Question

Errors in approximations Suppose \(f(x)=1 /(x+1)\) is to be approximated near \(x=0 .\) Find the linear approximation to \(f\) at 0 Then complete the following table, showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is 100 lapproximation \(-\) exact \(|/|\) exact \(| .\) Comment on the behavior of the errors as \(x\) approaches 0 \begin{array}{|l|l|l|l|} \hline x & \text { Linear approx. } & \text { Exact value } & \text { Percent error } \\ \hline 0.1 & & & \\ \hline 0.01 & & & \\ \hline 0.001 & & & \\ \hline 0.0001 & & & \\ \hline-0.0001 & & & \\ \hline-0.001 & & & \\ \hline-0.01 & & & \\ \hline-0.1 & & & \\ \hline \end{array}

Short answer

Answer: As x approaches 0, the percent error in the linear approximation decreases, indicating that the approximation is more accurate near \(x=0\).

Step-by-step solution

Calculate the derivative of the given function

To derive \(f(x) = \frac{1}{x+1}\), we'll use the power rule. First, rewrite the function as \(f(x) = (x+1)^{-1}\). Now, apply the power rule: \(f'(x) = -1(x+1)^{-2}\)

Find the linear approximation near \(x=0\)

Now, we'll use the formula for the linear approximation: \(L(x)=f(0)+f'(0)(x-0)\) Substitute the function and its derivative into the formula: \(L(x) = (0+1)^{-1} - 1(0+1)^{-2}(x-0)\) Simplify the equation: \(L(x) = 1 - x\)

Complete the table

We'll now evaluate the linear approximation \(L(x)\), the exact value \(f(x)\), and the percent error for each given value of \(x\). Percent error formula: \(\text{Percent error} = 100 \times \left|\frac{\text{Approximation} - \text{Exact}}{\text{Exact}}\right|\) \begin{array}{|l|l|l|l|} \hline x & \text { Linear approx. } & \text { Exact value } & \text { Percent error } \\ \hline 0.1 & L(0.1)=1-0.1=0.9 & f(0.1)=\frac{1}{0.1+1}=\frac{1}{1.1}\approx0.9091 & \approx1.01 \% \\ \hline 0.01 & L(0.01)=1-0.01=0.99 & f(0.01)=\frac{1}{0.01+1}\approx0.9901 & \approx0.01 \% \\ \hline 0.001 & L(0.001)=0.999 & f(0.001)=\frac{1}{0.001+1}\approx0.9990 & \approx0.001 \% \\ \hline 0.0001 & L(0.0001)=0.9999 & f(0.0001)\approx0.99990 & \approx0.0001 \% \\ \hline -0.0001 & L(-0.0001)=1.0001 & f(-0.0001)\approx1.00010 & \approx0.0001 \% \\ \hline -0.001 & L(-0.001)=1.001 & f(-0.001)\approx1.00100 & \approx0.001 \% \\ \hline -0.01 & L(-0.01)=1.01 & f(-0.01)\approx1.01010 & \approx0.01 \% \\ \hline -0.1 & L(-0.1)=1.1 & f(-0.1)=\frac{1}{-0.1+1}=1.1111 & \approx1.01 \% \\ \hline \end{array}

Comment on the behavior of the errors as \(x\) approaches 0

From the table, we can observe that the percent error decreases as \(x\) approaches 0. For both positive and negative values of \(x\), the percent error becomes smaller when \(x\) is closer to zero. This indicates that the linear approximation is more accurate near \(x=0\).

Key concepts

Derivative

When it comes to calculus, one of the most fundamental concepts is the derivative. In essence, the derivative of a function at a particular point measures how the function's value changes as its input changes. Think of the derivative as the instantaneous rate of change or the slope of the function at a specific point.

To find the derivative of a function, one can employ various rules, with the power rule being one of the most common and easiest to use. The power rule states that if you have a function of the form \( f(x) = x^n \), where \( n \) is a real number, the derivative is given by \( f'(x) = nx^{n-1} \). In our exercise, we applied the power rule to \( f(x) = (x+1)^{-1} \), leading to its derivative \( f'(x) = -1(x+1)^{-2} \). This derivative is crucial for several applications, including finding the linear approximation of the function which we'll explore next.

Percent Error

Percent error is an essential concept in not just calculus, but in all of science and engineering. It is used to measure how accurate an approximation or measurement is when compared to the exact or accepted value. The percent error is the absolute difference between the approximate value and the exact value, divided by the exact value, all multiplied by 100 to get a percentage. The formula looks like this: \( \text{Percent error} = 100 \times \left|\frac{\text{Approximation} - \text{Exact}}{\text{Exact}}\right| \).

In our problem, we calculated the percent error for several approximations of \( f(x) \) as \( x \) approaches zero. The exercise shows that the percent error decreases as \( x \) gets closer to zero, indicating that the linear approximation becomes increasingly accurate near the point of tangency, which is at \( x=0 \) in this case.

Calculus

Calculus is a branch of mathematics that studies continuous change and consists of two major parts: differential calculus and integral calculus. Differential calculus, which involves the concept of the derivative, focuses on rates of change and slopes of curves. Meanwhile, integral calculus deals with accumulation of quantities and the areas under and between curves.

In the context of our exercise, we employed differential calculus to approximate the value of a function near a given point and to analyze the error in this approximation. Calculus not only helps us to solve practical problems but also provides a framework for understanding patterns, systems, and changes in the real world. It's the language of engineers, scientists, and economists, allowing them to model and predict behaviors and outcomes meticulously.

Power Rule

The power rule is a quick and handy method within calculus used to differentiate functions of the form \( x^n \). As mentioned earlier, the rule states that the derivative of \( x^n \) is \( nx^{n-1} \). This process is vitally important because it allows us to find the rate of change of a function at any point, which is one of the primary goals of differential calculus.

In practical terms, once you know the derivative of a function, you can use it for many things, such as finding where a function is increasing or decreasing, where it has extremums, or creating linear approximations as we have seen in our exercise. When using the power rule, always remember to multiply by the current power and then reduce the power by one. It's a powerful tool that makes finding derivatives straightforward for polynomial functions.