Problem 96 - Calculus Volume 1

Question

Sketch the graph of a function \(y=f(x)\) with all of the following properties: a. \(f^{\prime}(x)>0\) for \(-2 \leq x<1\) b. \(f^{\prime}(2)=0\) c. \(f^{\prime}(x)>0\) for \(x>2\) d. \(f(2)=2\) and \(f(0)=1\) e. \(\quad \lim _{x \rightarrow-\infty} f(x)=0\) and \(\lim _{x \rightarrow \infty} f(x)=\infty\) f. \(f^{\prime}(1)\) does not exist.

Step-by-Step Solution

Answer

Sketch starts at \(y=0\), rises through \((0,1)\) with a cusp at \(x=1\), levels at \((2,2)\) before rising to infinity.

Step 1: Understand the Conditions

We need to sketch a graph of a function \(y=f(x)\) based on the given properties. Let's first understand what each condition implies about the graph of \(f(x)\).

Step 2: Calculate Increasing Intervals

The function is increasing, i.e., \(f^{\prime}(x)>0\), for \(-2 \leq x<1\) and also for \(x>2\). This tells us the slope of the function in these intervals is positive, indicating an upward sloping curve.

Step 3: Find Critical Points

At \(x = 2\), \(f^{\prime}(2)=0\) implies a critical point, where the slope of the tangent line is zero. This could indicate a local minimum or maximum or can be a saddle point depending on the surrounding function behavior.

Step 4: Consider Non-Existence of Derivative

When \(f^{\prime}(1)\) does not exist, it suggests a cusp or a vertical tangent at \(x = 1\). This means the graph of \(f(x)\) should have a sharp turn or endpoints at \(x = 1\).

Step 5: Apply Function Values

The conditions \(f(2)=2\) and \(f(0)=1\) give specific points through which the graph must pass: \((2,2)\) and \((0,1)\). Sketch the graph such that it goes through these coordinate points.

Step 6: Determine Asymptotic Behavior

As \(x\) approaches \(-\infty\), \(f(x)\) approaches 0, implying a horizontal asymptote along \(y=0\). As \(x\) approaches \(\infty\), the function \(f(x)\) grows unbounded, suggesting that the curve moves off towards infinity.

Step 7: Draw the Graph

Using the information above: start a curve near \(y=0\) as \(x\rightarrow-\infty\). Rise from \((-2,f(-2))\) to \(x=1\), ensuring a sharp turn at \(x=1\). Plot points \(f(0)=1\) and \(f(2)=2\), level off or dip at \(x=2\) due to \(f^{\prime}(2)=0\), and continue rising for \(x>2\) towards \(y=\infty\).

Step 8: Verify the Sketch

Ensure that the drawn graph adheres to all conditions set by the problem: increasing intervals, critical points, sharp turn, specified function values, and asymptotic behavior.

Key Concepts

Understanding Critical Points

Critical points of a function occur where the derivative equals zero or is undefined. At these points, the function may have a local maximum, a local minimum, or a saddle point (which is neither a maximum nor a minimum). In our problem, the derivative of the function, denoted as \(f'(x)\), is zero at \(x = 2\), indicating a critical point. This means that at \(x = 2\), the tangent to the function is horizontal. However, whether it signifies a peak, a trough, or a flat region depends on examining the surrounding behavior of the function. Critical points are essential in determining the shape of the graph, as they highlight possible turning points.

Exploring Derivatives

Derivatives are a central concept in calculus, providing insights into the rate of change of a function. The derivative of a function \(f(x)\), written as \(f'(x)\), tells us how fast or slow \(f(x)\) is changing at any point \(x\). When \(f'(x) > 0\), the function is increasing; when \(f'(x) < 0\), it is decreasing. If \(f'(x) = 0\), the function is flat at that point, representing a possible critical point.
In our problem, derivatives play multiple roles: indicating increasing intervals such as \(-2 \leq x < 1\) and \(x > 2\) where \(f'(x) > 0\), pinpointing critical points like \(x = 2\) where \(f'(x) = 0\), and showing undefined behavior at \(x = 1\) where \(f'(1)\) does not exist, suggesting a sharp feature in the graph.

Asymptotic Behavior of Functions

Asymptotic behavior describes how a function behaves as it approaches extreme values along the x-axis. It helps predict the function's end behavior without examining the entire graph. Often, you find horizontal or vertical asymptotes suggesting the function's long-term trends.
For our function, as \(x\) approaches minus infinity, \(f(x)\) approaches 0. This indicates a horizontal asymptote at \(y = 0\). Conversely, as \(x\) approaches plus infinity, \(f(x)\) heads towards infinity, suggesting that the graph keeps rising indefinitely. Understanding asymptotes is crucial for sketching an accurate graph, as they provide boundaries beyond which the function does not traverse.

Identifying Increasing Functions

Increasing functions are those where the output value rises as the input value increases. Mathematically, this is indicated by a positive first derivative, \(f'(x) > 0\). In our exercise, the function is identified as increasing between \(-2 \leq x < 1\) and \(x > 2\). This tells us that, in these intervals, the function will slope upwards.
Recognizing where a function is increasing helps in graphing because it dictates the direction of the curve in these intervals. In an increasing interval, any x-coordinate greater than another within this range will yield a higher y-value. This is crucial for accurately depicting the graph according to project requirements.