#$&* course Mth 173 9/30 16:30
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Given Solution: This derivative will tell us the rate at which the volume changes with respect to the diameter of the pile. On a graph of the y = k x^3 curve the slope of the tangent line is equal to the derivative. Through the given point we can sketch a line with the calculated slope; this will be the tangent line. Knowing the slope and the change in x we easily find the corresponding rise of the tangent line, which is the approximate change in the y = k x^3 function. In short you use y' = 3 k x^2 to calculate the slope, which you combine with the change `dx in x to get a good estimate of the change `dy in y. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q Query class notes #08What equation do we get from the statement 'the rate of temperature change is proportional to the difference between the temperature and the 20 degree room temperature'? What sort of graph do we get from this equation and why? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The statement “the rate of temperature change is proportional to the difference between the temperature and the 20 degree room temperature” states that, if the temperature is T, and dT / dt is the rate of change of temperature, and T - 20 is the difference between the temperature and the 20 degree room temperature. Thus we have dT / dt = k * ( T - 20 ) where k is the proportionality constant. The function obtained above is the function of the rate of change of temperature with time. If a graph of rate vs time is required it would be a straight line. But here we need the graph of temperature vs. time. Integrating the above function would give us the function of temperature vs. time. On Integrating the above function we get ln( T - 20 ) = kt + c , where c in the integration constant Rearranging the above equation we get T - 20 = e^( kt + c ) T = 20 + e ^ ( kt + c ) Since e ^ ( kt + c ) is always positive, the function will always have positive values. Most generally for temperature functions k is negative and depending upon the value of c the quantity kt + c changes. Since k is negative as t increases the quantity kt + c decrease and the value of e ^ ( kt + c ) decreases approaching 0. Thus the over function T = 20 + e ^ ( kt + c ) will be an exponential function decreasing and approaching the temperature of 20 with T = 20 being the asymptote confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: STUDENT SOLUTION AND INSTRUCTOR COMMENT: Would it be y = x-20 degrees., with y being the rate of temperature change and x being the temperature?You get a graph with a straight line and a slope of -20? INSTRUCTOR COMMENT: Not a bad attempt. However, you wouldn't use y to represent a rate, but rather dy /dt or y'. An in this example I'm going to use T for temperature, t for clock time. Read further. We need a graph of temperature vs. clock time, not rate of change of temperature vs. clock time. The difference between temperature and room temperature is indeed (T - 20). The rate of change of the temperature would be dT / dt. To say that these to our proportional is to say that dT / dt = k ( T - 20). To solve the situation we would need the proportionality constant k, just as with sandpiles and other examples you have encountered. Thus the relationship is dT / dt = k ( T - 20). Since dT / dt is the rate of change of T with respect to t, it tells you the slope of the graph of T vs. t. So the equation tells you that the slope of the graph is proportional to T - 20. Thus, for example, if T starts high, T - 20 will be a relatively large positive number. We might therefore expect k ( T - 20) to be a relatively large positive number, depending on what k is. For positive k this would give our graph a positive slope, and the temperature would move away from room temperature. If we are talking about something taken from the oven, this wouldn't happen--the temperature would move closer to room temperature. This could be accomplished using a negative value of k. As the temperature moves closer to room temperature, T - 20 becomes smaller, and the steepness of the graph will decrease--i.e., as temperature approaches room temperature, it will do so more and more slowly. So the graph approaches the T = 20 value more and more slowly, approaching as an asymptote. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q Query Inverse Functions and Logarithms, Problem 7. Construct table for the squaring function f(x) = x^2, using x values between 0 and 2 with a step of .5. Reverse the columns of this table to form a partial table for the inverse function. