/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 134 Graph \(f(x)=\tan ^{-1} x\) toge... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 134

Graph \(f(x)=\tan ^{-1} x\) together with its first two derivatives. Comment on the behavior of \(f\) and the shape of its graph in relation to the signs and values of \(f^{\prime}\) and \(f^{\prime \prime}\)

Short Answer

Expert verified
\( f(x) = \tan^{-1}(x) \) shows decreasing slope and changes from concave up to down across zero.

Step by step solution

01

Identify the function and its definition

The function given is \( f(x) = \tan^{-1}(x) \), which is the inverse tangent function. The inverse tangent function takes a real number \( x \) and returns an angle whose tangent is \( x \). Its graph resembles that of an arctangent curve.
02

Determine the first derivative

To find the first derivative \( f'(x) \), we use the derivative of the inverse tangent function:\[ f'(x) = \frac{d}{dx} \tan^{-1}(x) = \frac{1}{1+x^2} \].This derivative indicates the slope of the tangent to the curve at any given point \( x \).
03

Determine the second derivative

To find the second derivative \( f''(x) \), we differentiate \( f'(x) \):\[ f''(x) = \frac{d}{dx} \left( \frac{1}{1+x^2} \right) \].Using the quotient rule or chain rule, we get:\[ f''(x) = -\frac{2x}{(1+x^2)^2} \].This tells us about the concavity of the function.
04

Graph \( f(x) \), \( f'(x) \), and \( f''(x) \)

Plot the graph of \( f(x) = \tan^{-1}(x) \). Note its S-shaped curve approaching horizontal asymptotes at \( y = \frac{\pi}{2} \) as \( x \to \infty \) and \( y = -\frac{\pi}{2} \) as \( x \to -\infty \). Then, plot \( f'(x) = \frac{1}{1+x^2} \), which is positive and decreasing, indicating the slope of \( f \) lessens as \( x \) increases.Finally, plot \( f''(x) = -\frac{2x}{(1+x^2)^2} \). Since this is zero at \( x=0 \), positive for \( x<0 \), and negative for \( x>0 \), the graph of \( f \) is concave down for positive \( x \) and concave up for negative \( x \).
05

Analyze the behavior of \( f \)

The graph of \( f(x) = \tan^{-1}(x) \) starts with a high slope for small \( x \) values, indicating \( f'(x) \) is initially high but decreases as \( x \) increases. Since \( f''(x) \) changes sign, \( f \) transitions from concave up to concave down as \( x \) crosses zero. The asymptotic flattening of \( f \) and the reduction of \( f'(x) \) demonstrates steady leveling out as \( x \to \infty \) or \( x \to -\infty \).

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Key Concepts

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

Derivatives
Understanding derivatives helps us examine how a function changes at each point. For the inverse trigonometric function \( f(x) = \tan^{-1}(x) \), its first derivative is crucial in analyzing the graph's behavior. The first derivative \( f'(x) = \frac{1}{1+x^2} \) represents the slope of the tangent line to the curve at any given point. Here, because the denominator \( 1+x^2 \) is always positive, \( f'(x) \) is always positive too. This tells us that \( f(x) \) is a strictly increasing function.
  • As \( x \) becomes large in magnitude, \( f'(x) \) approaches zero. The slope of \( f(x) \) diminishes, meaning the curve flattens out.
  • Derivatives give insights into the growth rate of the function.
In summary, the first derivative is a tool to reveal how quickly or slowly a function's value changes as \( x \) varies.
Concavity
Concavity provides information about the curvature of a graph. For our inverse tangent function, using its second derivative \( f''(x) = -\frac{2x}{(1+x^2)^2} \) helps us determine its concavity. The sign of \( f''(x) \) tells us about the direction of the curve's bending:
  • When \( x < 0 \), \( f''(x) > 0 \), indicating that the function is concave up鈥攍ike a smile.
  • Conversely, when \( x > 0 \), \( f''(x) < 0 \), meaning the function is concave down鈥攍ike a frown.
  • At \( x = 0 \), the change in concavity suggests that the curve transitions smoothly.
Concavity analysis is vital as it helps predict the direction of a function's curve and can indicate possible inflection points, where the curve changes from stretching up to bending down.
Asymptotes
Asymptotes are theoretical lines that a graph approaches but never actually reaches. For the inverse tangent function \( f(x) = \tan^{-1}(x) \), horizontal asymptotes are crucial in understanding behavior at extreme values of \( x \).
  • As \( x \rightarrow \infty \), \( f(x) \) approaches the horizontal asymptote \( y = \frac{\pi}{2} \).
  • Likewise, as \( x \rightarrow -\infty \), \( f(x) \) approaches \( y = -\frac{\pi}{2} \).
  • These asymptotes indicate that the values of the function have definitive bounds, constrained within \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).
Understanding asymptotes allows one to see where a function will go infinitely large or small, giving insights into its end behavior. For \( \tan^{-1}(x) \), these asymptotes dictate the flow and the limiting behavior of the function's graph as it extends towards infinity.

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