3. One way to define the natural log, Ina, is as the definite integral dt for...

Transcribed Image Text:

3. One way to define the natural log, Ina, is as the definite integral dt for x>0; that is dt = ln x. (a) Sketch the graph of 1/t and and approximate In 2 numerically by partitioning the interval [1,2] into 4 equal pieces and computing L4 and R₁. Write out each sum long-hand, without using a calculator program. (b) Which one of the approximating sums, L₁ or R₁, do you expect is a better approximation of In 2? Explain. Hint: sketch and compare the areas that represent the errors. (c) Use the Error Bounding formula for L₁ and R, to find an upper bound on the error in the approximations you found in part (a). (d) Suppose we used n = 100 slices instead of 4. What is an upper bound on the error in our approximations? (You don't need to compute L100 or R100!) 3. One way to define the natural log, Ina, is as the definite integral dt for x>0; that is dt = ln x. (a) Sketch the graph of 1/t and and approximate In 2 numerically by partitioning the interval [1,2] into 4 equal pieces and computing L4 and R₁. Write out each sum long-hand, without using a calculator program. (b) Which one of the approximating sums, L₁ or R₁, do you expect is a better approximation of In 2? Explain. Hint: sketch and compare the areas that represent the errors. (c) Use the Error Bounding formula for L₁ and R, to find an upper bound on the error in the approximations you found in part (a). (d) Suppose we used n = 100 slices instead of 4. What is an upper bound on the error in our approximations? (You don't need to compute L100 or R100!)

Answer rating: 100% (QA)

A a beginalign L4 frac11frac12frac13frac14 R4 frac12frac13frac14frac15 endalign b I w... View the full answer