Problem 231 - Calculus Volume 1

Question

For the following exercises, determine $$ \begin{array}{l}{\text { a. intervals where } f \text { is increasing or decreasing, }} \\ {\text { b. local minima and maxima of } f} \\ {\text { c. intervals where } f \text { is concave up and concave }} \\ {\text { down, and }} \\ {\text { d. the inflection points of } f \text { . Sketch the curve, then }} \\ {\text { use a calculator to compare your answer. If you }} \\\ {\text { cannot determine the exact answer analytically, use }} \\ {\text { a calculator. }}\end{array} $$ $$ f(x)=\sin (\pi x)-\cos (\pi x) \text { over } x=[-1,1] $$

Step-by-Step Solution

Answer

The function is increasing on \((-1, -\frac{1}{4})\) and \((\frac{3}{4}, 1)\); decreasing on \((-\frac{1}{4}, \frac{3}{4})\). It has a local maximum at \(x = -\frac{1}{4}\) and a local minimum at \(x = \frac{3}{4}\). It is concave up on \((-\frac{3}{4}, \frac{1}{4})\) and concave down on \((-1, -\frac{3}{4})\) and \((\frac{1}{4}, 1)\), with inflection points at \(x = -\frac{3}{4}\) and \(x = \frac{1}{4}\).

Step 1: Find the First Derivative

To analyze where the function is increasing or decreasing, we first need to find its derivative. The function is given by \( f(x) = \sin(\pi x) - \cos(\pi x) \). The derivative is \( f'(x) = \pi \cos(\pi x) + \pi \sin(\pi x) \).

Step 2: Determine Increasing and Decreasing Intervals

Set \( f'(x) = 0 \) to find the critical points: \( \pi (\cos(\pi x) + \sin(\pi x)) = 0 \Rightarrow \cos(\pi x) + \sin(\pi x) = 0 \). This equation holds true for \( \pi x = \frac{3\pi}{4} \) and \( \pi x = -\frac{\pi}{4} \) within the interval \([-1, 1]\). These correspond to \( x = \frac{3}{4} \) and \( x = -\frac{1}{4} \). Evaluate \( f'(x) \) around these points to determine where \( f(x) \) is increasing or decreasing.

Step 3: Find Local Minima and Maxima

Use the First Derivative Test. At \( x = -\frac{1}{4} \), check to see if \( f'(x) \) changes sign around this point. \( f'(x) \) changes from positive to negative, indicating a local maximum. At \( x = \frac{3}{4} \), \( f'(x) \) changes from negative to positive, indicating a local minimum.

Step 4: Find the Second Derivative

The second derivative helps us find concavity. Calculate \( f''(x) = -\pi^2 \sin(\pi x) + \pi^2 \cos(\pi x) \).

Step 5: Determine Concavity Intervals

Set \( f''(x) = 0 \) to find possible points of inflection: \(-\pi^2 \sin(\pi x) + \pi^2 \cos(\pi x) = 0 \Rightarrow \tan(\pi x) = 1 \). The solution is \( \pi x = \frac{\pi}{4} + k\pi \), giving \( x = \frac{1}{4} \) and \( x = -\frac{3}{4} \) within the interval \([-1, 1]\). Evaluate \( f''(x) \) around these points to check for concavity.

Step 6: Identify Points of Inflection

At \( x = \frac{1}{4} \) and \( x = -\frac{3}{4} \), the concavity changes which indicates these points are inflection points.

Step 7: Sketch the Curve

Using the information about increasing/decreasing intervals, local extrema, concavity, and inflection points, sketch the curve of \( f(x) = \sin(\pi x) - \cos(\pi x) \). Check with a calculator for accuracy.

