5 So far we have been using an artificial example because we could calculate the...
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Calculus
5 So far we have been using an artificial example because we could calculate the actual value of 5x 3x 1 dx just by using the Fundamental theorem of calculus and finding an antiderivative Now we use a more motivated example 2 a Now consider 1 9x dx Does this function have an 0 elementary antiderivative i Answer ii Note this integral arises when trying to find the length of 3 the curve y x as x goes from 0 to 2 b Now try to estimate the integral of that function from a 0 to b 2 using Riemann sums then record how many subintervals you needed to convince you that your error is less than 001 In parts b c d you will have to experiment with values of n until you are convinced your answer is within 0 001 of the actual value i Answers c Repeat this process but using the trapezoidal rule now How many subintervals were necessary to reach this level of accuracy i Answer d Finally do this again with Simpson s rule but remember to only use even values of n for Simpson s rule i Answer e Of these methods which one converged the fastest Did this result surprise you Share your thoughts i Answer f In each of these cases you also did not have a means of knowing the exact value of the integral so how did you determine when your error was less than 001 Answer
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