Texas Instruments TI-85 Bedienungsanleitung

Getting started

Texas Instruments TI-85 and TI-86 calculators

Overview: Your graphing calculator or computer is a powerful and ﬂexible tool, which you would

probably be able to use fairly well without reading any instructions. It is important, however, to learn

how to take advantage of some of its not-so-obvious features and how to avoid making errors using it.

Study these instructions and be sure you can work the tune-up exercises at the end.

Topics:

•Basic operations

•Priority of operations in calculations

•The dangers of using improper parentheses

•Exact and approximate decimal values of functions

Basic operations

Press the ON key to start the calculator. Press 2nd followed by the up cursor key Nto increase

display contrast and by Hto decrease it. Change the four AAA batteries as soon as the screen dims

when graphs are generated. Press 2nd MODE . The screen should show

Normal Sci Eng Func Pol Param DifEq

Float 012345678901 Dec Bin Oct Hex

Radian Degree RectV CylV SphereV

RectC PolarC dxDer1 dxNDer

The words printed in bold type here should be highlighted on the screen. If another item is highlighted or

you want to change a selection, use the cursor keys to move the ﬂashing box to the correct item and press

ENTER .<Normal>denotes normal notation for decimals; <Sci>is for scientiﬁc notation; and <Eng>

for engineering notation. With <Float>selected decimals are printed with twelve digits. Choosing an

integer instead of <Float>causes that many digits be shown after decimal points. (Use the second 0

for ten digits and the second 1 for eleven digits.) <Radian>is for radians and <Degree>for degrees.

<Func>is selected to generate graphs y=f(x) of functions. <Pol>is for polar coordinates, <Par>is

used with parametric equations, and <DifEq>is for diﬀerential equations. The other selections will be

explained as needed.

Press EXIT or 2nd QUIT to return to the home screen and then GRAPH for the ﬁrst row of the

graph menu. Press MORE to see the second row of the menu and then press F3 for <FORMT>. The

screen should read

RectGC PolarGC GridOﬀ GridOn

CoordOn CoordOﬀ AxesOn AxesOﬀ

DrawLine DrawDot LabelOﬀ LabelOn

SeqG SimulG

with the words in bold highlighted. With <RecGC>and <CoordOn>selected, rectangular coordinates

are used and the coordinates of the cursor are displayed with graphs. Points on graphs are connected

if <DrawLine>is chosen and not with <DrawDot>. Use <SeqG>(sequential graphs) to have two

or more graphs drawn one after the other, and <SimulG>(simultaneous graphs) to have them drawn

at the same time. If <GridOn>were selected, dots would be placed on the screen at the points whose

coordinates correspond to the tickmarks on the axes. The axes would not be shown with <AxesOﬀ>

and labels would be displayed with <LabelOn>.

The key 2nd activates the yellow commands above the keys. EXIT is used to return to a previous

screen and to remove menus. 2nd QUIT returns you to the home screen where calculations are made.

CLEAR with the cursor on a blank line of the home screen clears the screen. In other cases it clears the

line with the cursor or removes a menu.

The key ALPHA puts the calculator in upper-case alpha mode, activating the blue letters and

Getting started TI-85/86 calculators, 2

other symbols above the keys. Pressing ALPHA ALPHA locks the calculator in upper-case alpha mode

and then pressing ALPHA or ENTER takes it out of upper-case alpha-lock mode. 2nd ALPHA puts

it in lower-case alpha mode. Entering a number followed by STOI, one or more letters, and ENTER

assigns that number to the letter or letters. The number can then be recalled by entering the letter or

letters. The calculator is locked in alpha mode after STOIis pressed.

If you make an error in a command or calculation, the type of error is given and a menu appears.

Select <GOTO>to go to the error to correct it or <QUIT>to cancel the incorrect command.

In the home screen, 2nd ENTRY recalls the last expression that was evaluated so it can be edited,

if necessary, and used again. The ON key stops the generation of graphs, the running of programs, and

other operations. The ENTER key can be used to interrupt and resume the generation of graphs.

Refer to the owner’s manual for further information.

Priority of operations

The meaning of a formula involving functions, powers, sums, diﬀerences, products, and quotients depends

on how the formula is interpreted to determine the order in which the operations are performed. Texas

Instruments TI-86 calculators in most instances interpret formulas with the following rules, which are

those generally used in manual calculations.

