Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \sqrt{103} $$

Step-by-Step Solution

Answer

The approximation of \(\sqrt{103}\) using a second degree Taylor Polynomial is 10.149375.

Step 1: Define the function and its first two derivatives

Let \(f(x) = \sqrt{x}\). The first derivative \(f'(x)\) is \(\frac{1}{2\sqrt{x}}\) and the second derivative \(f''(x)\) is \(-\frac{1}{4x^{3/2}}\).

Step 2: Compute the function and its derivatives at the center point

The center point is \(a = 100\). The function \(f(a) = 10\), the first derivative at the center \(f'(100) = 0.05\), and the second derivative at the center \(f''(100) = -0.00025\).

Step 3: Combine the elements into the Taylor polynomial

The second degree Taylor polynomial is \(T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!} (x - a)^2\). Plugging in the values we calculated, we get \(T_2(x) = 10 + 0.05(x - 100) - 0.000125(x - 100)^2\).

Step 4: Calculate the approximate value

Substitute \(x = 103\) into the second degree Taylor polynomial to get the approximation of the square root of 103, hence \(T_2(103) = 10 + 0.05(3) - 0.000125(3)^2 = 10.149375\).

Key Concepts

Second Degree Taylor Polynomial

A second degree Taylor polynomial offers a way to create an approximation of a function for values near a specific point. The process involves using a function's value and its first two derivatives at a chosen point, termed as the 'center'. To construct this approximation, we require the original function, its first derivative, which provides the slope or rate of change at the center, and its second derivative, which represents the concavity or the rate of change of the slope.

For function \( f(x) \), the approximation around the center \( a \) with a second degree Taylor polynomial can be expressed as: \[ T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!} (x - a)^2 \.\] This approximation becomes more accurate as \( x \) approaches \( a \) and is particularly useful for complex expressions where finding the exact value may be difficult or impossible with standard methods.

Derivative Calculations

Derivative calculations are at the heart of forming Taylor polynomials. They involve finding the rate at which the function's value changes with respect to its input. The first derivative \( f'(x) \) provides the slope of the tangent line at a point on the function, essentially telling us how the function is moving at that point. The second derivative \( f''(x) \) informs us about the curvature of the function, indicating if it's bending upwards or downwards and how quickly.

The calculation of derivatives follows specific rules depending on the function in question. For instance, for the square root function \( f(x) = \sqrt{x} \), the first derivative is \( f'(x) = \frac{1}{2\sqrt{x}} \) and the second derivative is \( f''(x) = -\frac{1}{4x^{3/2}} \). These calulations are crucial for forming the Taylor polynomial that we use for approximations.

Square Root Approximation

Square root approximation is a typical use case for Taylor polynomial approximation, especially when the number under the square root is not a perfect square making it hard to evaluate directly. By picking a 'center' close to the number of interest, and using a second degree Taylor polynomial, we achieve an estimate which is often sufficiently accurate for practical purposes.

By substituting the value we want to approximate into our Taylor polynomial—constructed from a square root function centered around a nearby perfect square—we obtain the approximation of the root without cumbersome calculations. For example, to approximate \( \sqrt{103} \), we choose \( a=100 \) as the center, since \( \sqrt{100} = 10 \) is simple to calculate and is very close to \( 103 \). This approach simplifies a potentially complex calculation to a manageable one.

Calculus

Calculus, the mathematical study of continuous change, is a versatile tool that has applications in multiple fields including physics, engineering, economics, and more. It is divided primarily into two branches: differential calculus and integral calculus. Differential calculus, which encompasses the concept of derivatives and their applications, is used to find rates of change and slopes of curves. Integral calculus, on the other hand deals with accumulations of quantities and the areas under and between curves.

Taylor polynomial approximations are an application of differential calculus, where understanding the behavior of functions through their derivatives enables us to make predictions and approximations. Whether we're aiming to anticipate the movement of celestial bodies or simplify complex square root calculations, calculus gives us the analytical power to work through these problems with precision and insight.