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) = cos x, a = π/4; approximate cos (0.24π)

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Step 1: Identify the function and the center point. Here, the function is \(f(x) = \cos x\) and the center point is \(a = \frac{\pi}{4}\).

Step 2: Find the value of the function and its derivatives at the center point \(a\). Calculate \(f(a) = \cos\left(\frac{\pi}{4}\right)\), the first derivative \(f'(x) = -\sin x\) and evaluate \(f'(a) = -\sin\left(\frac{\pi}{4}\right)\), and the second derivative \(f''(x) = -\cos x\) and evaluate \(f''(a) = -\cos\left(\frac{\pi}{4}\right)\).

Step 3: Write the linear approximating polynomial (the first-degree Taylor polynomial) centered at \(a\): \(L(x) = f(a) + f'(a)(x - a)\).

Step 4: Write the quadratic approximating polynomial (the second-degree Taylor polynomial) centered at \(a\): \(Q(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2\).

Step 5: Use the polynomials \(L(x)\) and \(Q(x)\) to approximate \(\cos(0.24\pi)\) by substituting \(x = 0.24\pi\) into each polynomial.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Linear Approximation (Tangent Line Approximation)

Linear approximation uses the tangent line at a point to estimate function values near that point. It is given by L(x) = f(a) + f'(a)(x - a), where f'(a) is the derivative at a. This method provides a simple, first-order estimate of the function close to a.

Quadratic Approximation (Second-Order Taylor Polynomial)

Quadratic approximation improves accuracy by including the second derivative term. The polynomial is Q(x) = f(a) + f'(a)(x - a) + (f''(a)/2)(x - a)^2, capturing curvature near a. It provides a better estimate than linear approximation for values near the center point.

Using Approximations to Estimate Function Values

Once linear and quadratic polynomials are found, they can approximate function values at points near a. This avoids complex calculations of the original function, especially for transcendental functions like cosine. Comparing both approximations shows the improvement in accuracy.