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Chapter 4: Problem 61

Suppose \(f(x)=\sqrt[3]{x}\) is to be approximated near \(x=8 .\) Find the linear approximation to \(f\) at 8 Then complete the following table, showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is \(100 |\) approximation \(-\) exact \(|/|\) exact \(| .\) Comment on the behavior of the errors as \(x\) approaches 8.

Short Answer

Expert verified
Answer: The linear approximation of the function \(f(x) = \sqrt[3]{x}\) at \(x=8\) is given by the equation \(y = \frac{1}{12}(x - 8) + 2\). As \(x\) approaches 8, the errors between the linear approximation and the exact values of the function decrease, indicating that the linear approximation is quite accurate for values close to \(x=8\).

Step by step solution

01

Find the derivative of the function

To find the derivative of the function \(f(x) = \sqrt[3]{x}\), we use the power rule. The power rule states that if \(f(x) = x^n\), then \(f'(x) = n x^{n-1}\). In this case, we can rewrite the function in the form of a power: \(f(x) = x^{\frac{1}{3}}\). Using the power rule, we find the derivative: $$ f'(x) = \frac{1}{3}x^{\frac{1}{3} - 1} = \frac{1}{3}x^{-\frac{2}{3}} $$
02

Calculate the tangent line equation at \(x=8\)

Now, we'll use the tangent line equation \(y = f'(a)(x - a) + f(a)\) at \(x=8\). Let \(a = 8\). First, let's find the value of the function and its derivative at \(a\): $$ f(8) = \sqrt[3]{8} = 2 \\ f'(8) = \frac{1}{3} \cdot 8^{-\frac{2}{3}} = \frac{1}{12} $$ Now plug these values into the tangent line equation: $$ y = \frac{1}{12}(x - 8) + 2 $$
03

Calculate the errors

We now have our linear approximation to \(f(x)\). To calculate the errors at various values of \(x\), we first determine the exact value of \(f(x)\) and then find the value of the approximation by substituting the value of \(x\) into the linear approximation (\(y\)). Afterward, we calculate the percent error using the formula provided in the exercise. Specific values to calculate the error at will depend on the table that you're asked to fill out. For illustration purposes, we'll consider \(x = 7\) and \(x = 9\).
04

Example: Calculation of error at \(x = 7\)

1. Calculate the exact value: \(f(7) = \sqrt[3]{7} \approx 1.9129\) 2. Calculate the approximation: \(y = \frac{1}{12}(7 - 8) + 2 \approx 1.9167\) 3. Calculate the percent error: \(\frac{100 |1.9167 - 1.9129|}{|1.9129|} \approx 0.198 \%\)
05

Example: Calculation of error at \(x = 9\)

1. Calculate the exact value: \(f(9) = \sqrt[3]{9} \approx 2.0801\) 2. Calculate the approximation: \(y = \frac{1}{12}(9 - 8) + 2 \approx 2.0833\) 3. Calculate the percent error: \(\frac{100 |2.0833 - 2.0801|}{|2.0801|} \approx 0.153 \%\) Calculate the errors for the rest of the values in the table following the same procedure.
06

Comment on the behavior of the errors as \(x\) approaches \(8\)

As \(x\) approaches 8, the errors between the linear approximation and the exact values of the function should decrease. This is because the linear approximation is the best local approximation at the point \(x=8\), and as \(x\) gets closer to 8, the tangent line becomes a better approximation. In the examples provided, we observed that the percent errors were relatively small, which indicates that the linear approximation is quite accurate for values close to \(x=8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is an essential branch of mathematics focused on the study of rates of change and accumulation. It鈥檚 divided into two main parts: differential calculus and integral calculus. Differential calculus centers on the derivative concept, which measures how a function changes as its input changes. Integral calculus, on the other hand, focuses on the accumulation of quantities, such as areas under curves, through integration. Calculus enables us to solve complex problems in engineering, economics, physics, and more by giving us the tools to model and analyze dynamic systems.
Derivative
In calculus, the derivative represents the rate at which a function's output changes as its input changes. Technically, it is the limit of the average rate of change of the function over a particular interval, as that interval shrinks to zero. For example, if we have a function that represents the distance a car travels over time, the derivative of that function at any given point in time would tell us the car's speed at that very moment. Derivatives are fundamental in finding local extrema, analyzing the behavior of functions, and in approximating functions using linear and higher-order polynomials.
Power Rule
The Power Rule is a shortcut in finding the derivative of a function when the function is a power of x, stated as \( f(x) = x^n \). The rule is simple: if \( n \) is a real number, the derivative of \( f(x) = x^n \) with respect to \( x \) is \( f'(x) = n \times x^{n-1} \). For instance, if \( f(x) = x^3 \), then according to the Power Rule, \( f'(x) = 3x^2 \). This rule substantially simplifies the process of differentiation and is a fundamental tool used for more complex calculus problems.
Tangent Line Equation
The tangent line equation at a specific point on a curve provides the best linear approximation of the curve at that point. It's computed using both the function's value and its derivative at that point. The equation for the tangent line to a function \( f \) at a point \( a \) is given by \( y = f'(a)(x - a) + f(a) \). This line touches the curve at only one point when the curve is smooth and without corners, and it has the same slope as the curve at that point. Using the tangent line, we can approximate the value of the function near the point \( a \), which is especially useful when calculating exact values is difficult or when we seek a quick estimate.
Approximation Error
Approximation error is the difference between the approximate value and the exact value of a function. When we use methods like linear approximation, we accept that there will be some error in our estimate versus the true value. In the context of our current problem, the approximation error is the difference between the value given by the tangent line equation, which serves as our approximation, and the actual value of the function \( f(x) \). Understanding approximation error is crucial because it helps us determine the effectiveness and limitations of our approximations.
Percent Error
Percent error is a measure that compares the size of the approximation error to the exact value, expressed as a percentage. It is a useful way to understand the significance of the approximation error in relation to the true value. Calculated as \( 100 \times \frac{|\text{approximation} - \text{exact value}|}{|\text{exact value}|} \), it's a relative measure which lets us compare the accuracy of different approximations regardless of the scale of the values being compared. A small percent error indicates a highly accurate approximation, which is particularly insightful when refining scientific measurements or evaluating different approximation methods.

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