course Mth 272 A much more comprehendable assignment.
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17:52:35 4.4.4 (was 4.3 #40 write ln(.056) = -2.8824 as an exponential equation
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RESPONSE --> e ^(-2.8824)= 0.056 confidence assessment: 3
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17:52:41 y = ln x is the same as e^y = x, so in exponential form the equation should read e^-2.8824 = .056 **
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RESPONSE --> ok self critique assessment: 3
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17:53:12 4.4.8 (was 4.3 #8) write e^(.25) = 1.2840 as a logarithmic equation
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RESPONSE --> ln(1.2840)= 0.25 confidence assessment: 3
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17:53:19 e^x = y is the same as x = ln(y) so the equation is .25 = ln(1.2840). **
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RESPONSE --> ok self critique assessment: 3
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17:53:49 4.4.16 (was 4.3 #16) Sketch the graph of y = 5 + ln x.
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RESPONSE --> So all I can do is describe the graph I see, right? Please offer advice as necessary. The graph is increasing at a decreasing rate in quadrant II of the graph. The graph is increasing along the x axis as it approaches the limit of 5 along the y axis. The x value begins around zero. The graph is concave down. I know that was a rough explanation, please tell me the proper way. This graph was a little different than the previous ones we had to describe.
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17:54:14 Plugging in values is a good start but we want to explain the graph and construct it without having to resort to much of that. The logarithmic function is the inverse of the exponential function, which is concave up, has asymptote at the negative x axis and passes thru (0, 1). Reversing the coordinates of the exponential graph gives you a graph which is concave down, asymptotic to the negative y axis and passes thru (1, 0). Adding 5 to get y = 5 + ln x raises this graph 5 units so that the graph passes thru (1, 5) and remains asymptotic to the negative y axis and also remains concave down. The graph is asymptotic to the negative y axis, and passes through (1,5). The function is increasing at a decreasing rate; another way of saying this is that it is increasing and concave down. STUDENT COMMENT: I had the calculator construct the graph. I can do it by hand but the calculator is much faster. INSTRUCTOR RESPONSE: The calculator is faster but you need to understand how different graphs are related, how each is constructed from one of a few basic functions, and how analysis reveals the shapes of graphs. The calculator doesn't teach you that, though it can be a nice reinforcing tool and it does give you details more precise than those you can imagine. Ideally you should be able to visualize these graphs without the use of the calculator. For example the logarithmic function is the inverse of the exponential function, which is concave up, has asymptote at the negative x axis and passes thru (0, 1). Reversing the coordinates of the exponential graph gives you a graph which is concave down, asymptotic to the negative y axis and passes thru (1, 0). Adding 5 to get y = 5 + ln x raises this graph 5 units so that the graph passes thru (1, 5) and remains asymptotic to the negative y axis and also remains concave down. **
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RESPONSE --> ""Asymptotic"" was the term I could not think of to save my life! I have a better idea of how to describe the graph and not rely on as much use of the calculator now. self critique assessment: 2
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17:56:34 4.4.22 (was 4.3 #22) Show e^(x/3) and ln(x^3) inverse functions
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RESPONSE --> The two functions can be manipulated without graphing to prove that they are inverses of each other. e^(x/3) y = e^(x/3) x = e^(y/3) = lnx^3 ln x^3 x = lny^3 = e^(x/3) confidence assessment: 2
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17:56:57 GOOD STUDENT RESPONSE: Natural logarithmic functions and natural exponential functions are inverses of each other. f(x) = e^(x/3) y = e^(x/3) x = e^(y/3) y = lnx^3 f(x) = lnx^3 y = ln x^3 x = lny^3 y = e^(x/3) INSTRUCTOR RESPONSE: Good. f(x) = e^(x/3) so f(ln(x^3)) = e^( ln(x^3) / 3) = e^(3 ln(x) / 3) = e^(ln x) = x would also answer the question MORE ELABORATION You have to show that applying one function to the other gives the identity function. If f(x) = e^(x/3) and g(x) = ln(x^3) then f(g(x)) = e^(ln(x^3) / 3) = e^( 3 ln(x) / 3) = e^(ln(x)) = x. **
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RESPONSE --> ok, was the ""more elaboration"" part of the answer necessary in my response? confidence assessment: 2
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17:59:03 4.4.46 (was query 4.3 #44) simplify 1/3 [ 2 ln(x+3) + ln x - ln(x^2-1) ]
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RESPONSE --> using logarithm rules in chapter 4: 1/3 [ 2 ln(x+3) + ln x - ln(x^2-1) ] multiply by 1/3 = 2/3 ln(x+3) + 1/3 ln x - 1/3 ln(x^2-1) rearrange ""ln"" and exponents = ln(x+3)^(2/3) + ln(x^(1/3)) - ln((x^2-1)^(1/3)) factor out ""ln"" and simplify by using addition/subtraction rules = ln [ (x+3)^(2/3) (x^(1/3) / (x^2-1)^(1/3) ] simplify = ln [ {(x+3)^2 * x / (x^2-1)}^(1/3) ] confidence assessment: 3
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17:59:12 1/3 [ 2 ln(x+3) + ln x - ln(x^2-1) ] = 2/3 ln(x+3) + 1/3 ln x - 1/3 ln(x^2-1) = ln(x+3)^(2/3) + ln(x^(1/3)) - ln((x^2-1)^(1/3)) = ln [ (x+3)^(2/3) (x^(1/3) / (x^2-1)^(1/3) ] = ln [ {(x+3)^2 * x / (x^2-1)}^(1/3) ] **
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RESPONSE --> 3 self critique assessment:
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17:59:54 4.4.58 (was 4.3 #58) solve 400 e^(-.0174 t) = 1000.
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RESPONSE --> 1000/400=e^(-.0174t) 2.5= e^(-.0174t) ln 2.5= -.0174t ln 2.5/-.0174=t t=- 52.660 confidence assessment: 3
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18:00:16 The equation can easily be arranged to the form e^(-.0174) = 2.5 We can convert the equation to logarithmic form: ln(2.5) = -.0174t. Thus t = ln(2.5) / -.0174 = 52.7 approx.. **
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RESPONSE --> ok, can you please explain why the negative sign was lost in the final answer? self critique assessment: 2
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18:02:33 4.4.72 (was 4.3 #68) p = 250 - .8 e^(.005x), price and demand; find demand for price $200 and $125
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RESPONSE --> p= price $200= 250- 0.8e^0.005x -50= -.8e^0.005x -50/-.8= e^0.005x 62.5= e^0.005x ln 62.5= 0.005x ln 62.5/ 0.005= x x= 827.03 units sold 125=250-0.8e^0.005x -125/-.8= e^0.005x 156.25= e^.005x ln 156.25/ .005=x x= 1,010.3 units confidence assessment: 3
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18:02:44 p = 250 - .8 e^(.005x) so p - 250 = - .8 e^(.005x) so e^(.005 x) = (p - 250) / (-.8) so e^(.005 x) = 312.5 - 1.25 p so .005 x = ln(312.5 - 1.25 p) and x = 200 ln(312.5 - 1.25 p) If p = 200 then x = 200 ln(312.5 - 1.25 * 200) = 200 ln(62.5) = 827.033. For p=125 the expression is easily evaluated to give x = 1010.29. **
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RESPONSE --> ok self critique assessment: 3
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