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# Applied 20S December 18, 2008

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Proportional relations and arithmetic sequences.

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### Applied 20S December 18, 2008

1. 1. What's Wrong With This Picture? SMALL is the new big by ﬂickr user assbach
2. 2. INTERPOLATION & EXTRAPOLATION HOMEWORK A student in electronics measured the current through a circuit with a constant resistance while the voltage was increased. The results are shown below, with the voltage measured in volts and the current in milliamps. a) Make a scatterplot of the data. Set the window to show the origin (0, 0) and the x-and y-axes. b) What is the equation for a line of best ﬁt? (Use three decimal places) c) Rewrite the equation, using the voltage as, V, and the current as, I. d) What is the slope of the line? INTE RPO LAT ION e) What would be the current if the voltage were 8.000 volts? EXTR APOL f) What would be the current if the voltage were 22.736 volts? ATIO N
3. 3. INTERPOLATION & EXTRAPOLATION a) Make a scatterplot of the data. Set the window to show the origin (0, 0) and the x-and y-axes. b) What is the equation for a line of best ﬁt? (Use three decimal places) c) Rewrite the equation, using the voltage as, V, and the current as, I. d) What is the slope of the line?
4. 4. INTERPOLATION & EXTRAPOLATION INTE RPO LAT e) What would be the current if the voltage were 8.000 volts? ION EXTR APOL ATIO f) What would be the current if the voltage were 22.736 volts? N 22.736 22.736
5. 5. In an experiment, the following masses were attached to a spring. As they were attached, the following elongations were recorded: HOMEWORK 1. What is the independent variable? 2. Use your graphing calculator to plot the elongation against mass. 3. Use your calculator to ﬁnd a regression line (line of best ﬁt) for the data. What is the y-intercept? 4. Select two points in the table and use them to calculate the slope of the line. 5. Is it the same in the equation of the line? 6. Do you think that this model for the elongation of the spring is always accurate? What do you think would happen if the mass were increases to 15 kg?
6. 6. Jimmy has observed that the distance to a thunderstorm can be estimated by counting the number of seconds between a ﬂash of lightning and the sound of the thunder. With further investigation, he obtains the following information: a. Complete the pattern shown in the chart up to 21 seconds. b. Graph the information using t as the independent variable and d as the dependent variable. c. What is the equation for this direct proportion? What is the constant of proportionality? This can be calculated in the same way as the slope of a line. d. Estimate the distance if the time is 10 seconds, 20 seconds, 30 seconds.
7. 7. Find the next three terms in each sequence of numbers ... 4, 7, 10, 13, , , 3, 6, 12, 24, , , 32, 16, 8, 4, , , 1, 1, 2, 3, 5, 8,13, , ,
8. 8. 4, 7, 10, 13, , ,
9. 9. Some Definitions Sequence: An ordered list of numbers that follow a certain pattern (or rule). Arithmetic Sequence:(i) Recursive Definition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given first term. (ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by a linear equation. Common Difference (d):(i) The number that is repeatedly added to successive terms in an arithmetic sequence. (ii) From the implicit definition, d is the slope of the linear equation.
10. 10. To Find The Common Difference d is the common difference d = tn - t(n - 1) tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence To Find the nth Term In an Arithmetic Sequence tn is the nth term tn = a + (n - 1)d a is the first term n is the quot;rankquot; of the nth term in the sequence d is the common difference Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ... Solution: a = 11 t51 = 11 + (51 - 1)(-6) Implicitly d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
11. 11. Which of the following sequences are arithmetic sequences? a) 1, 2, 6, 24, 120,… HOMEWORK b) 3, 9, 15, … c) 2, 4, 8, 16, 32,… d) 1, 2, 3, 5, 8, 13, … e) -4, -1, 2, 5, 8,…
12. 12. What is the pattern in the sequence 2, 8, 14, 20, 26…? Suggest an equation that could be used to generate such a list. HOMEWORK
13. 13. a) Why do the numbers 5, 8, 11, 14, 17… form an arithmetic sequence? HOMEWORK b) What is the deﬁning equation that produced them? c) What is the 27th term of this sequence?