Rule 1 Operations are performed from left to right, except as described in Rules 2 through 5 below.

Rule 2 Expressions inside parentheses are evaluated as soon as they are reached.

Rule 3 Addition and subtraction have the lowest priority. If an addition or subtraction is followed by

multiplication, division, a power, or a function, the addition or subtraction is postponed until another

addition or subtraction or the end of the expression is reached.

Rule 4 Multiplication and division have medium priority. If a multiplication or division is followed by

a power or a function, the multiplication or division is postponed until the power or function has been

evaluated.

Rule 5 The taking of powers and evaluation of functions have the highest priority and are performed

as soon as they are reached.

Example 1 (a) Calculating 5 + 2√9 involves addition, multiplication, and the taking of a square

root. In what order are these operations perfomed? (b) Find the value of 5+2√9 with

your calculator.

Solution (a) By Rule 5 above, ﬁnding the square root has the highest priority and is performed

ﬁrst, yielding 5 + 2√9 = 5 + 2(3). Multiplication has the next priority, by Rule 4, and

gives 5 + 2(3) = 5 + 6. The remaining addition gives 5 + 6 = 11.

(b) Press 5 + 2 2nd √9so the screen reads 5 + 2√9. Then press ENTER

for the answer 11. 

Example 2 (a) What steps would you use to evaluate 3(4)

√36 −10

8−3?(b) Find the value of

3(4)

√36 −10

8−3on your calculator.

Solution (a) Working from left to right, you would ﬁrst multiply the 3 and 4 to have 12

√36 −

8−3. Then you would evaluate the square root, yielding 12

6−10

8−3. Dividing 12 by

6 would give 2 −10

8−3. Next, you would perform the subtraction in the remaining

denominator to have 2 −10

5. Finally, you would divide 5 into 10 to obtain 2 −2 and

subtract for the answer 0.

Getting started TI-85/86 calculators, 3

(b) Press 3 ( 4 ) ÷2nd √( 3 6 ) −10÷( 8 −3 ) so

the screen reads 3(4)/√(36) −10/(8 −3). Then press ENTER for the value 0. 

The next example shows how using the negation symbol –for subtraction can lead to an error

message or give an incorrect result because a product is calculated instead of a diﬀerence.

Example 3 Evaluate the expressions 2π−π, 2π–π, 8 −5, and 8 –5, where −is the subtraction

symbol and –is the negation symbol. Explain the results.

Solution The calculator gives 3.14159265359 .

=πfor 2π−π, which it obtains by subtracting

πfrom 2π. It gives −19.7392088022 .

=−2π2for 2π–π, which it interprets as 2π

multiplied by −π. The expression 8 −5 equals 3, and you get −40 with 8 –5, which the

calculator either interprets as the product of 8 and −5.

The need to use ∗

Because TI-85 and TI-86 calculators allow words to be used for variables, multiplication signs (∗) must

be used between letters that represent numbers to be multiplied.

Example 4 Evaluate AB with A= 5 and B= 2 by ﬁrst storing the values of Aand B.

Solution Enter 5 STOIA ENTER 2 STOIB ENTER to store the values. Then enter

ALPHA A ∗ALPHA B ENTER for the answer A×B= 10. (Notice that using

ALPHA A ALPHA B to write AB and then ENTER yields an error message since

the variable AB has not been deﬁned.) 

The dangers of using improper parentheses

TI-85 and TI-86 calculators interpret certain expressions in unexpected ways because they use the

following modiﬁcation of Rules 3 through 5.

Rule 6 The taking of powers has priority over the evaluation of functions that appear before their

variables, such as the trigonometric functions, logarithms, ex,√, and negation. Also, the parentheses in

expressions such as sin(2) and e∧(2) are ignored.

Example 5 Evaluate sin3(2) = (sin(2))3.

Solution The seemingly logical expression sin(2)∧3 will not work. By Rule 6, the parentheses are

ignored, leaving sin 2 ∧3. Then the taking of the cube has priority over the evaluation

of the sine function, and the calculator gives 0.989358246623 .

= sin(23) = sin(8).†

For the correct answer, use an extra pair of parentheses by entering (sin(2)) ∧3.

This gives the correct value 0.751826944669. 

