Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$e^{0.06}$$

Short answer

Question: Use linear approximations to estimate the value of $$e^{0.06}$$. Answer: $$e^{0.06}$$ can be approximated as $$1.06$$ using linear approximations.

Step-by-step solution

Calculate the first-degree Taylor Polynomial of $$e^x$$

Recall that the first-degree Taylor polynomial for a function $$f(x)$$ about the point $$a$$ is given by: $$P_1(x) = f(a) + f'(a)(x-a)$$. For the exponential function $$e^x$$, the derivative is also $$e^x$$. Now, we can create the first-degree Taylor approximation of $$e^x$$ about the point $$a=0$$: $$P_1(x) = e^0 + e^0(x-0) = 1 + x$$

Use the approximation to estimate $$e^{0.06}$$

Replace $$x$$ with $$0.06$$ in the Taylor approximation: $$e^{0.06} \approx P_1(0.06) = 1+0.06$$

Simplify and write the approximate result

Simplify the expression: $$1+0.06 = 1.06$$ So, using linear approximation, we have: $$e^{0.06} \approx 1.06$$

Key concepts

Taylor Polynomial

The Taylor polynomial is an important tool for approximating certain functions, making it easier to work with them. Imagine a scenario where you want to approximate a complex function using a simpler polynomial. This is done by calculating a Taylor polynomial. A Taylor polynomial of degree n for a function f(x) is a polynomial that matches the function's value at a particular point, as well as its first n derivatives. The general formula for a Taylor polynomial of degree n, around a point a, is:\[P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n\]This formula provides a basis for creating an infinite series when n is infinity, known as a Taylor series. However, in many cases, we only need a first-degree or linear approximation, which simplifies many calculations.

• Utilizing smaller degree polynomials simplifies complex calculations.
• Perfect for functions with difficult differentiation or integration.
• Provides better understanding through approximations.

Exponential Function

An exponential function, like \(e^x\), is one of the fundamental functions in mathematics, especially known for its unique properties. These functions are crucial in modeling many natural phenomena, such as population growth and radioactive decay. The base of this function, e, is a special number approximately equal to 2.71828, and it was discovered due to its intriguing property of being its own derivative. This means:- Derivative of \(e^x\) is \(e^x\) itself, making it easy to differentiate.- Perfectly suited for growth and decay models.- Has applications in calculus, complex analysis, and financial mathematics.Knowing this property simplifies the process of finding approximations using Taylor polynomials. For instance, when approximating \(e^x\) near the point x = 0, things become much simpler as both the function and its derivative are equal to 1.

First-Degree Polynomial

A first-degree polynomial is also known as a linear polynomial. It's the simplest form of polynomial and can be easily identified by having only one term with a variable raised to the power of one. Its general form is:\[ P_1(x) = ax + b \]Compared to higher-degree polynomials, first-degree polynomials are straightforward and represent linear relationships without the effects of curves. They appear in various applications:

• Ideal for creating linear models and estimations.
• Frequently used in regression analysis.
• The linear approximation of complex functions simplifies initial estimates.

In the context of linear approximation, a first-degree Taylor polynomial can provide a quick and remarkably accurate estimate of a function, especially when the point of approximation is very close to the function value. Using this approach can transform complex calculations into manageable steps, like approximating an exponential function such as \(e^{0.06}\) simply as \(1 + 0.06\).