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x = 0, f(0) = 0^2 = 0 x = 0.5, f(0.5) = (0.5)^2 = 0.25 x = 1, f(1) = 1^2 = 1 x = 1.5, f(1.5) = (1.5)^2 = 2.25 x = 2, f(2) = 2^2 = 4 thus the points are ( 0 ,0 ) , ( 0.5 , 0.25) , (1,1) , (1.5,2.25) , (2,4) thus for the inverse function we will have x = 0, f-1(0) = 0 x = 0.25, f-1(0.25) = 0.5 x = 1, f-1(1) = 1 x = 2.25, f-1(2.25) = 1.5 x = 4, f-1(4) = 2 thus the points are ( 0 , 0 ) , ( 0.25 , 0.5 ) , ( 1 , 1 ) , ( 2.25 , 1.5 ) and ( 4 , 2 ) confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: STUDENT SOLUTION: We get the following ordered pairs: Table 1-- (0,0),(.5,.25),(1,1),(1.5,2.25),(2,4) Table2--(0,0),(.25,.5),(1,1),(2.25,1.5),(4,2). Plot the points corresponding to the table of the squaring function, and plot the points corresponding to the table of its inverse. Sketch a smooth curve corresponding to each function. The diagonal line on the graph is the line y = x. Connect each point on the graph of the squaring function to the corresponding point on the graph of its inverse function. How are these pairs of points positioned with respect to the y = x line? ** The segments connecting the graph points for function and for its inverse will cross the y = x line at a right angle, and the graph points for the function and for the inverse will lie and equal distances on either side of this line. The graph of the inverse is therefore a reflection of the graph of the original function through the line y = x. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q **** 8. If we reversed the columns of the 'complete' table of the squaring function from 0 to 12, precisely what table would we get? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given function y = x^2 Thus if we reverse the columns of the ‘complete’ table of the squaring function from 0 to 12 we obtain the inverse function which is y = f-1(x). thus the y values of the table becomes the x values and the x values become y. That is we get the inverse function as a square root function. Thus x = 0, f-1(0) = 0 x = 1, f-1(1) = 1 x = 4, f-1(4) = 2 x = 9, f-1(9) = 3 x = 16, f-1(16) = 4 x = 25, f-1(25) = 5 x = 36, f-1(36) = 6 x = 49, f-1(49) = 7 x = 64, f-1(64) = 8 x = 81, f-1(81) = 9 x = 100, f-1(100) = 10 x = 121, f-1(121) = 11 x = 144, f-1(144) = 12 This input values are only integers, the actual values run from 0 to 144 and values for it similarly increases from 0 to 12. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** We would get a table of the square root function with the first column running from 0 to 144, the second column consisting of the square roots of these numbers, which run from 0 to 12. ** Sketch the graphs of the functions described by both tables. 9. If we could construct the 'complete' table of the squaring function from 0 to infinity, listing all possible positive numbers in the x column, then why would we be certain that every possible positive number would appear exactly one time in the second column? &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q ** The table you constructed had only some of the possible x and y values. A complete table, which couldn't actually be written down but can to an extent be imagined, would contain all possible x values. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Since the original function was a squaring function thus all the y values were positive. Thus for the input value for the inverse function which is a square root function will contain all positive values. Thus the actual table will have all x values from 0 to + infinity. The y values corresponding to the input x values will range from - infinity to + infinity ( that is a set of all real numbers ) confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We could be sure because every number is the square of some other number. If the function was, for example, x / (x^2 + 1) there would be a great many positive numbers that wouldn't appear in the second column. But this is not the case for the squaring function. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q What number would appear in the second column next to the number 4.31 in the first column? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first column corresponds to the input values that are the x values and the second column are the y values. The function is the square function thus if we have x = 4.31, then y = (4.31) ^ 2 = 18.5761 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** In the original table the second column would read 18.57, approx.. This is the square of 4.31. