91Ó°ÊÓ

Graphing trigonometric functions like cosine and sine is about recognizing patterns and transformations. For the function \(y = 2\cos(2x)\), you start by identifying its key features:

• Amplitude: The amplitude here is 2, meaning the graph will reach heights of 2 and depths of -2.
• Frequency: This frequency of 2 indicates the cosine wave completes a cycle every \(\pi\) radians.
• Period: The period, the length of one complete cycle, is \(\pi\) in this case as calculated by \(\frac{2\pi}{b}\), where \(b=2\).

These aspects will help in plotting the wave correctly. You mark the key points where the cosine function starts, peaks, passes through zero, and reaches its lowest point again. Continue to graph these key points within the specified domain of \(-2 \leq x \leq 3.5\).
This gives you a visual representation of how the function behaves over the interval and prepares you for understanding changes in trigonometric graphs.

The difference quotient is a fundamental concept in calculus used to articulate how a function behaves between two points. It is defined as \(\frac{f(x+h) - f(x)}{h}\), and is particularly important for understanding derivatives over an interval as \(h\) approaches zero.
In this exercise, we use it for the function \(\sin(2x)\) with the difference quotient \(\frac{\sin\big(2(x+h)\big)-\sin(2x)}{h}\). Here's how to interpret it:

• Approximation: The difference quotient shows the average rate of change of the function over the interval \([x, x+h]\).
• Slope of Secant Line: It represents the slope of a secant line joining the points \((x, \sin(2x))\) and \((x+h, \sin(2(x+h)))\).
• Approaching Derivative: As \(h\) becomes very small, approaching zero, the difference quotient nears the instantaneous rate of change—essentially the derivative of \(\sin(2x)\).

Experimenting with various values of \(h\), including negative ones, allows students to observe how closely the quotient approximates the derivative when \(h\) is close to zero. It provides a practical way to see the theoretical concepts of calculus in action.

Derivatives in calculus are vital for understanding how functions change. A derivative gives you the slope of the tangent line to the curve of a function at any point, revealing the instantaneous rate of change.
For the function \(\sin(2x)\) in this problem, the derivative is calculated using the process of the difference quotient. As \(h\) approaches zero, the difference quotient \(\frac{\sin\big(2(x+h)\big)-\sin(2x)}{h}\) converges to \(2\cos(2x)\).
The steps to understanding the derivative include:

• Conceptualizing Instantaneous Change: This is a shift from the average rate of change (over an interval) to the specific rate at one point.
• Link to Graph: The derivative function \(2\cos(2x)\) reflects the slope of the sine wave \(\sin(2x)\), linking the steepness of the tangent to the shape of \(\sin(2x)\).
• Behavior as \(h \rightarrow 0\): This transition makes clear how the theoretical ideas of limits in calculus meet the practical calculations of derivatives.

By understanding derivatives, students see how calculus provides tools for analyzing dynamic systems, predicting changes, and solving real-world problems.