The TI-85 also uses the following two additional modiﬁcations of Rules 1 through 5

Rule 7 Multiplication by juxtaposition has priority over division and multiplication represented by ∗.

Example 6 Attempt to evaluate 1

5(10) = 2 by entering 1/5(10).

Solution The TI-85 evaluates 1/5(10) as 1/(50) = 0.02 because it uses Rule 7 and multiplies the

10 and the 5 before performing the division. Enter 1/5∗10 or (1/5)(10) instead.

The TI-86 gives 1

5(10) = 2, as expected, because it does not use Rule 7. 

Rule 8 Multiplication represented by juxtaposition, where the second term is a number or a variable,

has priority over the evaluation of functions that appear before their arguments.

†If you obtained 0.13917310096 here, then your calculator is using degrees instead of radians. Press 2nd MODE , put the

cursor on <Radian>and press ENTER to select radian mode. Press 2nd QUIT 2nd ENTRY to return to the home screen

and recall the last typed line, and then ENTER for the correct answer.

Getting started TI-85/86 calculators, 4

Example 7 Attempt to evaluate √4πby entering √4π

Solution The TI-85 calculator reads √4πas p(4 ∗π).

= 3.54490770181 because it uses Rule 8

and multiplies the 4 and the πbefore taking the square root. Enter (√4)πor √4∗π

instead to obtain the correct value 6.28318530718.

The TI-86 yields (√4)π= 2π.

= 6.28318530718, as expected, because it does not use

Rule 8. 

Example 8 Atempt to evaluate sin(5)(10) by entering this expression in the calculator.

Solution The TI-85 gives the wrong value sin(50) .

=−0.262374853704 because it uses Rule 8

and does the multiplication before evaluating the sine. Use sin 5 ∗(10) or (sin 5)(10)

instead.

The TI-86 yields the correct answer 10 sin(5) .

=−9.58924274663 because it does not

use Rule 8. 

Exact and approximate decimal values of functions

Since some but not all numbers can be represented exactly as ﬁnite decimals, it is important to distinguish

exact expressions, such as 1

3and π, from decimal approximations, such as 0.33333 and 3.14159. You also

need to recognize when coordinates obtained from graphs generated by calculators and computers are

approximations.

Example 4 Use your calculator to complete the table below of ten-digit values of 5x1/3=

√xat x=−27,−30,4,6,8,and 10. The value 5(−27)1/3= 5(−3) = −15 is

exact, but −15.53616253 is only a decimal approximation of 5(−30)1/3, which cannot

be represented by a ﬁnite decimal. Its value to 20 decimal places, for example, is

−15.53616252976929433439.Which y-values in the completed table in addition to −15

do you recognize as exact?

x y = 5x1/3.

=x y = 5x1/3.

−27 −15 −30 −15.5361625298

4 10

6 8

Solution You can do these calculations more eﬃciently by storing the formula for the function.

Press GRAPH F1 to access the y(x) = menu and CLEAR to erase any previous

formula for y1. Press 5 x−V AR ∧( 1 ÷3 ) to have y1 = 5x∧(1/3).

To ﬁnd the value of the function at x=−27, press 2nd QUIT to return to

the home screen and press (–) 2 7 STOIx−V AR 2nd : 2nd ALPHA Y

ALPHA ALPHA 1 so the screen reads −27 →x:y1. The colon (above the period

key) separates the two commands on the one line. Then press ENTER for the value

−15 of y1 at x=−27.

Press 2nd ENTRY to display the last line again, use Jto move the cursor to

the 2 and press 3 0 to have −30 →x:y1. Press ENTER for the approximate decimal

value −15.5361625298 of y1 at x=−30.

Press 2nd ENTRY to display the last line again, use Jto move the cursor

to the minus sign and press 4 DEL DEL to have 4 →x:y1. Press ENTER for the

approximate decimal value 7.93700525984 of y1 at x= 4.

Press 2nd ENTRY to display the last line again, use Jto move the cursor to

the 4 and press 1 2nd INS 0 to have 10 →x:y1. Press ENTER for the approximate

decimal value 10.7721734502 of y1 atx= 10.

Getting started TI-85/86 calculators, 5

Repeat this process for the other two values in the table below. Only the values

at −27 and x= 8 are exact because only −27 and 8 of the

x-values are perfect cubes.

x y = 5x1/3.

=x y = 5x1/3.