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q What number would appear in the second column next to the number `sqrt(18) in the first column? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first column corresponds to the input values that are the x values and the second column are the y values. The function is the square function thus if we have x = sqrt(18), then y = [sqrt(18)]^2 = 18 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** 18 would appear in the second column because the square of sqrt(18) is 18. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q What number would appear in the second column next to the number `pi in the first column? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first column corresponds to the input values that are the x values and the second column are the y values. The function is the square function thus if we have x = pi, then y = (pi) ^ 2 = 9.8596 approx confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The number would be `pi^2 ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q What would we obtain if we reversed the columns of this table? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If we reversed the columns of this tables, then the x values would become y and the y values x and correspondingly. The function which was the square function would now become the square root function. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: STUDENT ANSWER: We would obtain the inverse, the square roots of the squares being in the y colume and the squared numbers being in the x column. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q What number would appear in the second column next to the number 4.31 in the first column of this table? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first column corresponds to the input values that are the x values and the second column are the y values. The function is the square function thus if we have x = 4.31, then y = (4.31) ^ 2 = 18.5761 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: This number would be 4.31 squared,18.5761. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q What number would appear in the second column next to the number `pi^2 in the first column of this table? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first column corresponds to the input values that are the x values and the second column are the y values. If the function being considered in the question is the square function thus if we have x = pi ^2, then y = (pi ^ 2) ^ 2 = pi ^ 4 = 97.2117 approx The first column corresponds to the input values that are the x values and the second column are the y values. If the function being considered in the question is the square function thus if we have x = pi ^ 2, then y = sqrt [(pi) ^ 2] = pi confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: STUDENT ANSWER: This number would be the square root, 'pi &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I really do not understand the way this assignment is put up, the questions keep giving some value in the first column and ask me to find the value in the second column. But I really do not know which function is being considered, whether the original squaring function or the inverse square root function. I am following them upon the answer, and if I do not get it I am considering both the functions and giving answers. ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q What number would appear in the second column next to the number -3 in the first column of this table? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first column corresponds to the input values that are the x values and the second column are the y values. If the function being considered is the square root function thus if we have x = -3, then y = sqrt(-3) , but sqrt(-3) is not possible thus no such number is possible. The first column corresponds to the input values that are the x values and the second column are the y values. If the being considered is a square function thus if we have x = -3, then y = (-3) ^ 2 = 9 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: There is no such number. No real number has a square equal to -3, since the square of any number which is positive or negative is the product of two numbers of like sign and is therefore positive. Put another way: sqrt(-3) is not a real number, since the square of a real number cannot be negative. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: 13. Translate each of the following exponential equations into equations involving logarithms, and solve where possible: 2 ^ x = 18 2 ^ (4x) = 12 5 * 2^x = 52 2^(3x - 4) = 9. