Question

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

Find the Taylor polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ

Accepted Answer

Step 1: Understand the problem requires finding Taylor polynomials of degrees 1 through 5 for the function $$f(x) = e$$^{-x}$$ centered at a=0$. Recall that the Taylor polynomial of degree n$ centered at a$ is given by:

p_n(x$$) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k$$

where f$$^{(k)}$$(a) is $$the $$k-th $$derivative of $$f$$ evaluated at $$a. $$Step 2: Compute the derivatives of $$f(x) = e$$^{-x}$$ up to the 5th order. Note the pattern in derivatives:

- f(x$$) = $$e^{-x}$$
- $$f'(x) = -e$$^{-x}$$
- f$$''(x) = $$e^{-x}$$
- $$f^{(3)}(x$$) = $$-e^{-x}$$
- $$f^{(4)}(x$$) = $$e^{-x}$$
- $$f^{(5)}(x$$) = $$-e^{-x}$$

This alternating pattern will help in evaluating derivatives at $$x=0. $$Step 3: Evaluate each derivative at $$a=0. $$Since $$e^0 = 1$, the values will alternate between 1$ and -1$ depending on the order of the derivative:

- f(0) = 1$
- f$$'(0) = -1$$
- f$$''(0) = 1$$
- f$$^{(3)}$$(0) = -1$$
- $$f^{(4)}(0) = 1$
- f$$^{(5)}$$(0) = -1$$ Step 4: Write the Taylor polynomials $$p_1$$ through $$p_5 by $$substituting the derivative values into the Taylor polynomial formula:

$$p_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k$$

For example, the linear polynomial $$p_1(x$$)$$ includes terms up to k=1$, the quadratic p_2$(x) up to$$ $$k=2$$, and so on. Step 5: Use the polynomials $$p_1$$ and $$p_2 ($$linear and quadratic approximations) to approximate values of $$f(x)$$ near $$x=0 by $$plugging in the desired $$x$$ values into these polynomials. This provides an approximation of $$e^{-x}$$ using simpler polynomial expressions.

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Key Concepts

Taylor Polynomials

Linear and Quadratic Approximations

Evaluating and Using Taylor Polynomials for Approximation