Problem 36
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. $$ y=\left\\{\begin{array}{ll} 2 x+1, & x \leq-1 \\ x^{2}-2, & x>-1 \end{array}\right. $$
Step-by-Step Solution
Verified Answer
The function does not have any critical numbers. It is increasing in all its domain. There is no discontinuity, and the graph has a line with a positive slope for \(x \leq -1\) and a parabolic portion \(x>-1\) opening upwards.
1Step 1: Determine the critical numbers
Find the derivative of each piece of the function. For \(y=2x+1\), the derivative is \(y'=2\). There are no critical points in this function, because the derivative neither equals zero nor does it not exist. For \(y=x^2-2\), the derivative is \(y'=2x\). The critical point from this function is at \(x=0\), where \(y'=0\). However, since \(x=0\) is not within the domain of the function, we will disregard it.
2Step 2: Determine the intervals of increase or decrease
The derivative for the first function is always positive, so this function is always increasing. For the second function, the derivative is negative for \(x<0\) and positive for \(x>0\). However, since the domain of this function starts from \(x=-1\), we need only consider the interval where \(x>-1\), during which the derivative happens to be positive. Thus, it is increasing in this interval.
3Step 3: Check for discontinuities
We have to look at \(x=-1\), where the function changes definition. Evaluating both sides at \(x=-1\), they both give the value \(y=-1\), so there is no discontinuity.
4Step 4: Sketch the graph
Jot down all the important points and behaviours derived from previous steps, sketching a graph which includes the increasing line segment for \(x \leq -1\) and the increasing parabola for \(x>-1\), without any discontinuity at \(x=-1\).
Key Concepts
Increasing and Decreasing FunctionsDerivative ApplicationsFunction Sketching
Increasing and Decreasing Functions
Understanding when a function is increasing or decreasing is fundamental in calculus and helps us to sketch graphs and analyze the behavior of functions. A function is increasing on an interval if, as x values increase, y values also increase. Conversely, a function is decreasing on an interval if, as x values increase, y values decrease.
When looking at the given exercise, we apply this concept by taking the derivative of each piece of the function. The derivative gives us the slope of the tangent line to the curve at any point. For a constant derivative such as in the first piece of the function, where the derivative is 2, the function is continuously increasing since the slope is positive. In the second piece, the derivative changes sign based on the value of x, so we need to consider the values of x where this change occurs - the critical points.
To enhance the learning experience, students should visualize the slope by sketching a small line segment with the given slope at several points on the graph. For segments where the function is increasing, they should sketch segments with a positive slope, and for those that are decreasing, segments with a negative slope. By comparing these sketches, learners can develop a more intuitive understanding of how the derivative shows increasing and decreasing behavior.
When looking at the given exercise, we apply this concept by taking the derivative of each piece of the function. The derivative gives us the slope of the tangent line to the curve at any point. For a constant derivative such as in the first piece of the function, where the derivative is 2, the function is continuously increasing since the slope is positive. In the second piece, the derivative changes sign based on the value of x, so we need to consider the values of x where this change occurs - the critical points.
To enhance the learning experience, students should visualize the slope by sketching a small line segment with the given slope at several points on the graph. For segments where the function is increasing, they should sketch segments with a positive slope, and for those that are decreasing, segments with a negative slope. By comparing these sketches, learners can develop a more intuitive understanding of how the derivative shows increasing and decreasing behavior.
Derivative Applications
The derivative is a powerful tool in calculus with vast applications, especially when it comes to analyzing the properties of functions. Not only does it determine the slope of the tangent line at a given point, but it also reveals critical information about the function's behavior.
In our exercise, the application of derivatives involves finding critical numbers, which are the x-values where the function's derivative is zero or undefined. These numbers often correspond to local maxima, minima, or points of inflection. The derivative also allows us to determine on which intervals the function is increasing or decreasing - key to function analysis and sketching.
To improve comprehension, it's helpful for students to connect the critical numbers and the derivative's sign to the function's geometry. For example, where the derivative is zero, they might mark on the graph that the function could have a 'flat' tangent, indicating a possible maximum or minimum. Also, distinguishing between where the derivative is positive or negative can be linked with the rise or fall of the function's graph at those intervals.
In our exercise, the application of derivatives involves finding critical numbers, which are the x-values where the function's derivative is zero or undefined. These numbers often correspond to local maxima, minima, or points of inflection. The derivative also allows us to determine on which intervals the function is increasing or decreasing - key to function analysis and sketching.
To improve comprehension, it's helpful for students to connect the critical numbers and the derivative's sign to the function's geometry. For example, where the derivative is zero, they might mark on the graph that the function could have a 'flat' tangent, indicating a possible maximum or minimum. Also, distinguishing between where the derivative is positive or negative can be linked with the rise or fall of the function's graph at those intervals.
Function Sketching
Turning the abstract concepts of calculus into a visual graph is what function sketching is all about. Combining our understanding of derivatives and the behavior of functions, sketching provides a tangible representation of where the function is increasing, decreasing, or flat, and how it behaves around critical points.
From our exercise, once you determine the intervals of increase or decrease and identify discontinuities, you can begin to sketch the graph. Plotting the critical points, the points of increase and decrease, and drawing the function piece by piece leads to a comprehensive graph that visually explains the function's properties. For the function at hand, note the distinct pieces of the graph - a line and a parabola - and how they join at the point where x equals -1, without any gaps, jumps, or holes.
To further guide learning, students should be encouraged to use a step-by-step approach to graphing: first plotting the critical points, then noting where the function increases and decreases, and checking for continuity. This systematic approach avoids confusion and helps create an accurate graphical representation of the function.
From our exercise, once you determine the intervals of increase or decrease and identify discontinuities, you can begin to sketch the graph. Plotting the critical points, the points of increase and decrease, and drawing the function piece by piece leads to a comprehensive graph that visually explains the function's properties. For the function at hand, note the distinct pieces of the graph - a line and a parabola - and how they join at the point where x equals -1, without any gaps, jumps, or holes.
To further guide learning, students should be encouraged to use a step-by-step approach to graphing: first plotting the critical points, then noting where the function increases and decreases, and checking for continuity. This systematic approach avoids confusion and helps create an accurate graphical representation of the function.
Other exercises in this chapter
Problem 36
In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=-4 x^{3}-8 x^{2}+32 $$
View solution Problem 36
In Exercises, use a graphing utility to find graphically the absolute extrema of the function on the closed interval. $$ f(x)=4 \sqrt{x}-2 x+1, \quad[0,6] $$
View solution Problem 36
In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=3 x^{3}-9 x+1 $$
View solution Problem 37
In Exercises, find the point(s) of inflection of the graph of the function. $$ h(x)=(x-2)^{3}(x-1) $$
View solution