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given 2^x = 18 Taking log on both sides We get log ( 2^x) = log ( 18 ) x * log2 = log 18 x = log18 / log2 = log{base 2}(18) = 4.1699 approx given 2 ^ ( 4x) = 12 thus we obtain 4x = log{base 2}(12) x = (1/4)*( log{base2}(12) ) = (1/4)*( log(12) / log(2) ) = 0.8962 approx Given 5 * 2^x = 52 2^x = 52/5 = 10.4 On performing logarithmic rules we obtain x = log{base 2} ( 10.4 ) = log(10.4) / log(2) = 3.3785 given 2^(3x - 4) = 9 on performing logarithmic rules we obtain 3x - 4 = log{base 2}(9) Thus 3x = 4 + log{base 2}(9) And x = [ 4 + log{base 2}(9) ] / 3 = 2.3899 approx confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: b^x = c is translated into logarithmic notation as log{base b}(c) = x. So: 2^x = 18 translates directly to log{base 2}(18) = x. For 5 * 2^x = 52, divide both sides by 5 to get 2^x = 10.4. Now take logs: x = log{base 2}(10.4) You can easily evaluate this and the preceding solution on your calculator. 2^(3x-4) = 9 translates to log{base 2}(9) = 3x - 4. Solving for x we get x = (log(base 2)(9) + 4) / 3. This can be evaluated using a calculator. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q 14. Solve 2^(3x-5) + 4 = 0 ** YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given 2^(3x - 5) + 4 = 0 Further reducing using logarithmic rules we get 3x - 5 = log{base 2 }(-4) but log of any negative number is not defined thus there is no such value for log{base 2}(-4) and thus no real x satisfying this equation. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 2^(3x-5) + 4 = 0 rearranges to 2^(3x-5) =-4, which we translate as 3x-5 = log {base 2}(-4) = log(-4) / log (2). However log(-4) doesn't exist. When you invert the 10^x table you don't end up with any negative x values. So there is no solution to this problem. Be sure that you thoroughly understand the following rules: 10^x = b translates to x = log(b), where log is understood to be the base-10 log. e^x = b translates to x = ln(b), where ln is the natural log. a^x = b translates to x = log{base a} (b), where log{base a} would be written in your text as log with subscript a. log{base a}(b) = log(b) / log(a), where log is the base-10 log. It also works with the natural log: log{base a}(b) = ln(b) / ln(a). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q Solve 2^(1/x) - 3 = 0 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^( 1/ x) = 3 Using logarithmic rules we get ( 1 / x) = log { base 2 } ( 3) = log(3) / log(2) x = log(2) / log(3) = log{base 3}(2) = 0.63093 approx confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** Rearrange to 2^(1/x) = 3. Then take log of both sides: log(2^(1/x) ) = log(3). Use properties of logs: (1/x) log(2) = log(3). Solve for x: x = log(2) / log(3). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: Solve 2^x * 2^(1/x) = 15 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Rearranging we obtain 2 ^ ( x + 1/x ) = 15 x + 1/x = log(15) / log(2) x^2 + 1 = x * [ log(15) / log(2) ] x^2 - 3.90689x + 1 = 0 thus quadratic expression is approximated to x^2 - 3.91x + 1 = 0 using the quadratic root formula we obtain x as x = ( 3.91 +- 3.36 ) / 2 x = ( 3.91 + 3.36 ) / 2 or x = ( 3.91 - 3.36 ) / 2 x = 3.635 or x = 0.275 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** 2^x * 2^(1/x) = 15. By the laws of exponents we get 2^(x +1/x) = 15 so that x + 1/x = log {base2}(15) or x + 1/x =log(15) / log(2). Multiply both sides by x to get x^2 + 1 = [log(15) / log(2) ] * x. This is a quadratic equation. Rearrange to get x^2 - [ log(15) / log(2) ] * x + 1 = 0 or x^2 - 3.91 * x + 1 = 0. Solve using the quadratic fomula. ** Solve (2^x)^4 = 5 ** log( (2^x)^4 ) = log(5). Using laws of logarithms 4 log(2^x) = log(5) 4 * x log(2) = log(5) 4x = log(5) / log(2) etc.** STUDENT QUESTION OK Even with the solution I am missing a step as I work from x+(1/x)=log15/log2 to the final solution? INSTRUCTOR RESPONSE As stated, you multiply both sides of that equation by x. x * ( x + 1/x) = x^2 + 1, and multiplying the other side by x gives the obvious expression, so you get x^2 + 1 = [log(15) / log(2) ] * x &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q problem 1.3.20 5th; 1.3.22 4th. C=f(A) = cost for A sq ft. What do f(10k) and f^-1(20k) represent? