Problem 17 - Calculus with Concepts in Calculus

Question

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ f(x)=\sin 2 x $$

Step-by-Step Solution

Answer

The function is concave upward on \((-\infty, 0)\) and repeating intervals \([\pi, 2\pi)\). It is concave downward on \((0, \pi)\) and repeating intervals \([2\pi, 3\pi)\).

Step 1: Find the Second Derivative

To analyze concavity, we need the second derivative of the function. Start by finding the first derivative of \( f(x) = \sin(2x) \). The derivative of \( \sin(2x) \) is \( f'(x) = 2\cos(2x) \). Now, take the derivative of \( f'(x) \) to get the second derivative: \( f''(x) = -4\sin(2x) \).

Step 2: Determine Where the Second Derivative Equals Zero

To find potential inflection points and determine concavity, set the second derivative equal to zero: \(-4\sin(2x) = 0\). Solving this equation gives \(\sin(2x) = 0\). The solutions are where \(2x = n\pi\), resulting in \(x = \frac{n\pi}{2}\), where \(n\) is an integer.

Step 3: Divide the Number Line Into Test Intervals

Use the points found in Step 2, \(x = \frac{n\pi}{2}\), to divide the number line. Choose test points between these intervals to determine concavity: - For \(-\frac{\pi}{2} < x < 0\), use \(x = -\frac{\pi}{4}\).- For \(0 < x < \frac{\pi}{2}\), use \(x = \frac{\pi}{4}\).

Step 4: Test Each Interval

Substitute the test points into the second derivative to determine the sign:- For \(x = -\frac{\pi}{4}\), \(f''(x) = -4\sin(-\frac{\pi}{2}) = 4\), so the function is concave upward.- For \(x = \frac{\pi}{4}\), \(f''(x) = -4\sin(\frac{\pi}{2}) = -4\), so the function is concave downward.

Step 5: Identify Intervals of Concavity

From the previous tests, determine where the function is concave:- Concave upward on \((-\infty, 0)\) and \( (\pi, 2\pi) \) repeating every \(2\pi\).- Concave downward on \((0, \pi)\) and \( (2\pi, 3\pi) \) repeating every \(2\pi\).

Step 6: Sketch the Graph

Plot the function \( f(x) = \sin(2x) \) indicating its periodic nature and alternating intervals of concavity based on the previous intervals. Notice there are points where concavity changes, corresponding to \(x = \frac{n\pi}{2}\) with changing signs.

Key Concepts

Second Derivative

The second derivative is a crucial tool in analyzing the concavity of a function. For the function given as \( f(x) = \sin(2x) \), determining concavity requires us to first find the second derivative. This is because the second derivative, \( f''(x) \), reveals where the graph of the function is concave up or concave down.

• The first step involves finding the first derivative, \( f'(x) = 2\cos(2x) \).
• The second derivative is then derived from the first, resulting in \( f''(x) = -4\sin(2x) \).

When \( f''(x) > 0 \), the function is concave upward, while \( f''(x) < 0 \) indicates the function is concave downward.
Understanding this concept helps determine the behavior of the graph and its concavity over different intervals.

Inflection Points

Inflection points are critical points where the graph of a function changes its concavity. To find these points in our function \( f(x) = \sin(2x) \), we set the second derivative to zero.
This means solving \(-4\sin(2x) = 0\). The solutions to this equation occur at \( \sin(2x) = 0 \), which simplify to \( 2x = n\pi \), or \( x = \frac{n\pi}{2} \) where \( n \) is an integer.

• At each \( x = \frac{n\pi}{2} \), the graph may potentially change from being concave up to concave down, or vice versa.

These inflection points mark the vertices of concavity changes on the graph, giving essential insights into its shape and behavior.

Interval Testing

Once the inflection points are identified, interval testing helps determine the concavity of the function across these intervals. For the function \( f(x) = \sin(2x) \), the number line is divided based on the inflection points obtained, like \( x = \frac{n\pi}{2} \).
To determine concavity:

• Select test points between these inflection points, such as \( x = -\frac{\pi}{4} \) or \( x = \frac{\pi}{4} \).
• Substitute these test points into the second derivative \( f''(x) = -4\sin(2x) \).
• For these intervals, check the sign of the result: a positive result indicates concave up, while a negative one indicates concave down.

For instance:- When \( x = -\frac{\pi}{4} \), \( f''(x) = 4 \) means the function is concave up.- When \( x = \frac{\pi}{4} \), \( f''(x) = -4 \) means the function is concave down.
This testing method systematically helps sketch the graph more accurately and understand its curvature across different segments.

Graph Sketching

Graph sketching is a fundamental skill, enhanced by understanding the concept of concavity and using the second derivative. For the function \( f(x) = \sin(2x) \), sketching involves demonstrating its concavity along with its periodic nature.

• Draw the basic sine wave, but remember to adjust for the frequency by 2, indicating a shorter period.
• Identify sectors for concave up and concave down based on previous analysis.
• Include markers or vertical lines at each inflection point \( x = \frac{n\pi}{2} \) to show changes in concavity.

The visual representation should highlight how the wave alternates in its shape.
Understanding these sketching techniques helps bridge the function’s mathematical behavior with its graphical appearance, providing a holistic understanding of the function’s nature.