course Mth 271 6/5/2010 11:32 p.m. 005. Calculus ********************************************* Question: `q001. The graph of a certain function is a smooth curve passing through the points (3, 5), (7, 17) and (10, 29). Between which two points do you think the graph is steeper, on the average? Why do we say 'on the average'? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The graph is steeper between points (7, 17) and (10, 29) because the slope is greater. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q002. 2. Answer without using a calculator: As x takes the values 2.1, 2.01, 2.001 and 2.0001, what values are taken by the expression 1 / (x - 2)? 1. As the process continues, with x getting closer and closer to 2, what happens to the values of 1 / (x-2)? 2. Will the value ever exceed a billion? Will it ever exceed one trillion billions? 3. Will it ever exceed the number of particles in the known universe? 4. Is there any number it will never exceed? 5. What does the graph of y = 1 / (x-2) look like in the vicinity of x = 2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x = 2.1 = 1 / (2.1 - 2) = 1/.1 = 10 x = 2.01 = 1 / (2.01 - 2) = 1/.01 = 100 x = 2.001 = 1 / (2.001 - 2) = 1/.001 = 1,000 x = 2.0001 = 1 / (2.0001 - 2) = 1/.0001 = 10,000 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q003. One straight line segment connects the points (3,5) and (7,9) while another connects the points (10,2) and (50,4). From each of the four points a line segment is drawn directly down to the x axis, forming two trapezoids. Which trapezoid has the greater area? Try to justify your answer with something more precise than, for example, 'from a sketch I can see that this one is much bigger so it must have the greater area'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The second trapizod has the greater area. The first trapizod is only 4 points wide versus 40 points wide on the second. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q004. If f(x) = x^2 (meaning 'x raised to the power 2') then which is steeper, the line segment connecting the x = 2 and x = 5 points on the graph of f(x), or the line segment connecting the x = -1 and x = 7 points on the same graph? Explain the basisof your reasoning. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The line segment connecting x = 2 and x = 5 points is steeper because of a greater slope. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q005. Suppose that every week of the current millenium you go to the jewler and obtain a certain number of grams of pure gold, which you then place in an old sock and bury in your backyard. Assume that buried gold lasts a long, long time ( this is so), that the the gold remains undisturbed (maybe, maybe not so), that no other source adds gold to your backyard (probably so), and that there was no gold in your yard before.. 1. If you construct a graph of y = the number of grams of gold in your backyard vs. t = the number of weeks since Jan. 1, 2000, with the y axis pointing up and the t axis pointing to the right, will the points on your graph lie on a level straight line, a rising straight line, a falling straight line, a line which rises faster and faster, a line which rises but more and more slowly, a line which falls faster and faster, or a line which falls but more and more slowly? 2. Answer the same question assuming that every week you bury 1 more gram than you did the previous week. {}3. Answer the same question assuming that every week you bury half the amount you did the previous week. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The graph would make a straight line that would be increasing at an increasing rate. The graph would be increasing at a decreasing rate if you kept buying smaller amounts of gold every week. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q006. Suppose that every week you go to the jewler and obtain a certain number of grams of pure gold, which you then place in an old sock and bury in your backyard. Assume that buried gold lasts a long, long time, that the the gold remains undisturbed, and that no other source adds gold to your backyard. 1. If you graph the rate at which gold is accumulating from week to week vs. tne number of weeks since Jan 1, 2000, will the points on your graph lie on a level straight line, a rising straight line, a falling straight line, a line which rises faster and faster, a line which rises but more and more slowly, a line which falls faster and faster, or a line which falls but more and more slowly? 2. Answer the same question assuming that every week you bury 1 more gram than you did the previous week. 3. Answer the same question assuming that every week you bury half the amount you did the previous week. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We are collecting the same amount of gold every time, thus the graph should be a straight line that does not change. If we continuly buy more gold than what we did the prior week, the straight line would be increasing at an increasing rate. The line would be increasing at a descreasing rate. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3 ``q007. If the depth of water in a container is given, in centimeters, by 100 - 2 t + .01 t^2, where t is clock time in seconds, then what are the depths at clock times t = 30, t = 40 and t = 60? On the average is depth changing more rapidly during the first time interval or the second? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 100 - 2t +.01 t^2 --------- t = 30 = 100 - 2t +.01 * t^2 = 100 - 2(30) +.01 * 30^2 = 100 - 60 + .01 * 900 = 100 - 60 + 9 = 49 --------- t = 40 = 100 - 2t +.01 * t^2 = 100 - 2(40) +.01 * 30^2 = 100 - 80 + .01 * 1600 = 100 - 80 + 16 = 36 --------- t = 60 = 100 - 2t +.01 * t^2 = 100 - 2(60) +.01 * 30^2 = 100 - 120 + .01 * 3600 = 100 - 120 + 36 = 16 --------- The depth is changing more during the first time period. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q008. If the rate at which water descends in a container is given, in cm/s, by 10 - .1 t, where t is clock time in seconds, then at what rate is water descending when t = 10, and at what rate is it descending when t = 20? How much would you therefore expect the water level to change during this 10-second interval? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: rate = 10 - .1t --------- t = 10 = 10 - .1(10) = 10 - 1 = 9 --------- t = 20 = 10 - .1(20) = 10 - 2 = 8 --------- The water level should change between 80 cm to 90 cm. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ------------------------------------------------ Self-critique rating #$&* 3"