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Text problem 1.3.20 and 1.3.22 both submitted through text 07 assignment. If C = f(A) = cost for A sq ft f(10000) means the cost required of 10000 sq ft and f-1(20000) means the the square feet that can be covered by providing $20000 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** f(10,000) is the cost of 10,000 sq ft. f^-1(20,000) is the number of square feet you can cover for $20,000. ** STUDENT COMMENT Still not positive about the - 1. INSTRUCTOR RESPONSE f ^-1 (x) is the notation for the inverse function. If x is quantity A and f(x) is the value of quantity B, then when you invert the function x becomes quantity B and f ^-1 (x) becomes quantity A. In the original function x is the area and f(x) is the cost. When inverted to the form f ^-1 (x), x becomes the cost and f ^-1 (x) the area. You can think of inverting a function in terms of switching the columns of a table. We can also think of inverting a function in terms of switching the x and y coordinates on a graph, which reflects the graph through the line y = x. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q problem 1.3.38 4th edition (problem omitted from 5th edition but everyone should do this problem). Write an equation for the function if we vertically stretch y = x^2 by factor 2 then vertically shift the graph 1 unit upward. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Text problem 1.3.38 submitted through text 07 assignment. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** Vertically stretching y = x^2 we get y = 2 x^2. The vertical shift adds 1 to all y values, giving us the function y = 2 x^2 + 1. ** STUDENT QUESTION I graphed the solution for vertically stretching and I see the change in the graph but I am still confused on the idea of Vertically stretching. INSTRUCTOR RESPONSE This is summarized at the very beginning of the section in the text. When you multiply a function by c, you move every point | c | times as far from the x axis. It's as if you grabbed along of the top and bottom of the graph and stretched it out factor c (if | c | < 1 it's actually as if you compressed the graph). If c is negative the graph also reflects through the x axis. This is a typical precalculus topic. If your precalculus or analysis course didn't cover this you might want to consider at least reading through, and perhaps working through at least some of the relevant parts of the following documents, which among other things provide a detailed introduction to understanding the concepts of stretching and shifting graphs: http://vhcc2.vhcc.edu/dsmith/genInfo/qa_query_etc/pc1/pc1_qa5.htm http://vhcc2.vhcc.edu/pc1fall9/Assignments/assignment_98126_function%20_families.htm http://vhcc2.vhcc.edu/pc1fall9/pc1/basic_point_graphs_identifying_equation.htm http://vhcc2.vhcc.edu/dsmith/genInfo/qa_query_etc/pc1/pc1_qa6.htm http://vhcc2.vhcc.edu/pc1fall9/basic_function_families/basic_function_families.htm< &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q Give the equation of the function. Describe your sketch in detail. Explain what effect, if any, it would have on the graph if we were to reverse the order of the transformations. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The equation of the graph is 2x^2 + 1. The initial graph x^2 is initially stretched by 2 units, thus the every y value of the graph is doubled to reach the stretched graph. Thus points like ( 1 , 1) , ( -1 , 1 ) , ( 2 , 4 ) , ( -2 , 4 ) now become ( 1 , 2 ) , ( -1 , 2 ) , ( 2 , 8 ) and ( -2 , 8 ) respectively. The vertex remains the same. Next it is shifting the graph by 1 unit in the positive y direction ( that is in the upward direction ). This results in the graph to maintain its shape and structure, but just to shift itself by a unit, thus every y value gets added by 1 unit. Thus the vertex of the parabola gets shifted to ( 0 , 1 ). Similarly other points mentioned get shifted to ( 1 , 3) , ( -1 , 3 ) , ( 2 , 9 ) and ( -2 , 9 ). The parabola is a concave parabola. Now if the transformations are reversed we would first have the shift of the graph that it the equation of the graph would become y = x^2 + 1. Thus the vertex of the parabola gets shifted to ( 0 , 1 ) and every value on the parabola gets shifted by 1 unit. Next stretching the graph by 2 units the final parabola equation would become y = 2x^2 + 2. This case differs from the previous parabola by only one factor which is the shift along the y axis. Since in this case the shift was performed first and then stretched, the shift too got doubled and as a result the graph shifted by 2 units in the positive y direction. The parabola is same as in the previous case with the only difference that each value is shifted by an extra of 1 unit as compared to the previous graph, thus the vertex of the parabola becomes ( 0 , 2 ). Thus the final points would be ( 1 , 4) , ( -1 , 4 ) , ( 2 , 10 ) and ( -2 , 10 ). This parabola too is a concave one. