Problem 35
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. $$ y=\left\\{\begin{array}{ll} 4-x^{2}, & x \leq 0 \\ -2 x, & x>0 \end{array}\right. $$
Step-by-Step Solution
Verified Answer
The critical number is \(x = 0\). The function is decreasing on the intervals \((- \infty, 0]\) and \((0, \infty)\). The graph is a parabola until \(x = 0\), and a straight line for \(x > 0\).
1Step 1: Compute the Derivatives
We have two parts of the piecewise function, thus we calculate derivatives for each one. For \(4 - x^2\) over interval \(x \leq 0\), the derivative is \(-2x\). For \(-2x\) over interval \(x>0\), the derivative is \(-2\).
2Step 2: Find the Critical Numbers
Critical numbers of a function are numbers \(x\) in the domain of the function such that the derivative is zero or undefined. Here, over interval \(x \leq 0), the equation is \(-2x = 0\), so \(x = 0\). Over interval \(x>0), the derivative is \(-2\), which is a constant, so there are no critical numbers.
3Step 3: Determine the Intervals of Increase and Decrease
For \(x \leq 0\), the derivative is \(-2x\). The derivative is negative for all \(x < 0\), so the function is decreasing on the interval \((- \infty, 0]\). When \(x > 0\), the derivative is \(-2\), which is always negative. Thus, the function is decreasing on the interval \((0, \infty)\).
4Step 4: Sketch the Graph
Plotting the values of the function and noting the critical point and intervals of increase and decrease, will result in a parabola dropping at \(x = 0\) for \(x \leq 0\), then continuing in a straight line with a negative slope for \(x > 0\).
Key Concepts
Piecewise FunctionsDerivatives CalculationIncreasing and Decreasing Intervals
Piecewise Functions
Piecewise functions are mathematical expressions that have different formulas for different intervals of their domain. This means the rule that defines the function changes depending on the value of the independent variable, usually noted by 'x'. The function provided in the exercise is an example of a piecewise function:
\[ y = \left\{ \begin{array}{ll} 4 - x^2, & x \leq 0 \ -2x, & x > 0 \end{array} \right. \]
Understanding piecewise functions is crucial because they can model complex real-world situations where a single rule does not suffice. When working with them, it's important to consider each piece individually, especially when calculating derivatives or discussing the behavior on different intervals, like when finding critical numbers or intervals of increase and decrease.
\[ y = \left\{ \begin{array}{ll} 4 - x^2, & x \leq 0 \ -2x, & x > 0 \end{array} \right. \]
Understanding piecewise functions is crucial because they can model complex real-world situations where a single rule does not suffice. When working with them, it's important to consider each piece individually, especially when calculating derivatives or discussing the behavior on different intervals, like when finding critical numbers or intervals of increase and decrease.
Derivatives Calculation
Calculation of derivatives is a fundamental concept in calculus, which measures the sensitivity to change of the function value (output) with respect to a change in its argument (input). In other words, the derivative represents the rate of change or the slope of the function at a certain point. For piecewise functions, such as the one from the exercise, you must calculate the derivative separately for each segment, as the formula for the rate of change differs.
For the piecewise function provided:
\begin{itemize}For the first piece, \(4 - x^2\), applicable where \(x \leq 0\), the derivative is \(-2x\). The second piece, \(-2x\), which applies where \(x > 0\), has a constant derivative of \(-2\).
The next step is to use these derivatives to find critical numbers, which are potential locations for local maxima, minima, or inflection points.
For the piecewise function provided:
\begin{itemize}
The next step is to use these derivatives to find critical numbers, which are potential locations for local maxima, minima, or inflection points.
Increasing and Decreasing Intervals
The increasing and decreasing intervals of a function can be determined by analyzing the sign of the derivative. If the derivative is positive on an interval, the function is increasing there; conversely, if the derivative is negative, the function is decreasing. This principle helps us see how a function behaves without graphing it fully.
From the exercise:
\begin{itemize}For \(x \leq 0\), the derivative \(-2x\) is negative when \(x < 0\), indicating that the function is decreasing on the interval \((-fty, 0]\). For \(x > 0\), the constant derivative \(-2\) indicates the function is also decreasing, but on the interval \((0, fty)\).
As a result, the function is decreasing for all real numbers. Spotting these intervals is a key step not only in sketching graphs but also in understanding where a function reaches its highest and lowest values.
From the exercise:
\begin{itemize}
As a result, the function is decreasing for all real numbers. Spotting these intervals is a key step not only in sketching graphs but also in understanding where a function reaches its highest and lowest values.
Other exercises in this chapter
Problem 35
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