# Using Taylor's Theorem to approximate

• Shameel
In summary, Taylor series can be used to approximate values of e^{x} at x=0 by using only a few terms of the series. This method is useful for quickly finding approximations for values such as e^{0.01} or e^{0.17671234}.
Shameel
Hi Guys,

Is there any whay I can use the following theorem to approximate the value of e^x at x=0?

$$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ...$$

If the above function is not used for approximation, then what is it used to do?

Thanks heaps

Why bother approximating $$e^{0}?$$
You know what it is: $$e^{0}=1$$

However, since you know that, you are now in a position to approximate with only a couple of terms in the Taylor series function values like $$e^{0.01}, e^{0.17671234}$$

arildno said:
Why bother approximating $$e^{0}?$$
You know what it is: $$e^{0}=1$$

However, since you know that, you are now in a position to approximate with only a couple of terms in the Taylor series function values like $$e^{0.01}, e^{0.17671234}$$
I am a n00b. Could you give me an example how Taylor approximations work?

Sure.

$$e^{0.001}\simeq e^{0}+1(0.001-0)+\frac{1}{2}(0.001-0)^{2}$$

Daniel.

Let us try to approximate $$e^{0.01}$$ by forming the Taylor series about the origin:
Then, we have:
$$e^{x}=f(x)\approx{f}(0)+f'(0)(x-0)+\frac{1}{2}f''(0)(x-0)^{2}$$
where I've retained the 3 first terms in the Taylor series.
But, now, we have:
$$f(0)=e^{0}=1,f'(0)=e^{0}=1,f''(0)=e^{0}=1$$
since the derivative of the exponential function is itself
Thus, we have:
$$e^{0.01}\approx{1}+1*(0.01-0)+\frac{1}{2}*1*(0.01-0)^{2}=1+\frac{1}{100}+\frac{5}{100000}=1.01005$$

## 1. What is Taylor's Theorem?

Taylor's Theorem is a mathematical formula that allows us to approximate a function using a polynomial.

## 2. How does Taylor's Theorem work?

Taylor's Theorem uses derivatives of a function at a specific point to create a polynomial that closely approximates the function in a small interval around that point.

## 3. What is the purpose of using Taylor's Theorem?

The purpose of using Taylor's Theorem is to estimate the value of a function at a specific point or to approximate the behavior of a function in a small interval around that point.

## 4. What is the difference between Taylor's Theorem and Maclaurin's Theorem?

Taylor's Theorem is a more general form of Maclaurin's Theorem, which is a special case where the point of approximation is 0.

## 5. How accurate is the approximation using Taylor's Theorem?

The accuracy of the approximation using Taylor's Theorem depends on the number of terms used in the polynomial and the smoothness of the function. The more terms used, the closer the approximation will be to the actual function.

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