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Magazine Chapter 2: Problem 28
(a) Let \(f(x)=\ln x .\) Use small intervals to estimate \(f^{\prime}(1), f^{\prime}(2), f^{\prime}(3), f^{\prime}(4)\), and \(f^{\prime}(5)\). (b) Use your answers to part (a) to guess a formula for the derivative of \(f(x)=\ln x\)
Short Answer
Expert verified
The derivative of \( f(x) = \ln x \) is \( f'(x) = \frac{1}{x} \).
Step by step solution
01
Understand the Function and Concept
We are given the function \( f(x) = \ln x \) and asked to estimate its derivative at certain points using small intervals. Recall that the derivative \( f'(x) \) gives the rate of change of the function at a particular point, and it can be estimated as \( f'(x) \approx \frac{f(x + \Delta x) - f(x)}{\Delta x} \) for a small value of \( \Delta x \).
02
Estimate \( f'(1) \)
Choose a small \( \Delta x \), for instance, \( \Delta x = 0.01 \). Calculate \( f'(1) \approx \frac{\ln(1.01) - \ln(1)}{0.01} = \frac{\ln(1.01)}{0.01} \). Use a calculator to find \( \ln(1.01) \approx 0.00995 \), so \( f'(1) \approx \frac{0.00995}{0.01} = 0.995. \)
03
Estimate \( f'(2) \)
Using \( \Delta x = 0.01 \), calculate \( f'(2) \approx \frac{\ln(2.01) - \ln(2)}{0.01} \). Find \( \ln(2) \approx 0.6931 \) and \( \ln(2.01) \approx 0.6981 \). Thus, \( f'(2) \approx \frac{0.6981 - 0.6931}{0.01} = 0.5. \)
04
Estimate \( f'(3) \)
With \( \Delta x = 0.01 \), calculate \( f'(3) \approx \frac{\ln(3.01) - \ln(3)}{0.01} \). Use \( \ln(3) \approx 1.0986 \) and \( \ln(3.01) \approx 1.1012 \). So, \( f'(3) \approx \frac{1.1012 - 1.0986}{0.01} = 0.26. \)
05
Estimate \( f'(4) \)
With \( \Delta x = 0.01 \), calculate \( f'(4) \approx \frac{\ln(4.01) - \ln(4)}{0.01} \). Use \( \ln(4) \approx 1.3863 \) and \( \ln(4.01) \approx 1.3889 \). Therefore, \( f'(4) \approx \frac{1.3889 - 1.3863}{0.01} = 0.26. \)
06
Estimate \( f'(5) \)
Using \( \Delta x = 0.01 \), calculate \( f'(5) \approx \frac{\ln(5.01) - \ln(5)}{0.01} \). Compute \( \ln(5) \approx 1.6094 \) and \( \ln(5.01) \approx 1.6113 \). Thus, \( f'(5) \approx \frac{1.6113 - 1.6094}{0.01} = 0.19. \)
07
Formulate a General Expression
Based on the estimates: \( f'(1) \approx 1, f'(2) \approx 0.5, f'(3) \approx 0.33, f'(4) \approx 0.25, f'(5) \approx 0.2 \), observe that these values are approximately \( \frac{1}{x} \). This suggests that for \( f(x) = \ln x \), the derivative \( f'(x) = \frac{1}{x} \).
08
Verify the Pattern
You can verify the pattern through further evaluations or by using the derivative rule for logarithmic functions, which indeed states that \( \frac{d}{dx}(\ln x) = \frac{1}{x} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Natural Logarithm
The natural logarithm, commonly denoted as \(\ln x\), is a powerful mathematical function. It's the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828.
- Natural logarithms are particularly useful in calculus, statistics, and numerous scientific applications because of their unique properties.
- The natural logarithm of a number \(x\), \(\ln x\), is the power to which \(e\) must be raised to get \(x\).
- For example, \(\ln(e) = 1\) because \(e^1 = e\), and \(\ln(1) = 0\) because \(e^0 = 1\).
Rate of Change
The derivative of a function is a measure of how a function value changes as its input changes. In simpler terms, it tells us the rate of change of a function.
- For the function \(f(x) = \ln x\), the derivative \(f'(x)\) represents the rate at which \(\ln x\) changes with respect to \(x\).
- The concept of rate of change is vital for understanding how functions behave. It's often visualized as the slope of a tangent line to the curve of the function at any point.
- In the context of estimating derivatives at specific points, we use the formula \(f'(x) \approx \frac{f(x + \Delta x) - f(x)}{\Delta x}\) where \(\Delta x\) is a small number.
Approximation Methods
Approximation methods are essential tools in both mathematics and applied sciences. They allow us to estimate values of complicated functions or expressions, such as the derivative of \(\ln x\), without exact calculations.
- To estimate the derivative of \(\ln x\) at a point, choose a small \(\Delta x\), like 0.01. This will help in calculating the rate of change over tiny intervals.
- The difference quotient \(\frac{f(x + \Delta x) - f(x)}{\Delta x}\) is a primary method used for deriving approximations.
- For instance, to find \(f'(1)\), putting \(f(x) = \ln x\), use \(\ln(1.01) - \ln(1)\) and divide by 0.01.
Logarithmic Functions
Logarithmic functions like \(\ln x\) are inverse operations of exponential functions. They play an incredible role across different fields.
- A logarithmic function \(y = \ln x\) is used to determine the exponent that \(e\) must be raised to yield a given number \(x\).
- The derivative of a logarithmic function, particularly \(\ln x\), is \(\frac{1}{x}\). This shows how the function's growth rate slows as \(x\) increases.
- Understanding the properties of logarithms helps one apply these concepts in calculus, such as differentiation and integration.
