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Chapter 1: Problem 29

If \(f(x)=x^{1 / 3}\), compute \(f(8)\) and \(f^{\prime}(8)\).

Short Answer

Expert verified
f(8) = 2, f^{\text{'}}(8) = \frac{1}{12}.

Step by step solution

01

Identify the Function

The given function is \(f(x) = x^{1/3}\).
02

Compute \(f(8)\)

Substitute \(x = 8\) in the function.\[f(8) = 8^{1/3} = 2\]. So, \(f(8) = 2\).
03

Find the Derivative of the Function

To find the derivative of \(f(x) = x^{1/3}\), use the power rule for differentiation. \[f'(x) = \frac{d}{dx}(x^{1/3}) = \frac{1}{3} x^{-2/3}\]
04

Compute \(f^{\text{'}}(8)\)

Substitute \(x = 8\) in the derivative function.\[f^{\text{'}}(8) = \frac{1}{3} (8)^{-2/3}\].\[8^{-2/3} = (8^{1/3})^{-2} = 2^{-2} = \frac{1}{4}\]. Thus, \[f^{\text{'}}(8) = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}\].

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Key Concepts

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

Power Rule
The power rule is a basic rule in differentiation used to find the derivative of a function of the form \(f(x) = x^n\). According to the power rule, if you have a function \(x^n\) where \(n\) is any real number, the derivative of this function is obtained by bringing the exponent down as a coefficient and then subtracting one from the exponent. So the derivative of \(x^n\) is \(nx^{n-1}\).

In our exercise, we have the function \(f(x) = x^{1/3}\). Applying the power rule here involves:
  • Bringing down the exponent \(1/3\) as a coefficient
  • Subtracting one from the exponent
This means the derivative of \(x^{1/3}\) is \(\frac{1}{3}x^{-2/3}\). Essentially, you apply the power rule step-by-step to transform the function into its derivative form.
Function Evaluation
Function evaluation is the process of finding the value of a function at a particular point. For example, to evaluate \(f(x)\) at \(x=8\), you simply substitute 8 for every instance of \(x\) in the function.

In this exercise, the function given is \(f(x) = x^{1/3}\). To compute \(f(8)\), we substitute 8 for \(x\):
  • \(f(8) = 8^{1/3}\)
  • \(8^{1/3}\) simplifies to 2 (since 2 multiplied by itself three times equals 8)
  • So, \(f(8) = 2\)
Evaluating functions at specific points helps in understanding their behavior at those points. It is a straightforward procedure that involves substitution and arithmetic simplification.
Derivative Calculation
Calculating derivatives involves finding the rate at which a function changes at any given point. In this problem, once the derivative \(f'(x)\) was found using the power rule, it was evaluated at a specific point.

Let's walk through this calculation step:
  • First, we apply the power rule to get \(f'(x) = \frac{1}{3}x^{-2/3}\)
  • Then, to compute \(f'(8)\), we substitute \(x = 8\) into the derivative
This leads to:
  • \(f'(8) = \frac{1}{3} \cdot 8^{-2/3}\)
  • To simplify \(8^{-2/3}\), consider it as \((8^{1/3})^{-2}\)
  • Since \(8^{1/3} = 2\), then \(8^{-2/3} = 2^{-2} = \frac{1}{4}\)
  • Finally, \(f'(8) = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}\)
Derivative calculation helps in analyzing the slope and rate of change of the original function at any point. This is crucial for many applications in math and science, including physics, engineering, and economics.

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Most popular questions from this chapter

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=\frac{x-1}{3}\)

Apply the three-step method to compute the derivative of the given function. \(f(x)=7 x^{2}+x-1\)

Compute the following limits. \(\lim _{x \rightarrow-\infty} \frac{1}{x^{2}}\)

Compute the following. \(\frac{d}{d t}\left(\frac{d v}{d t}\right)\), where \(v=2 t^{2}+\frac{1}{t+1}\)

Each limit in Exercises 49-54 is a definition of \(f^{\prime}(a)\). Determine the function \(f(x)\) and the value of \(a\). \(\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h}\)
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