Key Concepts

Increasing and Decreasing Intervals

When analyzing where a function is increasing or decreasing, we begin by evaluating its first derivative. For the function \( f(x) = \sin(\pi x) - \cos(\pi x) \), the first derivative is found to be \( f'(x) = \pi \cos(\pi x) + \pi \sin(\pi x) \). To determine the intervals, set the derivative equal to zero: \( \pi (\cos(\pi x) + \sin(\pi x)) = 0 \). This simplifies to \( \cos(\pi x) + \sin(\pi x) = 0 \), leading to solutions \( x = \frac{3}{4} \) and \( x = -\frac{1}{4} \) within the interval \([-1, 1]\).
We then test intervals around these points:

• For \( x < -\frac{1}{4} \), take a test point like \( x = -\frac{1}{2} \). Here, \( f'(x) > 0 \), meaning the function is increasing.
• Between \( x = -\frac{1}{4} \) and \( x = \frac{3}{4} \), a test point like \( x = 0 \) gives \( f'(x) < 0 \), indicating the function is decreasing.
• For \( x > \frac{3}{4} \), use a test point like \( x = 1 \). Once again, \( f'(x) < 0 \), so the function is again increasing.

Knowing where a function increases or decreases aids in understanding its behavior and getting hints at where local extremas might occur.

Local Minima and Maxima

The discovery of local minima and maxima occurs with the help of the First Derivative Test. We already determined that the critical points are \( x = -\frac{1}{4} \) and \( x = \frac{3}{4} \) by setting \( f'(x) = 0 \). The First Derivative Test analyzes the sign of \( f'(x) \) around these points to detect any sign change.

At \( x = -\frac{1}{4} \):

• Before \( x = -\frac{1}{4} \), the derivative is positive.
• After \( x = -\frac{1}{4} \), the derivative becomes negative.
• This indicates a change from increasing to decreasing, confirming a local maximum at \( x = -\frac{1}{4} \).

At \( x = \frac{3}{4} \):

• Before \( x = \frac{3}{4} \), the derivative is negative.
• After \( x = \frac{3}{4} \), the derivative switches to positive.
• This suggests a change from decreasing to increasing, identifying a local minimum at \( x = \frac{3}{4} \).

Locating these points gives a better view of the peaking behaviors of functions, helping us predict their rise and fall.

Concavity and Inflection Points

Concavity of a function provides insight into its curve behavior—whether it's curving upwards or downwards. We find concavity by calculating the second derivative. For the function \( f(x) = \sin(\pi x) - \cos(\pi x) \), the second derivative is \( f''(x) = -\pi^2 \sin(\pi x) + \pi^2 \cos(\pi x) \).

By setting the second derivative equal to zero, we reveal points of potential change in concavity: \( -\pi^2 \sin(\pi x) + \pi^2 \cos(\pi x) = 0 \). Simplification leads to \( \tan(\pi x) = 1 \), giving solutions \( x = \frac{1}{4} \) and \( x = -\frac{3}{4} \) for our interval \([-1, 1]\).

• Evaluate \( f''(x) \) just before and after these points to determine changes:
• If \( f''(x) > 0 \), the graph of \( f(x) \) is concave up (bowl shape).
• If \( f''(x) < 0 \), the graph of \( f(x) \) is concave down (cap shape).

A change in sign of \( f''(x) \) at these points denotes inflection points, where the curve shifts from being concave up to concave down and vice versa, indicating the flexibility in the curve's shape.

Derivative Analysis

Derivative analysis forms the backbone of understanding function behaviors across calculus. In this multi-step process, derivatives explain where functions rise and fall, hit highs or lows, and how they bend or shift.

• **First Derivative:** Represents the slope of the tangent line to the curve of the function, indicating the lying direction and speed changes.
• **Second Derivative:** Illuminates the curvature or bending behavior of the function, showing whether the slope itself is growing steeper or leveling out.

For effective derivative analysis:

• Find the zero points of these derivatives—these zeros suggest where the function behavior shifts.
• Test intervals around these critical points to determine how the function behaves on those intervals.
• Reflect on these findings. They narrate a beautiful story of the graph’s journey—a detailed landscape sewn together by highs and culminations, curves, and directional bends.

Each aspect of derivative analysis forms a facet of function exploration, unveiling the secrets from graphs and bringing about a precise understanding of mathematical and real-world phenomena.