Problem 216 - Calculus Volume 1

Question

Draw a graph that satisfies the given specifications for the domain \(x=[-3,3]\). The function does not have to be continuous or differentiable. \(\quad f(x)>0, f^{\prime}(x)>0 \quad\) over \(x>1,-3

Step-by-Step Solution

Answer

The graph is positive, increasing from \(-3\) to \(0\) and \(x > 1\), and constant for \(0 < x < 1\).

Step 1: Understand the Domain and Conditions

The graph we need to draw is for the domain where \(x = [-3, 3]\). This means our function \(f(x)\) should be defined for all \(x\) values from \(-3\) to \(3\). Furthermore, we have different conditions over specific intervals of \(x\), like \(f(x) > 0\) and \(f'(x) > 0\) for different domains and intervals where \(f'(x) = 0\).

Step 2: Determine Intervals for Conditions

First, we have \(f(x) > 0\) for all \(x > 1\) and \(-3 < x < 0\). In these ranges, the function value must be positive. We also have \(f'(x) > 0\) for \(x > 1\) and \(-3 < x < 0\), indicating that the function is increasing in these intervals.

Step 3: Apply Conditions for Derivative Being Zero

In the interval \(0 < x < 1\), it is given that \(f'(x) = 0\), meaning the function is constant in this range. This doesn't change the condition that \(f(x) > 0\) here as it's implicitly given by continuity from \(-3 < x < 0\) to \(0 < x < 1\) and \(x > 1\).

Step 4: Sketch the Graph Based on Conditions

Start by sketching \(f(x)\) as an increasing function from \(-3\) to \(0\), since \(f'(x) > 0\) and \(f(x) > 0\) in \(-3 < x < 0\). \(f(x)\) should remain positive for \(0 < x < 1\) and be constant, depicting \(f'(x) = 0\). For \(x > 1\), sketch \(f(x)\) as increasing again, maintaining \(f(x) > 0\) and \(f'(x) > 0\). Ensure the graph appropriately distinguishes these intervals.

Key Concepts

Function Behavior Analysis

To understand how a function behaves, we look at how it changes over different parts of its domain. In this exercise, the behavior of the function is defined over several specific intervals. Below, we'll uncover these aspects in detail:

• First, observe that over the interval \(-3 < x < 0\), the function \(f(x) > 0\). This signals that the function is always above the x-axis in this range. Plus, the derivative \(f'(x) > 0\) implies the function is increasing here.


• For \(x > 1\), similar conditions apply: \(f(x) > 0\) and \(f'(x) > 0\). Again, the function stays positive and continues rising as \(x\) increases.


• In the interval \(0 < x < 1\), \(f'(x) = 0\) is given. This tells us the function has no slope change, thus it remains constant. Yet, we know \(f(x) > 0\) persists here.

By understanding where the function is increasing or constant, and where it stays positive, we form an overall picture of how it behaves across its domain.

Graphical Representation

Visually sketching a function helps us grasp the changes described by its mathematical conditions. Here is a step-by-step guide on representing the given function:

• In \(-3 < x < 0\), draw the graph rising above the x-axis, illustrating positivity and an increasing trend.


• From \(0\) to \(1\), draw a straight horizontal line, showing the function's constancy because \(f'(x) = 0\).


• Lastly, past \(x = 1\), continue the graph in an upward direction, maintaining it above the x-axis to honor \(f(x) > 0\) and \(f'(x) > 0\).

Incorporating these successive changes into a single graph allows us to depict the comprehensive function behavior accurately.

Function Intervals

Analyzing a function’s intervals involves determining where certain conditions apply. In this exercise, differing intervals show varied behaviors:

• In \(-3 < x < 0\) and again for \(x > 1\), \(f(x) > 0\). The function values remain positive, thus these segments lie above the x-axis.


• Also, in these ranges, \(f'(x) > 0\), making the function increase steadily for each respective interval.


• A unique case appears in \(0 < x < 1\) where \(f'(x) = 0\), confirming the function becomes flat (it neither rises nor falls).

Breaking down the function's intervals and their individual behaviors equips us to fully understand and predict the graph's look and path over its defined domain.

Derivative Conditions

Derivative conditions reflect how the function is evolving with respect to changes in \(x\):

• Where \(f'(x) > 0\), shown in intervals \(-3 < x < 0\) and \(x > 1\), the derivative's positivity means the function is climbing.


• In contrast, between \(0 < x < 1\), \(f'(x) = 0\) signifies a plateau in the function's surface. Here, no incline or decline in behavior exists as \(f(x)\) remains constant.

Realizing these derivative effects guides us in recognizing and predicting motion on the graph, whether as flat lines or inclined paths, in response to x-value alterations.