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Magazine Chapter 3: Problem 87
Use a calculator to graph \(f(x)\). Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$ f(x)=\frac{1}{\sqrt{2 x}} $$
Short Answer
Expert verified
Graph \( f(x) = \frac{1}{\sqrt{2x}} \) and its derivative \( f'(x) = -\frac{1}{(2x)^{3/2}} \).
Step by step solution
01
Understand the given function
The given function is \( f(x) = \frac{1}{\sqrt{2x}} \). It is a rational function with a square root in the denominator.
02
Graph the function
Use a graphing calculator to plot the function \( f(x) = \frac{1}{\sqrt{2x}} \). Ensure you input the function correctly and view the graph. You should see a curve that decreases as \( x \) increases, starting from right of the y-axis since \( x > 0 \) for real numbers due to the square root.
03
Differentiate the function
To find \( f'(x) \), we need to differentiate \( f(x) = \frac{1}{\sqrt{2x}} \). Rewrite the function using exponents: \( f(x) = (2x)^{-1/2} \) and then use the chain rule. First, differentiate the outer function: \( (2x)^{-1/2} \) gives us \( -\frac{1}{2}(2x)^{-3/2} \). Then differentiate the inner function: \( 2x \), which is \( 2 \). Combine these to find that \( f'(x) = -\frac{1}{2}(2x)^{-3/2} \cdot 2 = -\frac{1}{(2x)^{3/2}} \).
04
Simplify the derivative function
The derivative function is \( f'(x) = -\frac{1}{(2x)^{3/2}} \). Simplify further if needed, which in this case is already in a usable form.
05
Graph the derivative function
Similar to Step 2, use a graphing calculator to plot \( f'(x) = -\frac{1}{(2x)^{3/2}} \). Enter the function correctly and observe its behavior. The graph should show a monotonically increasing curve, which shows that the rate of change of \( f(x) \) becomes less negative as \( x \) increases.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Functions
Rational functions are a type of function represented by the division of two polynomials. In the context of this exercise, the function \( f(x) = \frac{1}{\sqrt{2x}} \) is considered a rational function with a special characteristic: the square root in the denominator. Understanding rational functions involves recognizing their behavior, including asymptotes and domain restrictions.
- The domain of \( f(x) = \frac{1}{\sqrt{2x}} \) is restricted because the expression under the square root, \( 2x \), must be positive. Thus, \( x > 0 \).
- As \( x \to 0^+ \) (from the right), the function \( f(x) \) approaches infinity, indicating a vertical asymptote at \( x=0 \).
- As \( x \to \infty \), the value of \( f(x) \) approaches zero, suggesting a horizontal asymptote along the x-axis.
Graphing Calculators
Graphing calculators are essential tools for visualizing functions like \( f(x) = \frac{1}{\sqrt{2x}} \). They provide a clear representation of how functions behave across their domains, making it easier to understand complex functions.
When using a graphing calculator:
When using a graphing calculator:
- Ensure that you input the function correctly. Parentheses and appropriate use of square roots or exponents are crucial.
- Adjust the window settings to ensure that the significant parts of the graph are visible.
- Utilize the calculator's features to explore function properties, like intercepts and asymptotes, which can quickly inform your understanding of the function's behavior.
Chain Rule
The chain rule is a fundamental concept in calculus used to differentiate composite functions. In this exercise, it is necessary to apply the chain rule to find the derivative of \( f(x) = \frac{1}{\sqrt{2x}} \).
Understanding the Chain Rule involves these steps:
Understanding the Chain Rule involves these steps:
- Identify the outer and inner functions. Here, the outer function is \( u^{-1/2} \) where \( u = 2x \), and the inner function is \( 2x \).
- Differentiating the outer function with respect to the inner function gives \( -\frac{1}{2}u^{-3/2} \).
- The derivative of the inner function \( 2x \) is \( 2 \).
- The chain rule combines these derivatives: \( f'(x) = -\frac{1}{2}(2x)^{-3/2} \cdot 2 \).
Derivative Graphs
Derivative graphs illustrate the rate of change of a function with respect to its variables. For the function \( f(x) = \frac{1}{\sqrt{2x}} \), the derivative \( f'(x) = -\frac{1}{(2x)^{3/2}} \) describes how the function changes as \( x \) increases.
Some aspects to consider when plotting the derivative graph:
Some aspects to consider when plotting the derivative graph:
- The graph of the derivative does not touch the x-axis, indicating the function \( f'(x) \) never crosses zero; thus, it remains negative throughout its domain.
- Observing the graph, you'll see it is increasing, but still negative, reflecting that the original function \( f(x) \) is decelerating as \( x \) increases.
- The steepness of the curve in \( f'(x) \) highlights how fast the original function's slope is changing, detailing the intricacies of its decreasing nature.
