Trace and the CALCULATE tools

10.1Trace a graph#

Press trace to put a cursor on the graph at the centre of the window. The equation is shown at the top and the coordinates at the bottom. ◀ and ▶ move one pixel step; ▲ and ▼ jump to another function. If you move past the edge the window scrolls with you.

Example 10.1

Trace a curve

1. Graph Y1 = X²−4 and press trace.

   y=X,T,θ,nx²−4▼trace

Example 10.2

Take one step

1. Graph Y1 = X, press trace, then right.

   y=X,T,θ,n▼trace▶

Example 10.3

Steps of exactly 0.1 in ZDecimal

1. Graph Y1 = X², zoom to ZDecimal, trace and press right twice.

   y=X,T,θ,nx²▼zoom4trace▶▶

Example 10.4

Jump to a value of X

1. Press trace, then simply type 1.5 and press enter.

   y=X,T,θ,nx²−4▼trace1.5enter

Example 10.5

Switch to the second function

1. Graph X² and X+2, trace to X = 1 and press down.

   y=X,T,θ,nx²▼X,T,θ,n+2▼trace1enter▼

While tracing, the value of the cursor is stored in X and Y, so you can use them on the home screen afterwards. Press enter while tracing to centre the window on the cursor.

Example 10.6

Use the traced Y afterwards

1. Trace Y1 = X² at X = 3.

   y=X,T,θ,nx²▼trace3enter

2. Go home and type Y.

   2ndQUITmodealphaY1enter

10.2The CALCULATE menu#

Press 2ndCALCtrace to open CALCULATE. Each tool works on the function that is shown at the top; use ▲ and ▼ to choose another one.

10.3Find the value of a function#

Example 10.7

Value at x = 3

$$f(3)=3^{2}-4=5$$

1. Graph Y1 = X²−4, press 2nd trace, 1, type 3 and press enter.

   y=X,T,θ,nx²−4▼2ndCALCtrace13enter

10.4Find a zero (x-intercept)#

A zero is where $f(x)=0$. The calculator asks for a Left Bound, a Right Bound and a Guess: the root must lie between the bounds. Move the cursor with the arrows, or type an x value, and press enter at each prompt.

Example 10.8

The negative zero of x² − 4

$$x^{2}-4=0\Rightarrow x=-2$$

1. Graph Y1 = X²−4, press 2nd trace and 2. Type −5 for the left bound and press enter.

   y=X,T,θ,nx²−4▼2ndCALCtrace2(−)5enter

   

2. Type 0 for the right bound, enter, then enter for the guess.

   0enterenter

Example 10.9

The positive zero

1. Use bounds 0 and 5.

   y=X,T,θ,nx²−4▼2ndCALCtrace20enter5enterenter

Example 10.10

A zero of sin(x)

1. Graph sin(X) in radians and use bounds 3 and 4.

   y=sinX,T,θ,n)▼2ndCALCtrace23enter4enterenter

Example 10.11

No zero in the interval

1. Graph X²+1, which never crosses the axis, and ask for a zero between 1 and 2.

   y=X,T,θ,nx²+1▼2ndCALCtrace21enter2enterenter

10.5Find a minimum or maximum#

Minimum and maximum work like zero: give a left bound, a right bound and a guess. They find the lowest or highest point of the graph between the bounds.

Example 10.12

Minimum of x² − 2x

$$f(x)=x^2-2x,\ f_{min}=f(1)=-1$$

1. Graph Y1 = X²−2X, press 2nd trace and 3. Use bounds −5 and 5.

   y=X,T,θ,nx²−2X,T,θ,n▼2ndCALCtrace3(−)5enter5enterenter

Example 10.13

Maximum of −x² + 2x

$$f(x)=-x^2+2x,\ f_{max}=f(1)=1$$

1. Graph Y1 = −X²+2X, press 2nd trace and 4. Use bounds −5 and 5.

   y=(−)X,T,θ,nx²+2X,T,θ,n▼2ndCALCtrace4(−)5enter5enterenter

10.6Find where two graphs intersect#

Graph both functions. Intersect first shows the first curve, then the second curve; press enter on each, then move to a guess near the intersection and press enter.

Example 10.14

Where does y = x² meet y = x + 2?

$$x^2=x+2\Rightarrow x=2\ \text{or}\ x=-1$$

1. Graph X² and X+2, press 2nd trace, 5, enter twice, then type 3 for the guess and press enter.

   y=X,T,θ,nx²▼X,T,θ,n+2▼2ndCALCtrace5enterenter3enter

Example 10.15

The other intersection

1. Give −2 as the guess instead.

   y=X,T,θ,nx²▼X,T,θ,n+2▼2ndCALCtrace5enterenter(−)2enter

10.7Find the slope at a point (dy/dx)#

Choose 6:dy/dx, type the x value and press enter. Because the slope is estimated numerically the result can differ from the exact answer in the last digits.

Example 10.16

Slope of x² at x = 1

$$\frac{d}{dx}x^{2}\Big|_{x=1}=2$$

1. Graph X², press 2nd trace, 6, type 1 and press enter.

   y=X,T,θ,nx²▼2ndCALCtrace61enter

10.8Find the area under a curve#

Example 10.17

Area under x² from 0 to 1

$$\int_{0}^{1}x^{2}\,dx=\frac13$$

1. Graph X², press 2nd trace and 7. Type 0 for the lower limit and press enter.

   y=X,T,θ,nx²▼2ndCALCtrace70enter

   

2. Type 1 for the upper limit and press enter. The area is shaded.

   1enter

The value is also stored in Ans, so you can use it straight away on the home screen.

Example 10.18

Use the integral as Ans

1. Integrate 2X from 0 to 3.

   y=2X,T,θ,n▼2ndCALCtrace70enter3enter

2. Go home and type Ans.

   2ndQUITmode2ndANS(−)enter