Problem 105 - University Calculus: Early Transcendentals

Question

Graph the function \(y=2 \cos 2 x\) for \(-2 \leq x \leq 3.5 .\) Then, on the same screen, graph $$y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$ for \(h=1.0,0.5,\) and \(0.2 .\) Experiment with other values of \(h\) including negative values. What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

Step-by-Step Solution

Answer

As \( h \rightarrow 0 \), the difference quotient approaches \( 2 \cos 2x \). It illustrates the derivative of \( \sin 2x \).

Step 1: Understand the Function

First, identify the primary function to graph, which is the cosine function given as \( y = 2 \cos 2x \). This is a transformed cosine function with an amplitude of 2 and a horizontal scaling factor of 2.

Step 2: Plot the Cosine Function

To plot \( y = 2 \cos 2x \) from \( x = -2 \) to \( x = 3.5 \), start by determining the period. The function has a period of \( \frac{2\pi}{2} = \pi \), and identify key points for one period and extend them within the given range to construct the graph.

Step 3: Determine the Difference Quotient

Now, consider the difference quotient \( y = \frac{\sin 2(x+h) - \sin 2x}{h} \). This function represents the discrete approximation of the derivative of \( \sin 2x \) with respect to \( x \).

Step 4: Experiment with Different h Values

Graph the difference quotient for each given \( h \) value: 1.0, 0.5, and 0.2. Notice how the graph changes. For each \( h \), calculate the function over the same \( x \) range \([-2, 3.5]\). Observe how as \( h \) becomes smaller, the approximation gets closer to the derivative of \( \sin 2x \).

Step 5: Explore with Negative h Values

Try negative \( h \) values (e.g., \(-0.5\), \(-0.2\)) to see how the graph behaves. Notice that \( y = \frac{\sin 2(x+h) - \sin 2x}{h} \) will reflect differently because of negative division.

Step 6: Analyze Behavior as h Approaches Zero

As \( h \rightarrow 0 \), the graph of the difference quotient approaches the graph of the derivative of \( \sin 2x \), which is \( 2 \cos 2x \). This illustrates the definition of the derivative and shows how finite difference approximates the derivative.

Key Concepts

Cosine Function

When graphing trigonometric functions, the cosine function often comes into play due to its periodic nature and symmetry. The cosine function is written as \( y = \cos x \). This function has a standard amplitude of 1, which means the maximum height from the central axis is 1.
However, in the exercise given, we're working with a transformed cosine function, \( y = 2 \cos 2x \). Here's what this transformation means:

• Amplitude Changes: The "2" outside the cosine function indicates the amplitude is now 2, stretching the graph vertically.
• Period Adjustment: The "2" inside the cosine function compresses the graph horizontally so that its period is now \( \frac{2\pi}{2} = \pi \). This means it completes one full cycle in the span of \( \pi \) rather than the usual \( 2\pi \).

To graph this function, understand that it will oscillate between -2 and 2 and will make complete cycles more frequently due to this adjustment. Begin plotting key values of \( x \) within the specified range \(-2 \leq x \leq 3.5 \), starting from significant points where cosine equals zero or reaches its maximum and minimum.

Difference Quotient

The difference quotient is a fundamental concept in calculus used to approximate the derivative of a function. It is represented as \( \frac{f(x+h) - f(x)}{h} \) for a given function \( f \), and seeks to measure the instantaneous rate of change.
In the exercise, the difference quotient for \( \sin 2x \) is given by \( \frac{\sin 2(x+h) - \sin 2x}{h} \). Here are key points to understand about this function:

• Purpose: It approximates how much \( \sin 2x \) changes as \( x \) changes by a small amount \( h \).
• Behavior with Different \( h \): As you vary \( h \), you can observe how this quotient correlates more closely with the derivative of \( \sin 2x \), especially as \( h \) approaches zero.
• Graphical Impact: When graphed concurrently with the cosine function, it provides a visual insight into the tangent's approach to being nearly equal to the derivative as \( h \) decreases.

Understanding the role of \( h \) is crucial. With smaller \( h \) values, the approximation becomes more accurate. Trying negative \( h \) values serves to highlight symmetry and the nature of derivatives.

Derivatives

Derivatives play a fundamental role in calculus and signify the rate at which a function is changing at any given point. In the context of trigonometric functions like \( \sin x \), the derivative can be explored through calculus rules and principles.
For \( \sin 2x \), its derivative is \( 2 \cos 2x \). This is derived by applying the chain rule in calculus to the function \( f(x) = \sin(2x) \). Here’s a breakdown of key points:

• Chain Rule Application: The chain rule involves differentiating \( \sin(2x) \) as if it were \( \sin(u) \), resulting in \( \cos(u) \), and then multiplying by the derivative of the inside function \( u = 2x \), which is 2.
• As \( h \rightarrow 0 \): In the difference quotient, as \( h \) tends to zero, it precisely approaches this derivative, \( 2 \cos 2x \). This is because the finite difference mimics the definition of the derivative.
• Graph Interpretation: Visually, this derivative can be seen as the slope of the tangent line to the graph of \( \sin 2x \) at any given point, which correlates with its corresponding cosine graph.

So, exploring derivatives through this graphical approach enhances understanding of how trigonometric functions behave and change over specified intervals, which are foundational concepts in calculus and essential for deeper mathematical studies.