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The function would be y = f(x) = 2 x^2 + 1. The factor 2 stretches the y = x^2 parabola vertically and +1 shifts every point of the stretched parabola 1 unit higher. The result is a parabola which is concave up with vertex at point (0,1). The parabola has been stretched by a factor of 2 as compared to a x^2 parabola, so that the x = +-1 points lie 2 units higher than the vertex, at the points (-1, 3) and (1, 3). If the transformations are reversed then the graph is shifted upward 1 unit before being stretched vertically by factor 2. The vertex, for example, shifts to (0, 1) then when stretched moves to (0, 2). The points (-1, 1) and (1, 1) shift to (-l, 2) and (1, 2), and the stretch then moves them to (-1, 4) and (1, 4). The shift would transform y = x^2 to y = x^2 + 1. The subsequent stretch would then transform this function to y = 2 ( x^2 + 1) = 2 x^2 + 2. The reversed pair of transformations results in a parabola with its vertex at (0, 2), as opposed to (0, 1) for the original pair of transformations. The vertex for the reversed pair of transformations lies 1 unit higher than the vertex for the original pair. The shapes of both graphs are parabolic, and are identical except. The reversed pair of transformations results in a parabola that is everywhere 1 unit 'higher' than that associated with the first pair. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q problem 1.3.45 5th; 1.3.43 4th (was 1.8.30) estimate f(g(1))what is your estimate of f(g(1))?Explain how you look at the graphs of f and g to get this result YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Text problem 1.3.45 and 1.3.43 already submitted with the text _ 07 assignment. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** You have to first find g(1), then apply f to that value. To find g(1), you note that this is g(x) for x = 1. So you look on the x-axis where x = 1. Then you move up or down to find the point on the graph where x = 1 and determine the corresponding y value. On this graph, the x = 1 point lies at about y = 2. Then you look at the graph of f(x). You are trying to find f(g(1)), which we now see is f(2). So we look at the x = 2 point on the x-axis and then look up or down until we find the graph, which for x = 2 lies between 0 and 1, closer to 0. The value is about .3, give or take .1.. ** We can summarize this using function notation: • f(g(1)) = f(2) = .3 (estimated) Further note: It appears from the graph that g(1) might be a little greater than 2, in which case f(g(1)) might be closer to .4. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q problem Graph the function f(x) = x^2 + 3^x for x > 0. Decide if this function has an inverse. If so, find the approximate value of the inverse function at x = 20. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: For a function to have an inverse the function should be one to one (that means each x value should be associated with only 1 value of unique y.) we definitely know that x^2 is an increasing function and 3^x is also an increasing function, thus as x values increase the value of y will also increase. Since y will be ever increasing thus every x will be associated to a unique value of y. Thus the function definitely has an inverse. In order to track the inverse we plug in f(x) = 20 and find x corresponding to the value of x. 20 = x^2 + 3^x The value of the function at x=2 will be f(2) = 2^2 + 3^2 = 13 The value of the function at x=3 will be f(3) = 3^2 + 3^3 = 36 The value of the function at x=2.5 will be f(2.5) = 2.5^2 + 3^2.5 = 6.25 + 15.58845 = 21.83845 Since the value 20 is between 13 and 21.83845 the value of x will be between 2 and 2.5 more closer to 2.5 as 21.83845 is much closer to 20 as compared to 13. Thus x would be approximately 2.41 which when checked in the function results in 19.92. Thus f^-1(20) = 2.41 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The inverse of a function at a certain value is the x that would give you that value when plugged into the function. At x = 20 for g(x) = x^2 + 3^x is the x value for which x^3 + 3^x = 20. The double use of x is confusing and way the problem is stated in the text isn't as clear as we might wish, but what you have to do is estimate the required value of x. It would be helpful to sketch the graph of the inverse function by reflecting the graph of the original function through the line y = x, or alternatively and equivalently by making an extensive table for the function, then reversing the columns. ** BEGINNING OF STUDENT SOLUTION I started by graphing the function of y=x^2+3^x and y=x. INSTRUCTOR RESPONSE You don't say how