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11.1 Approximating Functions with Polynomials

NOT on exam 3

Power Series

\[ \text{Power Series: } \sum_{k=0}^{\infty} C_k (x-a)^k \]\[ = C_0 + C_1(x-a) + C_2(x-a)^2 + C_3(x-a)^3 + C_4(x-a)^4 + \dots \]

\( a \): center of power series
\( C_k \): coefficients of the \( k^{\text{th}} \) order term

The power series we will investigate is the Taylor Series

idea: write a power series that behaves like a function \( f(x) \) of our choice

Taylor series matches the function value and all derivatives at \( x = a \)

so near \( x = a \), Taylor series acts like the real \( f(x) \)

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Taylor series of \( f(x) \)

\[ f(x) = C_0 + C_1(x-a) + C_2(x-a)^2 + C_3(x-a)^3 + C_4(x-a)^4 + \dots \]

match function value at \( x = a \)

\[ f(a) = C_0 + C_1(a-a) + C_2(a-a)^2 + C_3(a-a)^3 + \dots \]\[ \rightarrow \]

\( f(a) = C_0 \)

\( = 0! C_0 \)

now match derivatives at \( x = a \)

\[ f'(x) = C_1 + 2C_2(x-a) + 3C_3(x-a)^2 + 4C_4(x-a)^3 + \dots \]\[ f'(a) = C_1 \rightarrow \]

\( f'(a) = C_1 \)

\( = 1! C_1 \)

\[ f''(x) = 2C_2 + 3 \cdot 2 C_3(x-a) + 4 \cdot 3 C_4(x-a)^2 + \dots \]\[ f''(a) = 2C_2 \rightarrow \]

\( f''(a) = 2C_2 \)

\( = 2! C_2 \)

\[ f'''(x) = 3 \cdot 2 C_3 + 4 \cdot 3 \cdot 2 C_4(x-a) + \dots \]\[ f'''(a) = 3 \cdot 2 C_3 \rightarrow \]

\( f'''(a) = 3 \cdot 2 C_3 \)

\( = 3! C_3 \)

\[ f^{(4)}(x) = 4 \cdot 3 \cdot 2 C_4 + \dots \]\[ f^{(4)}(a) = 4 \cdot 3 \cdot 2 C_4 \rightarrow \]

\( f^{(4)}(a) = 4 \cdot 3 \cdot 2 C_4 \)

\( = 4! C_4 \)

generalize: \( f^{(k)}(a) = k! C_k \rightarrow \)

\[ C_k = \frac{f^{(k)}(a)}{k!} \]

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Taylor series of \( f(x) \) at \( x = a \) is

\[ \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x-a)^k \]

\[ = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!} (x-a)^2 + \frac{f'''(a)}{3!} (x-a)^3 + \frac{f^{(4)}(a)}{4!} (x-a)^4 + \dots \]

if we stop at \( k=1 \) \( \rightarrow \) Linear approximation

\[ f(x) \approx f(a) + f'(a)(x-a) \]

think of Taylor series as an extension of linear approx. more shape features added each \( k \)

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example

Find the Taylor series of \( f(x) = e^x \) at \( x = a \), \( a = 0 \)

\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x-a)^k \]

\[ = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!} (x-a)^2 + \frac{f'''(a)}{3!} (x-a)^3 + \dots \]

\( f(x) = e^x \) at \( a = 0 \)

\( f(0) = e^0 = 1 \) (substitutes into \( f(a) \))

\( f'(x) = e^x \)

\( f'(0) = e^0 = 1 \) (substitutes into \( f'(a) \))

\( f''(x) = e^x \)

\( f''(0) = 1 \) (substitutes into \( f''(a) \))

\( \vdots \)

\( f^{(k)}(0) = 1 \)

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so we get

\[ f(x) = 1 + 1 \cdot (x-0) + \frac{1}{2!} (x-0)^2 + \frac{1}{3!} (x-0)^3 + \dots \]\[ = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \dots \]\[ = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \dots \approx e^x \text{ near } x=0 \]

the more terms we include, the better the approx.

\( k \to \infty \implies \text{series converges to } e^x \)

Taylor Polynomials

If we cut off after \( k \), we get the \( k \text{th-order} \) Taylor Polynomial \( (P_k) \)

\( P_0 = 1 \)

\( P_1 = 1 + x \) (linear approx)

\( P_2 = 1 + x + \frac{x^2}{2} \)

\( P_3 = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} \)

\( \} \approx e^x \text{ near } x=0 \)

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Application: Approximating \( e^{0.5} \)

one use of this: approx. \( e^{0.5} \)

w/o calculator, \( e^{0.5} = ? \)

but \( P_1(x) \approx e^x \approx 1 + x \)

\[ \text{so } e^{0.5} \approx 1 + 0.5 \approx 1.5 \]

(true value is \( e^{0.5} = 1.6487 \dots \))

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Visualizing Taylor Polynomial Approximations

This graph illustrates the approximation of the exponential function \( e^x \) using Taylor polynomials of various orders centered at \( x = 0 \).

