Problem 33
Question
In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ f(x)=\frac{2 x}{16-x^{2}} $$
Step-by-Step Solution
Verified Answer
The function \(f(x) = \frac{2x}{16 - x^2}\) has critical points at \(x=-4\) and \(x=4\). It is increasing on the intervals \((-\infty, -4)\) and \((-4, 4)\) and decreasing on the interval \((4, \infty)\).
1Step 1: Find the Derivative
To find the critical numbers, we first need to find the derivative of the function \(f(x) = \frac{2x}{16 - x^2} = 2x(16 - x^2)^{-1}\). Using the chain rule and the rule of differentiation for polynomials, the derivative is \(f'(x) = 2(16 - x^2)^{-1} - 2x(2x)(16 - x^2)^{-2}\). This simplifies to \(f'(x) = \frac{2}{16-x^2} - \frac{4x^2}{(16-x^2)^2}\).
2Step 2: Find the Critical Numbers
Critical numbers are the values of \(x\) that make the derivative \(f'(x)\) zero or undefined. Setting \(f'(x)\) equal to zero and solving for \(x\) gives no solution. Thus, we check for values that make \(f'(x)\) undefined. The function is undefined at \(x^2=16\) which results in \(x=-4,x=4\). Hence, the critical values are -4 and 4.
3Step 3: Determine the Increasing or Decreasing Intervals
We define some test points in the intervals \(x < -4, -4 < x < 4, x > 4\), namely -5, 0, 5 and evaluate \(f'(x)\) at these points. \(f'(-5)\) is positive, \(f'(0)\) is positive, and \(f'(5)\) is negative. Thus, the function \(f(x)\) is increasing on the interval \((-\infty, -4)\) and \((-4, 4)\), and decreasing on the interval \((4, \infty)\).
4Step 4: Sketch the Graph
To sketch the graph, we note the critical points, the intervals of increase and decrease, as well as the vertical asymptotes of \(f(x)\) at \(x=-4\) and \(x=4\). It is increasing from \(-\infty\) to \(4\), decreasing from \(4\) to \(\infty\) with vertical asymptotes at \(x=-4\) and \(x=4\).
Key Concepts
Increasing and Decreasing IntervalsDerivative of a FunctionChain Rule in CalculusSketching Graphs of FunctionsAsymptotes of a Function
Increasing and Decreasing Intervals
Understanding when a function is increasing or decreasing is crucial in calculus. To determine this, one must look at the function’s slope at various intervals, which is done by analyzing the derivative of the function.
If the derivative of a function, symbolically represented as \(f'(x)\), is positive over an interval, it means the function is increasing in that interval. Conversely, if the derivative is negative, the function is decreasing. Where \(f'(x)\) is zero, it may indicate a peak or a trough, where the function changes from increasing to decreasing or vice versa. These points are critical for understanding the behavior of functions and are helpful for graph sketching.
If the derivative of a function, symbolically represented as \(f'(x)\), is positive over an interval, it means the function is increasing in that interval. Conversely, if the derivative is negative, the function is decreasing. Where \(f'(x)\) is zero, it may indicate a peak or a trough, where the function changes from increasing to decreasing or vice versa. These points are critical for understanding the behavior of functions and are helpful for graph sketching.
Derivative of a Function
The derivative of a function \(f(x)\) represents the rate at which the function is changing at any given point. Formally, it's the limit of the average rate of change as the interval approaches zero.
Uniting algebra with the concept of limits, derivatives flesh out the behavior of graphs and inform us about slope at any given point. It serves as a cornerstone for finding critical numbers, analyzing functions, and predicting their behavior. Calculating derivatives may involve rules such as the power rule, product rule, quotient rule, and the chain rule, which the next section explains.
Uniting algebra with the concept of limits, derivatives flesh out the behavior of graphs and inform us about slope at any given point. It serves as a cornerstone for finding critical numbers, analyzing functions, and predicting their behavior. Calculating derivatives may involve rules such as the power rule, product rule, quotient rule, and the chain rule, which the next section explains.
Chain Rule in Calculus
The chain rule is a formula to compute the derivative of a composite function. In essence, if a variable is being affected by multiple functions, the chain rule helps unravel this complexity.
For instance, if \(h(x) = f(g(x))\), then the derivative of \(h\) with respect to \(x\) is \(h'(x) = f'(g(x)) \cdot g'(x)\). This simplifies the process of finding the derivative when the function is not just a single expression but a combination of functions affecting each other sequentially.
For instance, if \(h(x) = f(g(x))\), then the derivative of \(h\) with respect to \(x\) is \(h'(x) = f'(g(x)) \cdot g'(x)\). This simplifies the process of finding the derivative when the function is not just a single expression but a combination of functions affecting each other sequentially.
Sketching Graphs of Functions
Graphical representations of functions are an invaluable tool for visualizing their characteristics, such as continuity, extrema, and concavity.
When sketching the graph of a function, one should consider the function's critical points, its increasing and decreasing intervals, and asymptotes. By plotting these features, along with the evaluation of the function's value at select points, it becomes easier to predict and draw the shape of the function's curve across the coordinate plane.
When sketching the graph of a function, one should consider the function's critical points, its increasing and decreasing intervals, and asymptotes. By plotting these features, along with the evaluation of the function's value at select points, it becomes easier to predict and draw the shape of the function's curve across the coordinate plane.
Asymptotes of a Function
Asymptotes are lines that the graph of a function approaches but never quite touches. They depict the behavior of functions as the inputs grow without bound. The common types of asymptotes are horizontal, vertical, and oblique.
Vertical asymptotes typically occur at points where the function is undefined, which can be due to a zero in the denominator, signaling that the function values are approaching infinity. Identifying these can guide us in graph sketching and in understanding the limits and bounds of the function's domain.
Vertical asymptotes typically occur at points where the function is undefined, which can be due to a zero in the denominator, signaling that the function values are approaching infinity. Identifying these can guide us in graph sketching and in understanding the limits and bounds of the function's domain.
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