−27 −15 −30 −15.5361625298

4 7.93700525984 10 10.7721734502

6 9.08560296416 8 10

Exercises

Use your calculator or computer to ﬁnd the approximate decimal values of the expressions in Exercises T1

through T8. Do not simplify the expressions before entering them and be sure your machine is in radian

mode for the trigonometric function in Exercise T1. In some cases extra parentheses are needed to express

numerators, denominators, and exponents.

1.O(a) √6 cos(9/7) (b) 6 cos(π/73) (Edit the expression from part (a).)

2.O(−5−1.63 ×10−2)−1

3.O2 + 8

4−6−35−1

4.OA+BCDwith A= 7, B= 6, C= 5,and D= 4 (Store the values ﬁrst.)

5.O1

2log10 (7)

6.O1.34 ×106−4×105

7.12 ×10−8

7.O√4 + 78−10

8.O(−32)4/5

Outlines of solutions

1a. 0.688885143177 •(If your result is 2.44887304686, your calculator is not in radian mode.) •

Press 2nd √6 cos ( 9 ÷7 ) to have the screen read √6 cos (9/7). Then press

ENTER for the answer.

1b. 5.999748331 •If your last operation was the calculation of √6 cos (9/7), press 2nd ENTRY

to put it back on the screen. If you performed other calculations, type √6 cos (9/7) again. Press

Juntil the cursor is over the square root sign and press DEL to delete it. Move the cursor

to the 9 and press 2nd πto replace the 9 with π. Put the cursor on the close parenthesis and

press 2nd INS . Press ∧3to insert ∧3 before the close parenthesis, so the screen reads

6 cos(π/7∧3), and press ENTER for the answer.

2. −0.199350118613 •Press ( (−) 5 −1.6 3 EE (−) 2 ) 2nd x−1to display

(–5−1.63e–2)−1. (Aenstands for A×10n.) Press ENTER for the answer. Notice that (−)is

for negation, −is for subtraction, and x−1is for taking reciprocals.

3. −86 •Use ( 2 + 8 ) ÷( 4 −6 ) −3∧( 5 −1 ) to display

(2 + 8)/(4 −6) −3∧(5 −1). Press ENTER for the answer.

Getting started TI-85/86 calculators, 6

43757 •To put the commands on one line for possible later editing, press 7 STOIA

ALPHA 2nd : 6 STOIB ALPHA 2nd : 5 STOIC ALPHA 2nd : 4 STOI

D ALPHA 2nd : ALPHA A + ALPHA B ×ALPHA C ∧ALPHA D so the screen

reads 7 →A: 6 →B: 5 →C: 4 →D:A+B∗C∧D. Press ENTER to store the values of A, B, C

and Dand calculate the answer. The colon (above the period key) separates two commands on

one line. The ALPHA keys are needed because STOIlocks the calculator in upper-case alpha

mode.

5. 0.422549020007 •Use ( 1 ÷2 ) LOG ( 7 ) ENTER .

6. 1.3202247191 ×1013 •Use ( 1 .34EE6−4 EE 5 ) ÷( 7 .1 2 EE

(–) 8 ) ENTER .

7. 2.00509554966 •Use 2nd √( 4 + 7 ∧( 8 −1 0 ) ) ENTER .

8. 16 •TI-85 and TI-86 calculators evaluate x1/n for negative xand odd integers n, but do

not always give the correct, real value of xm/n with negative x, odd n, and man integer

>1. It is generally best to write xm/n as (xm)1/n for odd n. The keys (-) 3 2 ∧(

4÷5 ) ENTER give the wrong result(−16) since the negation is perfomed after the

power. The symbols ( (-) 3 2 ) ∧( 4 ÷5 ) ENTER probably yield, on a TI-

85, (−12.94427191,9.40456403668) representing −12.94427191 + 9.40456403668 i, which is the

approximate decimal value of one of the complex four-ﬁfth roots of −32, rather than the real

four-ﬁfth root 16. These symbols, however, give the correct result on a TI-86. The expressions

( (-) 3 2 ) ∧4∧( 1 ÷5 ) ENTER and ( (-) 3 2 ) ∧( 1 ÷5 )

∧4 ENTER give the correct result, since for any positive odd integer nand negative x, the

calculator interprets x∧(1/n) as the negative nth root of x.

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