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Problems

Problems

Problem : Approximate e0.1 to within 10-3 using Taylor's formula.

The error term in Taylor's formula is

rn(x) = xn = xn < xn    

where c is some number in the interval (0, 0.1). Thus r3(0.1) < 1/3000 < 10-3, so it is sufficient to go up to the degree 2 term in approximating e0.1. We have

e0.1 1 + 0.1 + = 1.105    

Problem : Approximate sin(- 0.1) to within 10-5. The Taylor series for sin(x) begins

x - + - + ...    

The absolute value of the error term in Taylor's formula is

| rn(x)| = xn = | x|n    

Noting that | r4(- 0.1)|≤| 0.1|4/4!≤10-5, we see that we need to use up to the degree 4 term in the Taylor series. We have

sin(- 0.1) (- 0.1) - –0.09983