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Chapter 4: Problem 28

\(24-29\) Sketch the graph of a function that satisfies all of the given conditions. $$f^{\prime}(x)>0 \text { if }|x|<2, \quad f^{\prime}(x)<0 \text { if }|x|>2$$ $$f^{\prime}(2)=0, \quad \lim _{x \rightarrow \infty} f(x)=1, \quad f(-x)=-f(x)$$ $$f^{\prime \prime}(x) < 0 \text { if } 0 < x < 3, \quad f^{\prime \prime}(x) > 0 \text { if } x > 3$$

Short Answer

Expert verified
Sketch an odd function that has a maximum at \( x = 2 \) and approaches \( y = 1 \) as \( x \to \infty \).

Step by step solution

01

Analyze Derivative Sign

The problem states that \( f'(x) > 0 \) when \( |x| < 2 \), meaning the function \( f(x) \) is increasing on the interval \((-2, 2)\). Also, \( f'(x) < 0 \) when \( |x| > 2 \), so \( f(x) \) is decreasing when \( x < -2 \) and \( x > 2 \). Furthermore, at \( x = 2 \), \( f'(2) = 0 \), indicating a critical point at \( x = 2 \).
02

Investigate Limits and Symmetry

The limit \( \lim_{x \to \infty} f(x) = 1 \) indicates that the function approaches the horizontal line \( y = 1 \) as \( x \to \infty \). The condition \( f(-x) = -f(x) \) tells us that \( f \) is an odd function, implying symmetry about the origin.
03

Examine Concavity

The condition \( f''(x) < 0 \) for \( 0 < x < 3 \) means \( f(x) \) is concave down in this interval. For \( x > 3 \), since \( f''(x) > 0 \), \( f(x) \) is concave up.
04

Combine Information and Sketch

The function is increasing and concave down from \( x = 0 \) to \( x = 2 \) with a maximum point at \( x = 2 \). It has a local maximum at \( x = 2 \) and then starts decreasing as it approaches infinity (concave up after \( x = 3 \)). It will mirror these behaviors in the negative x-direction due to symmetry, leading to a decreasing concave up portion from \( x = -3 \) to \( x = -2 \) and an increasing concave down portion on \( x = -2 \) to \( x = 0 \).

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Key Concepts

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

Derivative
To start understanding function sketching, let's delve into derivatives. When we talk about the derivative, we refer to the rate at which the function's value changes with respect to change in input. Mathematically, the derivative at a point gives us the slope of the tangent to the function at that point. Here:
  • The condition \( f'(x) > 0 \) for \( |x| < 2 \) indicates that the function is increasing when \( -2 < x < 2 \).
  • For \( |x| > 2 \), \( f'(x) < 0 \) means the function decreases.
  • At \( x = 2 \), \( f'(2) = 0 \) shows there's a critical point; the slope is flat.
In summary, understanding derivatives is crucial for determining where functions increase or decrease and identifying potential critical points.
Critical Points
Critical points are where the derivative equals zero or is undefined. At these points, the function may have local extrema (maxima or minima) or even inflection points. In this exercise:
  • Since \( f'(2) = 0 \), the function levels off at \( x = 2 \), pointing to a critical point.
This is significant because critical points could represent peak values or valleys in the graph, aiding in sketching the function's general shape. Recognizing these points is essential as they might contribute to understanding the overall structure and behavior of the function as it progresses along the domain.
Concavity
Concavity informs us about the "curvature" of the graph. It tells us whether the function bends upwards or downwards within certain intervals. The concavity of a function is determined by its second derivative. For this function:
  • \( f''(x) < 0 \) for \( 0 < x < 3 \) tells us that the graph is concave down in this interval. The curve bends downwards like a frown.
  • Conversely, \( f''(x) > 0 \) for \( x > 3 \) indicates that the graph is concave up, so it bends upwards like a smile.
Concavity helps in determining where a function has inflection points, where the curvature changes from concave up to concave down or vice versa, giving specific insight into how the graph flows through these regions.
Limits
Limits are a foundational concept in calculus that describe the behavior of a function as it approaches a particular value. Here, the limit \( \lim_{x \to \infty} f(x) = 1 \) means that as \( x \) moves towards infinity, the function \( f(x) \) gets infinitely close to the horizontal line \( y = 1 \). This is crucial because:
  • It suggests the presence of a horizontal asymptote at \( y = 1 \).
  • This behavior needs to be reflected in your graph as \( x \) progresses towards infinity, helping shape the tail end of the function.
Understanding limits and their implications on function behavior assist greatly in sketching the accurate and complete picture of the function across its domain.
Symmetry
Symmetry in functions can simplify understanding and graphing complex functions. The condition \( f(-x) = -f(x) \) reveals a special symmetry: odd symmetry, which implies the graph is symmetric about the origin. This is visually significant because:
  • For any point \((x, y)\) on the function, the point \((-x, -y)\) will also lie on the function.
  • The function's upward and downward curves on one side of the x-axis are mirrored inverted across the origin onto the other side.
Recognizing symmetry can simplify the graphing process, as when you know the shape of one part of the graph, you can deduce the shape of the mirrored section.

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Most popular questions from this chapter

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x)=\frac{c x}{1+c^{2} x^{2}}\)

since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A rain-drop has an initial downward velocity of 10 \(\mathrm{m} / \mathrm{s}\) and its downward acceleration is $$a=\left\\{\begin{array}{ll}{9-0.9 t} & {\text { if } 0 \leq t \leq 10} \\\ {0} & {\text { if } t>10}\end{array}\right.$$ If the raindrop is initially 500 \(\mathrm{m}\) above the ground, how long does it take to fall?

Use a computer algebra system to graph \(f\) and to find \(f^{\prime}\) and \(f^{\prime \prime} .\) Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x)=\sqrt{x+5 \sin x}, \quad x \leqslant 20\)

(a) Graph the function. (b) Explain the shape of the graph by computing the limit as \(x \rightarrow 0^{+}\) or as \(x \rightarrow \infty.\) (c) Estimate the maximum and minimum values and then use calculus to find the exact values. (d) Use a graph of \(f "\) to estimate the \(x\)-coordinates of the inflection points. \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{1 / \mathrm{x}}\)

\(57-62\) A particle is moving with the given data. Find the position of the particle. \(a(t)=t^{2}-4 t+6, \quad s(0)=0, \quad s(1)=20\)
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