you constructed the graph. If you used a graphing calculator, your solution would not be acceptable. It would be OK to use the calculator to check out your construction, but you would have to explain your construction of the graph. One description of the construction: From knowledge of basic power and exponential functions, you should know that x^2 and 3^x are both increasing functions for x > 0 (x^2 is decreasing for x < 0, but for the purpose of this question that doesn't matter). It should be clear why this is so. The greater the positive number you square, the greater the result. And the greater the value of x, the greater the power to which you raise 3, so the greater will be the value of 3^x. The graph of x^2 goes through (0, 0) and (1, 1). The graph of 3^x goes through (0, 1) and (1, 3). As x values continue to increase, the value of x^2 quickly becomes insignificant compared to that of 3^x (e.g., for x = 4 the function x^2 takes value 16 while 3^x takes value 81; for x = 8 we have x^2 = 64 and 3^x = 8261 (check my mental arithmetic on that one) ). So the graph of x^2 + 3^x passes through (0, 1) and (1, 4), and continues increasing more and more quickly as x increases. The resulting function is clearly invertible. It would also be OK to make a table. You should be able to do enough mental arithmetic to do so without relying on a calculator. Certainly you can square the integers from, say, 0 to 5. And you should be able to start with 3 and triple your result, repeating at least 5 times. x x^2 3^x x^2 + 3^x 0 0 1 1 1 1 3 4 4 2 4 9 13 13 3 9 27 36 36 4 16 81 77 77 5 25 343 368 368 A small portion of the graph is depicted below: The values of x and x^2 + 3^x can easily be reversed, giving us a partial table of the inverse function: x inv fn 1 0 4 1 13 2 37 3 77 4 368 5 It should be clear that the inverse function is single-valued (i.e., there is are no two x values greater than 0 for which the inverse function takes the same value). A partial graph of the inverse function (not to exactly the same scale as the original) is depicted below: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Do I need to mention the values of the function in columns in the way you presented it in the above solution to the student’s response to prove that it is increasing when it can be explained without inputting values as for this function it is very obvious that x^2 and 3^x both for positive x will be increasing and thus the sum of both will also be increasing. ------------------------------------------------ Self-critique Rating: OK
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Given Solution: H is the temperature, t is the clock time. H(30) is the temperature at clock time t = 30, so H(30) = 10 tells us that a clock time t = 30 the temperature was 10 degrees. The vertical coordinate is the temperature, and the vertical intercept of the graph occurs when t = 0 so the vertical intercept gives us the temperature at clock time 0. The vertical coordinate is the clock time, and the horizontal intercept occurs when H = 0, so the horizontal intercept gives us the clock time when temperature is 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q If H = f(t) describes the temperature H of an object at clock time t, then what does it mean to say that H(30)=10? What information would you get from thevertical and horizontal intercepts of the graph of the function? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given function H = f(t) where H is he temperature of an object and t is the clock time. H(30) = 10 means that at clock time 30 temperature units the temperature of the body will be 10 temperature units. The vertical intercept which is also the y intercept basically indicates the temperature of the body at the initial time that is the temperature at clock time 0. The horizontal intercept or the x intercept basically indicates the time when the temperature of the body reduces from initial to 0, that is the time when the temperature of the body becomes 0. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: H is the temperature, t is the clock time. H(30) is the temperature at clock time t = 30, so H(30) = 10 tells us that a clock time t = 30 the temperature was 10 degrees. The vertical coordinate is the temperature, and the vertical intercept of the graph occurs when t = 0 so the vertical intercept gives us the temperature at clock time 0. The vertical coordinate is the clock time, and the horizontal intercept occurs when H = 0, so the horizontal intercept gives us the clock time when temperature is 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK #*&!