a) Why can we not take the derivative of y(t) to find y'(0.2)? b) In this situation, we can still approximate the derivative. We approxiamate limits by plugging in values close to the limiting value. For example, if f(x) = x ln(x), we can approxiamate f'(0.2) using the limit definition: f'(2) = lim as h approaches 0 (f(2+h)-f(2))/h Approximate f'(2) by using h = 0.001 in the ratio above. Use h = -0.001 as well. Round to 4 decimal places. c) Why can we not approxiamate the derivative y'(0.2) using h = 0.001 (we are talking about the data in the table again) d) What is the smallest value of h we can use? Use that to approximate y'(0.2). This is called forward difference approximation
Question (sorry it's so long)
Often in science and enginnering, we do not have a function. We have data. Here's a table of data obtained by attatching motion sesnors to someone who was walking. This data is an angle, but we need the angular velocity.
| t (seconds) | 0.1 | 0.2 | 0.3 | 0.4 |
| y(t) (degrees) | 15.99 | 16.24 | 16.38 | 16.41 |
a) Why can we not take the derivative of y(t) to find y'(0.2)?
b) In this situation, we can still approximate the derivative. We approxiamate limits by plugging in values close to the limiting value. For example, if f(x) = x ln(x), we can approxiamate f'(0.2) using the limit definition:
f'(2) = lim as h approaches 0 (f(2+h)-f(2))/h
Approximate f'(2) by using h = 0.001 in the ratio above. Use h = -0.001 as well. Round to 4 decimal places.
c) Why can we not approxiamate the derivative y'(0.2) using h = 0.001 (we are talking about the data in the table again)
d) What is the smallest value of h we can use? Use that to approximate y'(0.2). This is called forward difference approximation.
e) Use h = -0.1 to approximate y'(0.2). this is called a backward differnce approximation.
f) Average your answers in parts (d) and (e). This is callled a centered difference approximation. Why would you think the centered differnce approximation is the best of our three approxiamtions?
Please check step 2 and 3 for solutions.
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