\( P_0 \): Constant approximation (horizontal line)
\( P_1 \): Linear approximation (tangent line)
\( P_2 \): Quadratic approximation (parabola)
\( P_3 \): Cubic approximation

The points labeled show the comparison between the true value \( e^{0.5} \) and the approximation from the first-order Taylor polynomial \( P_1 \) at \( x = 0.5 \).

Figure: Coordinate graph showing the curve e^x and its Taylor polynomial approximations P0 through P3.

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Example: Taylor Polynomial Calculation

Find the 4th-order Taylor polynomial of \( f(x) = \cos(2x) \) near \( x = a = \frac{\pi}{8} \).

(so we want a 4th order polynomial that behaves like \( \cos(2x) \) near \( x = \pi/8 \))

\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x-a)^k \quad \text{with } a = \pi/8 \]

Up to \( k = 4 \):

\[ f(x) = f\left(\frac{\pi}{8}\right) + f'\left(\frac{\pi}{8}\right)\left(x-\frac{\pi}{8}\right) + \frac{1}{2!} f''\left(\frac{\pi}{8}\right)\left(x-\frac{\pi}{8}\right)^2 \]\[ + \frac{1}{3!} f'''\left(\frac{\pi}{8}\right)\left(x-\frac{\pi}{8}\right)^3 + \frac{1}{4!} f^{(4)}\left(\frac{\pi}{8}\right)\left(x-\frac{\pi}{8}\right)^4 \]

Derivatives and Evaluations:

\( f(x) = \cos(2x) \rightarrow f\left(\frac{\pi}{8}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)

\( f'(x) = -2 \sin(2x) \rightarrow f'\left(\frac{\pi}{8}\right) = -2 \sin\left(\frac{\pi}{4}\right) = -\sqrt{2} \)

\( f''(x) = -4 \cos(2x) \rightarrow f''\left(\frac{\pi}{8}\right) = -4 \cos\left(\frac{\pi}{4}\right) = -2\sqrt{2} \)

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Taylor Series Expansion of \(\cos(2x)\)

Calculating higher-order derivatives at the point \(x = \frac{\pi}{8}\):

\[ f'''(x) = 8 \sin(2x) \rightarrow f'''\left(\frac{\pi}{8}\right) = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2} \]\[ f^{(4)}(x) = 16 \cos(2x) \rightarrow f^{(4)}\left(\frac{\pi}{8}\right) = 8\sqrt{2} \]

So, near \(x = \frac{\pi}{8}\), \(\cos(2x)\) behaves like:

\[ \underbrace{\frac{\sqrt{2}}{2}}_{f(\frac{\pi}{8})} - \underbrace{\sqrt{2}}_{f'(\frac{\pi}{8})} \left(x - \frac{\pi}{8}\right) - \frac{2\sqrt{2}}{2!} \left(x - \frac{\pi}{8}\right)^2 + \frac{4\sqrt{2}}{3!} \left(x - \frac{\pi}{8}\right)^3 + \frac{8\sqrt{2}}{4!} \left(x - \frac{\pi}{8}\right)^4 \]

Simplifying the coefficients, we obtain the Taylor polynomials \(P_0\) through \(P_4\):

\[ = \underbrace{\frac{\sqrt{2}}{2}}_{P_0} - \sqrt{2} \left(x - \frac{\pi}{8}\right) - \sqrt{2} \left(x - \frac{\pi}{8}\right)^2 + \frac{2\sqrt{2}}{3} \left(x - \frac{\pi}{8}\right)^3 + \frac{\sqrt{2}}{3} \left(x - \frac{\pi}{8}\right)^4 \]

\(P_1\) includes terms up to the first power.

\(P_2\) includes terms up to the second power.

\(P_3\) includes terms up to the third power.

\(P_4\) includes terms up to the fourth power.

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Visualizing Taylor Polynomial Approximations

The following graph illustrates the function \(f(x) = \cos(2x)\) (shown in red) and its various Taylor polynomial approximations centered at \(x = \frac{\pi}{8}\). The horizontal blue line represents the constant approximation \(P_0\), while the green line shows the linear approximation \(P_1\). The dark blue curve represents the quadratic approximation \(P_2\), which closely follows the cosine curve near the center point.

Figure: Coordinate graph showing red curve cos(2x) and polynomial approximations P0, P1, P2 on a grid.

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Exam Review

MON 11/6

6:30 – 8:30 pm

WTHR 200

Figure: A small, round inset image of a fluffy animal, possibly an otter or seal, looking forward.

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