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PRACTICE EXERCISES - Do these before the lab period. Answers at bottom of page. 1) Given f = 200 x^3, a measured value of x is 3.00 and the known error in x is 0.03. What is the % error in f? 2) With the same function as in Question 1, suppose that 0.03 is the standard deviation in the measured x's and 3.00 is the mean of the measured x's. What is the standard deviation in f? What is the fractional standard deviation in f? What is the standard deviation in the mean for f, if there were 16 readings of x? 3) Given f = 10.0(x - y), with x = 2.0 ± 0.1 and y = 5.0 ± 0.2, known errors. What is the maximum possible % error in f? Convince yourself that the answer is correct by taking the values of x and y so as to cause a maximum error in f, and checking to see what the error is. What is the absolute error in f? 4) Given the same function as in Question 3, and given that the x and y values are the mean and standard deviation for several readings of x and y, what is the standard deviation in f? How does it compare with the maximum absolute error in f, assuming the errors known (Question 3)? What is the standard deviation in the mean for f, if there are 100 readings each of x and y? 5) Given f = A cos(x). If the known error in x is ?x, what is the maximum fractional error in f(x)? Around what angle will the error in f be greatest for a given ?x? 6) Given the same function as in Question 5. If the standard deviation in x, obtained from several readings, is ?_x, what is the standard deviation in f(x)? What is the fractional standard deviation in f(x)? 7) Given f = x sin(ln y). If ?x and ?y are assumed to be known errors in x and y, what is the fractional error in f(x,y)? 8) Given the same function as in Question 7. Assume that ?_x and ?_y are the standard deviations in x and y. What is the standard deviation in f(x,y)? 9) Given f(x,y) = 3x^2y. Measurements of x are 1.9, 2.0, 2.2, 2.0, 2.4, 2.1. Measurements of y give y_best = y_ave = 10.0 and ?_y = 0.30. a) What is ?_x? b) What is ?_f (leave arithmetic in root un-worked out)? WORK IN THE LAB: Each student will be given a sheet of exercises to be worked out. They will be similar to those above. Answers: 1. 3% 2. ?_f = 162, 0.03, 40.5 3. 10% 4. 10?.05 = 2.2, .22 5. tan(x) ?x, 90° 6. Asin(x) ?_x, tan(x) ?_x 7. ?x/x + cot(ln y) ?y/y 8. ?_f = ?[sin^2(ln y) ?_x^2 + (x/y)^2 cos^2(ln y) ?_y^2] 9. a) 0.40/?6 b) ?[6(2.1)^2(10)^2(.40)^2 + 9(2.1)^4(.30)^2]

