Problem 31
Question
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{x}{\sqrt{x^{2}+1}}$$
Step-by-Step Solution
Verified Answer
No local extrema; inflection point at (0, 0); horizontal asymptotes y = ±1.
1Step 1: Find the First Derivative
\(y = \frac{x}{\sqrt{x^2+1}}\)
Using the quotient rule:
\(y' = \frac{\sqrt{x^2+1} - x \cdot \frac{x}{\sqrt{x^2+1}}}{x^2+1} = \frac{x^2+1-x^2}{(x^2+1)^{3/2}} = \frac{1}{(x^2+1)^{3/2}}\)
Using the quotient rule:
\(y' = \frac{\sqrt{x^2+1} - x \cdot \frac{x}{\sqrt{x^2+1}}}{x^2+1} = \frac{x^2+1-x^2}{(x^2+1)^{3/2}} = \frac{1}{(x^2+1)^{3/2}}\)
2Step 2: Analyze the First Derivative
Since \(y' = \frac{1}{(x^2+1)^{3/2}} > 0\) for all \(x\), the function is always increasing. There are no local extrema.
3Step 3: Find the Second Derivative
\(y'' = -\frac{3x}{(x^2+1)^{5/2}}\)
\(y'' = 0\) when \(x = 0\), and changes sign there, so \((0, 0)\) is an inflection point.
\(y'' = 0\) when \(x = 0\), and changes sign there, so \((0, 0)\) is an inflection point.
4Step 4: Determine Global Behavior
As \(x \to \infty\), \(y \to 1\); as \(x \to -\infty\), \(y \to -1\). Horizontal asymptotes at \(y = \pm 1\). No absolute extrema (approaches but never reaches \(\pm 1\)).
Key Concepts
ExtremaInflection PointsGraphing FunctionsDerivative Rules
Extrema
In calculus, the term **extrema** refers to the maximum and minimum values of a function. There are two main types of extrema: **local extrema**, which are the highest or lowest points in a region around a certain point, and **absolute extrema**, which are the highest or lowest points over the entire domain of the function. Finding extrema is crucial since these points can indicate optimal values in practical scenarios, such as the highest profit or lowest cost.
To find these points, one must first calculate the function's derivative, which involves using rules like the quotient rule in this scenario. Once the first derivative is known, setting it equal to zero and solving for the variable will reveal potential extrema. These correspond to critical points where the function's slope is zero, indicating a potential peak or valley. It's crucial to verify these points using the second derivative test or by examining the function's behavior around them to confirm if they are indeed maxima or minima. By identifying these key points, you understand how the function behaves and changes, providing deeper insight into its characteristics.
To find these points, one must first calculate the function's derivative, which involves using rules like the quotient rule in this scenario. Once the first derivative is known, setting it equal to zero and solving for the variable will reveal potential extrema. These correspond to critical points where the function's slope is zero, indicating a potential peak or valley. It's crucial to verify these points using the second derivative test or by examining the function's behavior around them to confirm if they are indeed maxima or minima. By identifying these key points, you understand how the function behaves and changes, providing deeper insight into its characteristics.
Inflection Points
An **inflection point** occurs where a curve changes its direction of concavity, shifting from being concave up ("U" shaped) to concave down ("n" shaped), or vice versa. Understanding and finding inflection points provides valuable insight into the behavior of a graph since they mark significant changes in the slope's direction and the overall trend of the function.
To find inflection points, it is necessary to calculate the second derivative of the function. The second derivative, denoted as \( y'' \), tells us about the concavity of the function. The next step is setting this second derivative equal to zero and solving for \( x \). If the second derivative changes sign around these solutions, an inflection point is confirmed.
These points are crucial for accurately sketching the graph of a function as they highlight areas of potential momentum shifts in the graph's slope. Knowing inflection points provides a more complete picture of how the function curves across its domain, further enriching the understanding and analysis of the function.
To find inflection points, it is necessary to calculate the second derivative of the function. The second derivative, denoted as \( y'' \), tells us about the concavity of the function. The next step is setting this second derivative equal to zero and solving for \( x \). If the second derivative changes sign around these solutions, an inflection point is confirmed.
These points are crucial for accurately sketching the graph of a function as they highlight areas of potential momentum shifts in the graph's slope. Knowing inflection points provides a more complete picture of how the function curves across its domain, further enriching the understanding and analysis of the function.
Graphing Functions
Graphing functions in calculus involves plotting the function's data points to visually understand its behavior across its domain. It's an integral part of analyzing functions as it provides a graphical visualization to accompany the algebraic work.
When plotting a function like \( y = \frac{x}{\sqrt{x^2 + 1}} \), it's important first to consider the key features like extrema, inflection points, and asymptotic behavior. These elements can be determined using derivatives, which assist in understanding the function's growth, decay, and potential plateaus.
Begin graphing a function by identifying critical points and plotting them. Next, evaluate and mark any inflection points. If asymptotes exist, draw these as well, since they indicate boundaries the function approaches but does not cross. By reviewing these components, one can effectively sketch the function, revealing helpful insights into its overall structure and directing further analysis or application.
When plotting a function like \( y = \frac{x}{\sqrt{x^2 + 1}} \), it's important first to consider the key features like extrema, inflection points, and asymptotic behavior. These elements can be determined using derivatives, which assist in understanding the function's growth, decay, and potential plateaus.
Begin graphing a function by identifying critical points and plotting them. Next, evaluate and mark any inflection points. If asymptotes exist, draw these as well, since they indicate boundaries the function approaches but does not cross. By reviewing these components, one can effectively sketch the function, revealing helpful insights into its overall structure and directing further analysis or application.
Derivative Rules
Derivative rules are fundamental in calculus, helping to determine the slope of a function at any point, which is essential for analyzing changes, optimizing functions, and solving real-world problems.
One common rule used in calculating derivatives is the **quotient rule**, suitable for expressions of the form \( \frac{u}{v} \). According to the quotient rule, if you have a function \( y = \frac{u}{v} \), then its derivative is given by:
Additionally, the **product rule** is applied when the function is a product, and the **chain rule** is used for compounded functions. Mastery of these rules allows for efficient manipulation of functions and the extraction of critical information such as slope, curvature, and points of inflection. By understanding derivative rules, one can glean important insights about the function's tendencies and responses to changes in its input.
One common rule used in calculating derivatives is the **quotient rule**, suitable for expressions of the form \( \frac{u}{v} \). According to the quotient rule, if you have a function \( y = \frac{u}{v} \), then its derivative is given by:
- \( y' = \frac{u'v - uv'}{v^2} \)
Additionally, the **product rule** is applied when the function is a product, and the **chain rule** is used for compounded functions. Mastery of these rules allows for efficient manipulation of functions and the extraction of critical information such as slope, curvature, and points of inflection. By understanding derivative rules, one can glean important insights about the function's tendencies and responses to changes in its input.
Other exercises in this chapter
Problem 31
Find all possible functions with the given derivative. a. \(y^{\prime}=x\) b. \(y^{\prime}=x^{2}\) c. \(y^{\prime}=x^{3}\)
View solution Problem 31
a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying wher
View solution Problem 31
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolut
View solution Problem 32
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $
View solution