Problem 69 - Single Variable Calculus

Question

Let \(f(x)=3 x^{5}-8 x^{3}+x\). a. Plot the graph of \(f\) using the viewing window \([-2,2] \times[-6,6]\). Can you determine from the graph of \(f\) the intervals where \(f\) is increasing or decreasing? b. Plot the graph of \(f\) using the viewing window \([-0.5,0.5] \times[-0.5,0.5]\). Using this graph and the result of part (a), determine the intervals where \(f\) is increasing and where \(f\) is decreasing.

Step-by-Step Solution

Answer

In this exercise, we analyzed the given function \(f(x)=3x^5-8x^3+x\) by plotting it in two different viewing windows and determining where it is increasing and decreasing based on the graphs. The function is increasing in the intervals \((-2,-1.3) \cup (-0.3, 0) \cup (1.3,2)\), and it is decreasing in the intervals \((-1.3,-0.3) \cup (0,1.3)\).

Step 1: Plot the function in the viewing window [-2,2] × [-6,6]

Firstly, let's plot the given function \(f(x)=3x^5-8x^3+x\) in the specified viewing window, which is \([-2,2] \times [-6,6]\). Make sure to carefully analyze the graph of the function in this window to gain insights into its general shape and behavior within this range of values for \(x\).

Step 2: Determine the intervals where f is increasing or decreasing

Analyze the plotted graph in the previous step and try to determine the intervals where the function is increasing or decreasing. Note that a function is increasing where the slope is positive and decreasing where the slope is negative. From the given graph in step 1, make an educated guess about where the function appears to be increasing and decreasing.

Step 3: Plot the function in the viewing window [-0.5,0.5] × [-0.5,0.5]

Now, plot the function \(f(x)=3x^5-8x^3+x\) in the second specified viewing window, which is \([-0.5,0.5] \times [-0.5,0.5]\). This step allows you to analyze the behavior of the function in a more refined manner, focusing on the range around zero.

Step 4: Determine the intervals where f is increasing and where f is decreasing using the two graphs

Finally, using the results from steps 1 and 3, determine the intervals where the function is increasing and where it is decreasing. It's important to combine the insights from the first graph, which gave an overall view of the function, with the second graph, which provided a more detailed look at the behavior around smaller values for \(x\). Make sure to provide a clear description of the increasing and decreasing intervals based on the combination of these two graphs.

Key Concepts

Plotting Functions

Before delving into the intricacies of calculus, it's essential to become familiar with plotting functions. This is an indispensable skill for visualizing the relationship between variables. When plotting the function f(x)=3x^5-8x^3+x, one should consider the viewing window carefully. Selecting the range [-2,2] x [-6,6] provides a broad overview, showcasing the general shape and turning points of the function. A smaller window like [-0.5,0.5] x [-0.5,0.5] offers a zoomed-in view, which is crucial for observing nuances near the origin.
Understanding how to adjust the viewing window based on the features you're looking for is key to effective plotting. A graphing calculator or software can help generate these plots, allowing students to clearly see how the function behaves across different intervals.

Analyzing Graphs

Analyzing graphs transmutes raw data into valuable insights. By looking at the plotted graph of our subject function, f(x)=3x^5-8x^3+x, we embark on a visual discovery of its behavior. The challenge, however, lies in deciphering the trends and patterns amidst the curves. When working with the interval [-2,2], one can observe the peaks and troughs, which are indicative of the function's increasing and decreasing tendencies.
In the pursuit of understanding function intervals, the areas where the graph ascends from left to right pinpoint where the function is increasing, while a descent indicates a decrease. Scrutinizing these trends becomes manageable through graph analysis, an essential skill in interpreting complex calculus concepts.

Function Behavior

Function behavior is all about how a mathematical expression's output varies with its input. For instance, take our polynomial function f(x)=3x^5-8x^3+x. What really matters is how the function acts over certain intervals, be it increasing, decreasing, or remaining constant within particular ranges. A function is said to be increasing on an interval if any two numbers chosen within that interval result in larger outputs for larger inputs. Conversely, it's decreasing when larger inputs lead to smaller outputs. The two different viewing windows provided earlier help us distinguish these behaviors over different ranges, confirming our suspicions about the rising and falling segments of the graph.

Slope of a Function

Finally, the concept of the slope of a function is the linchpin in understanding where a function increases or decreases. The slope refers to the steepness and direction of a line or curve at any point. For a curve like f(x)=3x^5-8x^3+x, the slope can vary at each point, and it's determined by the function's derivative.
A positive slope indicates an upward trend (increasing function), while a negative slope points to a downward trend (decreasing function). By examining the slope through the derivative of f(x), we can refine our analysis of the increasing and decreasing intervals, making our predictions more accurate and grounded in calculus fundamentals.