          PRACTICE EXERCISES - Do these before the lab period. Answers at bottom of page.
1) Given f = 200 x^3, a measured value of x is 3.00 and the known error in x is 0.03. What is the % error in f?
2) With the same function as in Question 1, suppose that 0.03 is the standard deviation in the measured x's and 3.00 is the mean of the measured x's. What is the standard deviation in f? What is the fractional standard deviation in f? What is the standard deviation in the mean for f, if there were 16 readings of x?
3) Given f = 10.0(x - y), with x = 2.0 ± 0.1 and y = 5.0 ± 0.2, known errors. What is the maximum possible % error in f? Convince yourself that the answer is correct by taking the values of x and y so as to cause a maximum error in f, and checking to see what the error is. What is the absolute error in f?
4) Given the same function as in Question 3, and given that the x and y values are the mean and standard deviation for several readings of x and y, what is the standard deviation in f? How does it compare with the maximum absolute error in f, assuming the errors known (Question 3)? What is the standard deviation in the mean for f, if there are 100 readings each of x and y?
5) Given f = A cos(x). If the known error in x is ?x, what is the maximum fractional error in f(x)? Around what angle will the error in f be greatest for a given ?x?
6) Given the same function as in Question 5. If the standard deviation in x, obtained from several readings, is ?_x, what is the standard deviation in f(x)? What is the fractional standard deviation in f(x)?
7) Given f = x sin(ln y). If ?x and ?y are assumed to be known errors in x and y, what is the fractional error in f(x,y)?
8) Given the same function as in Question 7. Assume that ?_x and ?_y are the standard deviations in x and y. What is the standard deviation in f(x,y)?
9) Given f(x,y) = 3x^2y. Measurements of x are 1.9, 2.0, 2.2, 2.0, 2.4, 2.1. Measurements of y give y_best = y_ave = 10.0 and ?_y = 0.30.
a) What is ?_x? b) What is ?_f (leave arithmetic in root un-worked out)?
WORK IN THE LAB: Each student will be given a sheet of exercises to be worked out. They will be similar to those above.
Answers:
1. 3%
2. ?_f = 162, 0.03, 40.5
3. 10%
4. 10?.05 = 2.2, .22
5. tan(x) ?x, 90°
6. Asin(x) ?_x, tan(x) ?_x
7. ?x/x + cot(ln y) ?y/y
8. ?_f = ?[sin^2(ln y) ?_x^2 + (x/y)^2 cos^2(ln y) ?_y^2]
9. a) 0.40/?6 b) ?[6(2.1)^2(10)^2(.40)^2 + 9(2.1)^4(.30)^2]

$PRACTICE EXERCISES - Do these before the lab period. Answers at bottom of page. 1) Given f = 200 x^3, a measured value of x is 3.00 and the known error in x is 0.03. What is the % error in f? 2) With the same function as in Question 1, suppose that 0.03 is the standard deviation in the measured x's and 3.00 is the mean of the measured x's. What is the standard deviation in f? What is the fractional standard deviation in f? What is the standard deviation in the mean for f, if there were 16 readings of x? 3) Given f = 10.0(x - y), with x = 2.0 ± 0.1 and y = 5.0 ± 0.2, known errors. What is the maximum possible % error in f? Convince yourself that the answer is correct by taking the values of x and y so as to cause a maximum error in f, and checking to see what the error is. What is the absolute error in f? 4) Given the same function as in Question 3, and given that the x and y values are the mean and standard deviation for several readings of x and y, what is the standard deviation in f? How does it compare with the maximum absolute error in f, assuming the errors known (Question 3)? What is the standard deviation in the mean for f, if there are 100 readings each of x and y? 5) Given f = A cos(x). If the known error in x is ?x, what is the maximum fractional error in f(x)? Around what angle will the error in f be greatest for a given ?x? 6) Given the same function as in Question 5. If the standard deviation in x, obtained from several readings, is ?x, what is the standard deviation in f(x)? What is the fractional standard deviation in f(x)? 7) Given f = x sin(ln y). If ?x and ?y are assumed to be known errors in x and y, what is the fractional error in f(x,y)? 8) Given the same function as in Question 7. Assume that ?x and ?y are the standard deviations in x and y. What is the standard deviation in f(x,y)? 9) Given f(x,y) = 3x^2y. Measurements of x are 1.9, 2.0, 2.2, 2.0, 2.4, 2.1. Measurements of y give ybest = yave = 10.0 and ?y = 0.30. a) What is ?x? b) What is ?f (leave arithmetic in root un-worked out)? WORK IN THE LAB: Each student will be given a sheet of exercises to be worked out. They will be similar to those above. Answers: 1. 3% 2. ?f = 162, 0.03, 40.5 3. 10% 4. 10?.05 = 2.2, .22 5. tan(x) ?x, 90° 6. Asin(x) ?x, tan(x) ?x 7. ?x/x + cot(ln y) ?y/y 8. ?f = ?[sin^2(ln y) ?x^2 + (x/y)^2 cos^2(ln y) ?y^2] 9. a) 0.40/?6 b) ?[6(2.1)^2(10)^2(.40)^2 + 9(2.1)^4(.30)^2]$

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