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Magazine Chapter 2: Problem 16
Make a careful sketch of the graph of \(f\) and below it sketch the graph of \(f^{\prime}\) in the same manner as in Exercises \(4-11\) . Can you guess a formula for \(f^{\prime}(x)\) from its graph? \(f(x)=\ln x\)
Short Answer
Expert verified
The derivative of the graph is \(f'(x) = \frac{1}{x}\).
Step by step solution
01
Understand the Function
The function given is \( f(x) = \ln x \). This function is the natural logarithm and is defined for all positive \(x\). It has a vertical asymptote at \(x = 0\) and is continuous and increasing for \(x > 0\).
02
Sketch the Graph of \(f(x) = \ln x\)
To sketch the graph, note that: \(f(x)\) is increasing through all positive \(x\); it passes through the point \((1, 0)\) because \(\ln 1 = 0\); for \(x>1\), \(f(x)\) is positive and for \(0 < x < 1\), \(f(x)\) is negative. Sketch an increasing curve starting from \(-\infty\) as \(x\) approaches \(0^+\) continuing smoothly to positive infinity as \(x\) increases.
03
Calculate the Derivative
The derivative of the function \(f(x) = \ln x\) is \(f'(x) = \frac{1}{x}\) using the basic derivative rule for the natural logarithm.
04
Sketch the Graph of \(f'(x)\)
\(f'(x) = \frac{1}{x}\) is defined for all \(x > 0\). The graph should reflect this: it is a hyperbola confined to the first quadrant as \(x > 0\), decreasing towards zero as \(x\) moves away from zero. The curve should approach the x-axis asymptotically but will never reach it.
05
Analyze the Graph of \(f'(x)\)
Notice that the graph of \(f'(x) = \frac{1}{x}\) is always positive for \(x > 0\) and decreases as \(x\) increases, confirming that \(f(x) = \ln x\) is an increasing function. In the context of the problem, the formula for \(f'(x)\) aligns with what we see in the graph.
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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, denoted as \( \ln x \), is a powerful concept in calculus. It is the inverse operation of taking an exponent of the mathematical constant \( e \), which is approximately 2.71828. When you see \( \ln x \), it simply asks, 'What power must \( e \) be raised to, to get \( x \)?' This makes it very important in various fields such as calculus and complex numbers.
Key features of the natural logarithm function include:
Key features of the natural logarithm function include:
- It is only defined for positive \( x \), meaning \( f(x) = \ln x \) is only valid when \( x > 0 \).
- The natural logarithm has a vertical asymptote at \( x = 0 \). As \( x \) approaches zero from the positive side, \( \ln x \) approaches negative infinity.
- The natural logarithm is continuous and increases steadily for values of \( x \) greater than zero.
- It passes through the point \( (1, 0) \) since \( \ln 1 = 0 \).
Derivative
The derivative of a function is a central concept in calculus that determines the rate at which a function is changing at any given point. For the natural logarithm function \( f(x) = \ln x \), its derivative is computed using a basic rule in calculus: the derivative of \( \ln x \) is \( f'(x) = \frac{1}{x} \).
This derivative, \( \frac{1}{x} \), has distinctive characteristics:
This derivative, \( \frac{1}{x} \), has distinctive characteristics:
- It is defined for all \( x > 0 \), reflecting that \( \ln x \) is also only defined for positive \( x \).
- Unlike the natural logarithm function, the derivative itself does not have a vertical asymptote at \( x = 0 \) since it only exists where \( x > 0 \).
- The derivative is always positive for \( x > 0 \), consistent with the fact that \( \ln x \) is an increasing function.
- It decreases towards zero as \( x \) increases, which implies that the growth rate of \( \ln x \) slows down as \( x \) becomes larger.
Graph Sketching
Graph sketching is an essential skill in calculus that helps you visualize the behavior of functions across their domains. When sketching the graph of \( f(x) = \ln x \), several key points should be considered to achieve an accurate representation.
For \( \ln x \):
For \( \ln x \):
- The graph is only defined for \( x > 0 \).
- Starting near \( x = 0^+ \), the function approaches negative infinity, reflecting its vertical asymptote.
- At \( x = 1 \), the graph passes through the point \( (1, 0) \).
- As \( x \) increases, the graph continues to climb, reflecting the increasing nature of \( \ln x \).
- The curve is hyperbolic and confined to the first quadrant since \( x > 0 \).
- It approaches the x-axis asymptotically, never quite touching it, illustrating a decrease in growth rate.
Asymptote
An asymptote is a line that a graph approaches but never actually touches or crosses. In the context of the function \( f(x) = \ln x \), we encounter a vertical asymptote.
Here's how it appears:
Here's how it appears:
- The vertical asymptote occurs at \( x = 0 \), meaning as \( x \) gets closer to zero from the positive end, \( \ln x \) dives towards negative infinity.
- This is a crucial characteristic of the natural logarithm and greatly influences the graph's shape.
- While \( f'(x) \) also has a vertical asymptote at \( x = 0 \), indicating it cannot touch or cross this line, it has a horizontal asymptotic behavior as it approaches the x-axis as \( x \) increases.
- This tells us that the impact of changes in \( x \) dampens, slowing the growth rate of change in \( \ln x \).
