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What Is Degree N (Nth Order) Taylor Approximation?

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Explanations 1

The constant, linear and quadratic approximations to a function fff about a point aaa do pretty well, if we stay close to the value of aaa. However, sometimes we may want to go …
By Maxime Serrano, Alex Federspiel

Problems 3

Use the fifth-order Taylor polynomial of sin⁡(x)\sin(x)sin(x) about x=0x=0x=0 to approximate sin⁡(2)\sin(2)sin(2). Give your answer as a decimal, rounded to the nearest hundredth.
Use the third-order Taylor approximation of exe^xex about x=0x=0x=0 to approximate e\sqrt{e}e. Give your answer as a decimal rounded to the nearest hundredth.
Let f(x)=11−xf(x) = \frac{1}{1-x}f(x)=1−x1. We know that f(13)=32=1.5f(\frac{1}{3}) = \frac{3}{2} = 1.5f(31)=23=1.5. So let's test the accuracy of our Taylor approximation wi…