Using the GDC

A graphic display calculator (GDC) is expected in Paper 2 and, for HL, Paper 3. The approved models draw graphs, find zeros, intersections, maxima and minima, evaluate derivatives at a point and definite integrals numerically, solve equations, analyse data, calculate probabilities from distributions and run a finance solver. They have no computer algebra system, so they cannot simplify, factorise, differentiate or integrate symbolically: every result is a number.

The GDC is a tool for the arithmetic, not a replacement for the method. Each GDC result in an answer must be supported by working that shows what was calculated and why. Menu names differ between models, so the descriptions here use generic names ("the zero function", "the normal cdf function"). Practise each one on your own GDC until it can be found without searching.

Setting up before the paper

A few settings decide whether every later answer is right.

• Angle mode. Set radians for the paper and switch to degrees only for a triangle question given in degrees. Calculus with trigonometric functions always uses radians.

• Number display. Use the normal floating-point display, not a fixed number of decimal places, so values are not rounded on screen before they are written down.

• Exam mode. Many models have an exam (or press-to-test) mode that disables stored programs and files. Use it as the rules for the session require, and practise with it beforehand, because some features may behave differently.

• Batteries. Charge or replace them before the day; a spare set is a sensible precaution.

Graphs and windows

The graph screen is the starting point for most GDC work: solving equations, finding turning points and areas, and checking the shape of a curve. Its main risk is the viewing window. A standard window shows $-10\le x\le 10$ and $-10\le y\le 10$, which hides most of the shape of many functions.

To choose a window, use the domain given in the question for the $x$-range, then a table of values or the GDC's automatic fitting option for the $y$-range. Then zoom out once to check that nothing lies outside it.

Zeros and intersections

Equations that cannot be solved algebraically are solved on the graph screen, in one of two ways:

1. Graph each side as a separate function and find the points of intersection.

2. Rearrange to the form $g(x)=0$, graph $y=g(x)$ and find the zeros.

Both give the same $x$-values. The first is easier to sketch in the answer; the second avoids errors when the two curves are almost tangent. For each zero or intersection, the GDC asks for a left bound and a right bound (or a starting guess), and finds the one solution inside them. Repeat for each solution.

Turning points

The maximum and minimum functions find local maxima and minima of a graph inside a chosen interval. They are useful for optimisation in context, for finding the range of a function on a restricted domain, and for checking a stationary point found by calculus.

Two cautions apply. First, the GDC finds a local maximum inside the bounds given; it does not check the end points of a domain, where the greatest value of a function may lie. Second, the results are approximate, so a value such as $x=0.99999987$ should be recognised as $x=1$ when an exact answer is expected.

Numerical derivatives and integrals

The derivative at a point

The numerical derivative function gives the value of $\frac{dy}{dx}$ at a specific value of $x$. It answers questions such as "find the gradient of the curve at $x=2$" or "find the rate of change when $t=3$".

For $f(x)=\ln (x^2+1)$, the GDC gives $f^{\mathrm{\prime }}(2)=0.8$. By hand, $f^{\mathrm{\prime }}(x)=\frac{2x}{x^2+1}$ and $f^{\mathrm{\prime }}(2)=\frac{4}{5}=0.8$, which agrees.

It cannot give $f^{\mathrm{\prime }}(x)$ as a formula. When a question asks for $f^{\mathrm{\prime }}(x)$, or for a stationary point in terms of a parameter, differentiation must be done by hand. The GDC can still check the result at one value of $x$.

Definite integrals

The numerical integral function evaluates ${\int }_a^{b}f(x)dx$ for numerical limits. It is the usual method on Paper 2 for areas, volumes of revolution (HL), distances travelled and probabilities from a continuous random variable.

Solving equations

Besides the graph screen, most GDCs have three solving tools.

• The numerical solver solves an equation in one unknown from a starting value. It returns one solution, the one nearest the starting value, so a graph is needed to find how many solutions there are and where to start.

• The polynomial root finder gives all the roots of a polynomial equation up to a certain degree, including complex roots on some models.

• The simultaneous equation solver solves systems of linear equations. For example, it solves

to give $x=1$, $y=2$, $z=3$. If the system has no solution or infinitely many, the GDC reports an error or a general solution, and the algebra explains which case applies (see 1.16).

Statistics

One-variable statistics

Enter the data in a list (with a frequency list if the data are grouped or repeated) and use the one-variable statistics function. It gives the mean, the standard deviation, the quartiles, the median and the sum.

For the data $4,7,7,9,12,15$, the GDC gives $\bar{x}=9$, median $8$, and two standard deviations: ${\sigma }_x=3.61$ and $s_x=3.95$. In IB maths, the standard deviation of a data set is $\sigma$, so use ${\sigma }_x=3.61$.

Regression

The linear regression function takes two lists, $x$ and $y$, and returns the equation of the regression line of $y$ on $x$ and Pearson's correlation coefficient $r$ (on some models $r$ must be switched on in the settings first). Check which list is which: the regression line of $x$ on $y$ needs the lists swapped (see 4.10).

Probability distributions

The distribution menu holds the functions for the binomial and normal distributions.

The binomial cdf only gives $P(X\le r)$, so other inequalities must be rewritten first. For a discrete variable, "fewer than 4" means $X\le 3$ and "at least 4" means $1-P(X\le 3)$.

For the inverse normal function, check whether the GDC wants the area to the left of $k$ or offers a choice of tail. For $P(X>k)=0.05$, either set the tail to the right with area $0.05$, or use area $0.95$ to the left.

The finance solver

The finance solver (time value of money) handles compound interest and depreciation. It has fields for the number of periods ($N$), the annual interest rate ($I\mathrm{\%}$), the present value ($PV$), the regular payment ($PMT$, which is $0$ in AA questions), the future value ($FV$), and the numbers of payments ($P\mathrm{/}Y$) and compounding periods ($C\mathrm{/}Y$) per year.

On most models $N$ is the total number of periods, not the number of years: with monthly periods ($P\mathrm{/}Y=12$), eight years is entered as $N=96$. Learn which convention your GDC uses.

For $5000 invested for 8 years at 3.5% per year compounded monthly, enter $N=96$, $I\mathrm{\%}=3.5$, $PV=-5000$, $PMT=0$, $P\mathrm{/}Y=12$, $C\mathrm{/}Y=12$ and solve for $FV$: the result is $6612.95. Solving instead for $N$ with $FV=10000$ gives $N=238.0$ months, so the money doubles after about $19.8$ years.

Communicating GDC work

Most of the marks lost with a GDC are lost through communication, not wrong buttons. An answer that shows only a number gives the examiner nothing to award if the number is wrong, and may not earn full marks even when it is right. A complete line of GDC working states three things:

1. What was set up: the equation, function, integral or distribution, in mathematical notation.

2. How the GDC was used: "from the graph", "using the GDC", or a labelled sketch for a graphical solution.

3. The result, to 3 significant figures or as the question asks, with any intermediate values to more figures.

Examples of good GDC working:

• "$P(X\ge 4)=1-P(X\le 3)$, where $X\sim B(12,0.3)$; $=0.507$."

• "${\displaystyle {\int }_0^5\mathrm{\mid }v(t)\mathrm{\mid }dt=11.2}$ m."

• "The graphs of $y=e^x$ and $y=4-x^2$ intersect at $x=-1.96$ and $x=1.06$" with a sketch of both curves and the two points marked.

Common errors