\alpha\) as required above.
- μ: The \(\mu\) parameter in the normal distribution as above.
- σ: The \(\sigma\) parameter in the normal distribution as above.
- Tail: Technically speaking asks for which tail of the normal distribution we're considering. For our purposes, ignore CENTER and use::
  - LEFT corresponds to a lower quantile.
  - RIGHT corresponds to an upper quantile

For example, suppose I wish to find the value \(z_{0.05}\), such that \(P(Z > z_{0.05}) = 0.05\), where \(Z\sim N(0,1)\). The answer I get is the anticipated value of approximately \(1.645\).

For example, suppose I wish to find the value of \(x\), such that \(P(X< 35) = 0.75\), where \(Z\sim N(29, 144)\). The standard deviation is 12 and here I need a lower quantile, so I consider:

The specialist maths supplement
To be added in the future