Texas Instruments TI-85 Getting Started

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 ▲ to increase display contrast and by ▼ to 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 . denotes normal notation for decimals; is for scientific notation; and for engineering notation. With selected decimals are printed with twelve digits. Choosing an integer instead of causes that many digits be shown after decimal points. (Use the second 0 for ten digits and the second 1 for eleven digits.) is for radians and for degrees. is selected to generate graphs y = f(x) of functions. is for polar coordinates, is used with parametric equations, and 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 . 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 and selected, rectangular coordinates are used and the coordinates of the cursor are displayed with graphs. Points on graphs are connected if is chosen and not with . Use (sequential graphs) to have two or more graphs drawn one after the other, and (simultaneous graphs) to have them drawn at the same time. If 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 and labels would be displayed with . 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 1

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 STO▶ , 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 STO▶ is pressed. If you make an error in a command or calculation, the type of error is given and a menu appears. Select to go to the error to correct it or 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 √ 9 so 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 − 3 on your calculator. Solution (a) Working from left to right, you would first multiply the 3 and 4 to have 12 √ 36 − 10 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.

(b) Press 3 ( 4 ) ÷ 2nd √ ( 3 6 ) − 1 0 ÷ ( 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 .= π for 2π − π, which it obtains by subtracting π from 2π. It gives −19.7392088022 .= −2π2 for 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 A and B. Solution Enter 5 STO▶ A ENTER 2 STO▶ B 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 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.

Example 7 Attempt to evaluate √ 4π by entering √ 4π Solution The TI-85 calculator reads √ 4π as � (4 ∗ π) .= 3.54490770181 because it uses Rule 8 and multiplies the 4 and the π before taking the square root. Enter ( √ 4)π or √ 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 3 and 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 = 5 3√x at 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 STO▶ x−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 ◀ to 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 ◀ to 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 ◀ to 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.

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.O 2 + 8 4 − 6 − 35−1 4.O A + BCD with A = 7, B = 6, C = 5, and D = 4 (Store the values first.) 5.O 1 2 log10(7) 6.O 1.34 × 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 ◀ until the cursor is over the square root sign and press DEL to delete it. Move the cursor to the 9 and press 2nd π to replace the 9 with π. Put the cursor on the close parenthesis and press 2nd INS . Press ∧ 3 to 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−1 to display (–5 − 1.63e–2)−1. (A e n stands for A × 10n.) Press ENTER for the answer. Notice that (−) is for negation, − is for subtraction, and x−1 is 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.

4 3757 • To put the commands on one line for possible later editing, press 7 STO▶ A ALPHA 2nd : 6 STO▶ B ALPHA 2nd : 5 STO▶ C ALPHA 2nd : 4 STO▶ 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 D and calculate the answer. The colon (above the period key) separates two commands on one line. The ALPHA keys are needed because STO▶ locks the calculator in upper-case alpha mode. 5. 0.422549020007 • Use ( 1 ÷ 2 ) LOG ( 7 ) ENTER . 6. 1.3202247191 × 1013 • Use ( 1 . 3 4 EE 6 − 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 x and odd integers n, but do not always give the correct, real value of xm/n with negative x, odd n, and m an 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 n and negative x, the calculator interprets x ∧ (1/n) as the negative nth root of x.