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course MTH 163
Time and Date Stamps (logged): 16:14:24 11-14-2011 °΅°³±³°°°³±―°° Precalculus I Test 2
Completely document your work and your reasoning.
You will be graded on your documentation, your reasoning, and the correctness of your conclusions.
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Instructions:
Test is to be taken without reference to text or outside notes.
Graphing Calculator is allowed, as is blank paper or testing center paper.
No time limit but test is to be taken in one sitting.
Please place completed test in Dave Smith's folder, OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va., 24212-0828 OR email copy of document to dsmith@vhcc.edu, OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken. Student may, if proctor deems it feasible, make and retain a copy of the test..
Directions for Student:
Completely document your work.
Numerical answers should be correct to 3 significant figures. You may round off given numerical information to a precision consistent with this standard.
Undocumented and unjustified answers may be counted wrong, and in the case of two-choice or limited-choice answers (e.g., true-false or yes-no) will be counted wrong. Undocumented and unjustified answers, if wrong, never get partial credit. So show your work and explain your reasoning.
Due to a scanner malfunction and other errors some test items may be hard to read, incomplete or even illegible. If this is judged by the instructor to be the case you will not be penalized for these items, but if you complete them and if they help your grade they will be counted. Therefore it is to your advantage to attempt to complete them, if necessary sensibly filling in any questionable parts.
Please write on one side of paper only, and staple test pages together.
Test Problems:
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Problem Number 1
For the function y = f(t) = t^2 construct a table of y vs. t for t running from t = 3.9 to t = 4.14 in four equal increments. Using appropriate transformation(s) on the y column, the t column, or both, linearize this data set and demonstrate that the data set has in fact been linearized.
For this one, I would make a table like this
Y=f(t)=t^2 t
15.21 3.9
17.14 4.14
But, what is meant by in four equal parts? Do I just need four numbers in between these two? And then I would graph these numbers correct (t,,y)? And to show that it has been linearized I must make a function out of the slope and y intercept y=mx+b. Is that correct?
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Right idea.
Between the two numbers the difference is 4.14 - 3.9 = .24. Four equal increments would each have length .24 / 4 = .06.
So your t column would get
3.9
3.96
4.02
4.08
4.14
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Problem Number 2
State the laws of exponents, and give an example of each.
x1 = x 61 = 6
x0 = 1 70 = 1
x-1 = 1/x 4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5
xm/xn = xm-n x6/x2 = x6-2 = x4
(xm)n = xmn (x2)3 = x2Χ3 = x6
(xy)n = xnyn (xy)3 = x3y3
(x/y)n = xn/yn (x/y)2 = x2 / y2
x-n = 1/xn x-3 = 1/x3
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Problem Number 3
What equation must be solved to find the doubling time tDoub for the function y = f(x) = 3 * 1.02^x?
.For this, you would double 3*1.02, which is 6. So 6=3*1.02^x and x= 2.67.
I am not sure if I did this correct. How do I solve for the x as an exponent exactly. I tried to do ln(2)/ln(1.02) and I got a huge number. But with this, I just did the e^(2)/e^(1.02) on the calculator.
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ln(2) / ln(1.02) is correct, and it comes out about 35.
Try it again on your calculator and be careful about your parentheses.
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Problem Number 4
What specific function y = A b^x + c fits the data points ( 9, 4.485) and ( 17, 4.072) with asymptote y = 0?
We would find the slope first, right? And that is -0.05. The asymptote is y=0. So the function would be -0.05*4.072^x+0. Because the A is the slope, but Im not really sure what b is? And then you keep your x and then c is your asymptote. Right?
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You plug the coordinates of your points into the function, obtaining
4.485 = A * b^9 + c
and
4.072 = A * b^17 + c.
The asymptote is zero provided c is zero, so the equations become
4.485 = A * b^9
4.072 = A * b^17.
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Problem Number 5
Sketch a graph of y = (x + 2.5) ^ 3 (x - 2) ( 3 x^2 + 5 x + 2.5).
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.I am not sure how to do this. Everything after the ^ is an exponent, correct? I get huge numbers when I try to graph it on the calculator to check myself. I get the point 2,0, but that is the honly definite point. Help with this one please?
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x = 0 makes (x - 2) zero, which when multiplied by the rest of the expression gives you zero.
Similarly x = -2.5 makes (x + 2.5) zero, which when multiplied by everything else makes the entire expression zero.
What values of x satisfy
3 x^2 + 5 x + 2.5 = 0?
What are your conclusions?
If x = 100 then approximately wat is y? Note that for example x + 2.5 is 102.5, which is close to 100. Also x - 2 is close to 100. What is 3 x^2 close to? What therefore do you get when you multiply the three factors, rounding everything to multiples of 100?
Answer the same question for x = -100.
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Problem Number 6
Explain the difference between growth rate and growth factor.
.Growth factor is the factor of how much something multiplies itself over time.
Growth rate is the total rate that it grows over time.
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If the growth rate is r then the growth factor is 1 + r.
For example for your previous funciton y = 3 * 1.02^x, the growth factor is 1.02 and the growth rate is .02.
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Problem Number 7
Express the function y = f(x) = 8 * 2^( .75 x) in the form y = A b^x.
.I understand that if this were x^2, I would take the square root, but I am not sure about this one? Do I take the log of both sides? How would I go about this?
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2^(.75 x) = (2^.75) ^ x = 1.68^x, approx.
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Problem Number 8
By how many decibels does a sound whose intensity is 100000 times threshold intensity exceed a sound whose intensity if 100 times threshold intensity?
.Would I use proportionality for this one? This one has me stumped as well.
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You need to review the worksheets to find the definition of decibels and a discussion of how to use it.
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Problem Number 9
Sketch the graph of a degree 3 polynomial with zeros at x = - 5, 4 and 5, with y taking on increasingly large positive values for large positive values of x.
Give the y = (x-x1)(x-x2)(x-x3) form of this function has well as the y = ax^3 + bx^2 + cx + d form.
.y=(x+5)(x-4) (x-5)
The graph would contain these three points and be a polynomial graph. I am not sure how to form this other function? I have looked in my notes, but just cant seem to get it. Help is very appreciated!
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.y=(x+5)(x-4) (x-5) is correct.
Multiply this out using the distributive law:
.y=(x+5)(x-4) (x-5) =
( (x+5) ( x - 4) ) * (x - 4) =
( x ( x - 4) + 5 ( x - 4) ) * (x - 4) =
(x^2 - 4 x + 5 x - 20) ( x - 4 ) =
(x^2 + x - 20) * (x - 4) =
(x^2 + x - 20) * x - (x^2 + x - 20) * 4 =
etc..
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Problem Number 10
Explain why the following statement must be true: No polynomial of degree 2 can be the product of three or more polynomials of degree 1.
Because it will only have a product of two polynomials at the most. It would have to be a degree 3 for it to have 3 polynomials.
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Check my notes and let me know if you have additional questions.
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