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) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)

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Step 1: Identify the function and the point of approximation. Here, the function is \(f(x) = 8x^{\frac{3}{2}}\) and the center point is \(a = 1\).

Step 2: Compute the value of the function at \(a\): calculate \(f(1) = 8 \times 1^{\frac{3}{2}}\).

Step 3: Find the first derivative of the function, \(f'(x)\). Use the power rule: \(f'(x) = 8 \times \frac{3}{2} x^{\frac{3}{2} - 1} = 12 x^{\frac{1}{2}}\).

Step 4: Evaluate the first derivative at \(a\): calculate \(f'(1) = 12 \times 1^{\frac{1}{2}}\).

Step 5: Write the linear approximating polynomial (the linearization) centered at \(a\): \(L(x) = f(a) + f'(a)(x - a)\).

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주요 개념

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

Linear Approximation (Linearization)

Linear approximation uses the tangent line at a point to estimate the value of a function 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 simplifies complex functions to linear ones for easier calculation of nearby values.

Quadratic Approximation (Second-Order Taylor Polynomial)

Quadratic approximation extends linear approximation by including the second derivative, providing a better estimate near the point a. It is expressed as Q(x) = f(a) + f'(a)(x - a) + (f''(a)/2)(x - a)^2, capturing curvature and improving accuracy over linearization.

Derivative and Higher-Order Derivatives

Derivatives measure the rate of change of a function and are essential for constructing approximations. The first derivative gives the slope for linear approximation, while the second derivative indicates concavity, crucial for quadratic approximation. Calculating these derivatives at the point a is key to forming the approximating polynomials.