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.

f(x)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².

Accepted Answer

Identify the function and the point of approximation: here, the function is $$f(x) = e$$^{-2x}$$ and the center point is a = 0$. Find the value of the function at a$: calculate f(0$$) = $$e^{-2 \cdot 0}$$ = $$e^0 = 1$. Compute the first derivative f$$'(x)$$ and evaluate it at a$: since f(x$$) = $$e^{-2x}$$, then $$f'(x) = -2 e$$^{-2x}$$, so f$$'(0) = -2 $$e^0 = -2$. For the linear approximation (first-degree Taylor polynomial) centered at a=0$, use the formula: L(x) = f(a) + f$$'(a)(x - a)$$, which becomes L(x) = 1 - 2(x - 0) = 1 - 2x$. For the quadratic approximation (second-degree Taylor polynomial), compute the second derivative f$$''(x)$$ and evaluate at a$: f$$''(x) = 4 $$e^{-2x}$$, so $$f''(0) = 4. $$Then use the formula $$Q(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2$, which becomes Q(x) = 1 - 2x$$ + $$2x^2. $$Finally, substitute $$x = 0.1 ($$since $$e^{-0.2}$$ = $$e^{-2 \cdot 0.1}) $$into both $$L(x)$$ and $$Q(x) to $$approximate $$e^{-0.2}.$$

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Key Concepts

Linear Approximation (Linearization)

Quadratic Approximation (Second-Order Taylor Polynomial)

Using Approximations to Estimate Function Values