Well use the form f t pa t and determine the unknown

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We’ll use the form f ( t ) = Pa t and determine the unknown constants P and a . To find P , evaluate f ( 0 ) in two ways. First We were told that f ( 0 ) = 3. Second Evaluate the general formula at t=0. f ( 0 ) = Pa 0 = P · 1 = P . Combining these we find that P = 3. Then, setting P = 3, we have: f ( t ) = 3 a t . To find the value of a , evaluate f ( 4 ) in two ways. First f ( 4 ) = 109. Second f ( 4 ) = 3 a 4 . Combining, we have 109 = 3 a 4 and we can solve for a : a 4 = 109 3 , and therefore a = 109 3 1 / 4 2.455140276. (This decimal value is the calculator’s best approximation to the correct value.) Now let’s see how rounding too soon affects calculations. . . 3
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Problem 2. In this problem you will calculate f ( 9.5 ) by first using a rounded-off value for a and then by using a more precise value of a . a. (i) Using the rounded-off value a = 2.5 , write the equation for f ( t ) . (ii) Use this equation (with the rounded value of a ) to find f ( 9.5 ) . Round the result to the nearest integer. b. (i) Compute ( 109 3 ) 1 / 4 and store that value in A . [See notes on next page.] Similarly store 3 into P . (ii) Compute f ( 9.5 ) = Pa 9.5 by typing: P × A x y x 9.5 Round the result to the nearest integer. c. Subtract your answer to part (ii) of (a) from your answer to part (ii) of (b). This number is the error introduced by rounding a . Notice that the value that you found in 2(c) is not zero! Lesson: Before calculating f ( 9.5 ) , you first found P and a . Errors in the calculations are magnified when larger exponents are involved. Small errors in a and P can lead to large errors in the final answer. It makes sense to avoid rounding values that you use in another calculation. From now on, If you calculate a value, don’t retype that value in your calculator later. Use the calculator memory! Round off only after the final value is computed! 4
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3 Using Calculator Memories These instructions are designed for TI-83 and TI-84 calculators. If you have a different calcu- lator, look in the calculator’s instruction book. Suppose you finish a calculation. The “current value” is the quantity just computed, appearing on the bottom line of the calculator display. The symbol ANS appears near the bottom, right corner of the keyboard and is accessed by pressing the two keys 2nd and (-) , one after the other. We will write A NS instead of those two keystrokes. Type 3.45678 E NTER to get that number as the current value. To compute e 3.45678 we can type e x A NS ) E NTER . The displayed answer is 31.71469062. Typing the same keystrokes again won’t yield the same result, because A NS now stands for 31.71469062.
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