Problem 52 - Thomas Calculus

Question

Graph \(y=-\sin x\) for \(-\pi \leq x \leq 2 \pi .\) On the same screen, graph $$ y=\frac{\cos (x+h)-\cos x}{h} $$ for \(h=1,0.5,0.3,\) and \(0.1 .\) Then, in a new window, try \(h=-1,-0.5,\) and \(-0.3 .\) What happens as \(h \rightarrow 0^{+} ?\) As \(h \rightarrow 0^{-} ?\) What phenomenon is being illustrated here?

Step-by-Step Solution

Answer

As \( h \rightarrow 0^{+} \) and \( h \rightarrow 0^{-}, \) the graphs approach \( y = -\sin x \), illustrating the derivative of \( \cos x \).

Step 1: Graph the Function y = -sin(x)

Begin by graphing the function \( y = -\sin x \) over the interval \( -\pi \leq x \leq 2\pi \). This graph will have a wave-like form with peaks and troughs. The wave will start from zero at \( x = -\pi \), reach a minimum at \( x = -\pi/2 \), return to zero at \( x = 0 \), have a maximum at \( x = \pi/2 \), return to zero at \( x = \pi \) and continue in the same manner towards \( x = 2\pi \).

Step 2: Graph for Positive h Values in Derivative Approximation

Next, graph \( y = \frac{\cos(x+h) - \cos(x)}{h} \) for \( h = 1, 0.5, 0.3, 0.1 \). This is a discrete approximation of the derivative of \( \cos x \), evaluated differently as \( h \) changes. Plot each graph on the same set of axes over the interval \( -\pi \leq x \leq 2\pi \). As \( h \) decreases towards zero from the positive side, observe how these graphs are approaching \( y = -\sin x \).

Step 3: Graph for Negative h Values in Derivative Approximation

In another window, plot \( y = \frac{\cos(x+h) - \cos(x)}{h} \) for \( h = -1, -0.5, -0.3 \). Here, \( h \) moves towards zero from the negative side. Observe that as \( h \) approaches zero, the graphs again seem to converge towards \( y = -\sin x \), showing the consistency of this approximation from both directions.

Step 4: Interpret the Results

As \( h \rightarrow 0^{+} \), the function \( \frac{\cos(x+h) - \cos(x)}{h} \) converges to \( -\sin x \). Similarly, as \( h \rightarrow 0^{-} \), the same convergence is observed. This behavior illustrates the geometric and analytical concept of a derivative, where \( \frac{d}{dx} \cos x = -\sin x \). The approximation converges to the actual derivative, visualizing both limits \( h \rightarrow 0^{+} \) and \( h \rightarrow 0^{-} \).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions helps us understand their behavior over a specific interval. Let's consider the graph of the function \( y = -\sin x \) over the interval \( -\pi \leq x \leq 2\pi \). This function results in a wave-like graph called a sinusoidal wave.

Key points and behavior:

• The function starts from zero at \( x = -\pi \).
• It reaches its minimum value of -1 at \( x = -\pi/2 \).
• It climbs back to zero at \( x = 0 \), then further peaks at 1 at \( x = \pi/2 \), creating a wave pattern.
• This oscillating behavior continues, showing similar patterns up to \( x = 2\pi \).

Trigonometric function graphs such as sine and cosine describe periodic behavior and are crucial in various applications, including physics and engineering.

Limits and Convergence

In calculus, limits and convergence are foundational concepts to understand behaviors of functions as their variables approach certain points. In our exercise, this is highlighted by exploring the expression \( y = \frac{\cos(x+h) - \cos(x)}{h} \).

Understanding the role of \( h \):

• As \( h \to 0^+ \) (approaching zero from the positive side), the function approximations become more accurate.
• This results in the graphs appearing closer to the graph of \( y = -\sin x \).
• Similarly, \( h \to 0^- \) yields the same outcome. The graphs from negative \( h \) values converge in the same manner.

This idea illustrates the geometric concept that as \( h \) approaches closer to zero, the function itself mimics the derivative of another function, showcasing the power of limits in improving the precision of function behavior approximations.

Derivative of Cosine Function

The task beautifully illustrates the process of approximating derivatives using limits. Understanding \( \frac{\cos(x+h) - \cos(x)}{h} \) is vital as it visually represents approximating the derivative of the cosine function.

Importance of derivatives:

• Derivatives represent the rate of change of a function. For cosine, the derivative is \( -\sin x \).
• When approximating using the difference quotient as \( h \) approaches zero, the value essentially becomes the function's derivative.
• Both positive and negative approaches of \( h \) illustrate this standardized convergence, emphasizing universal nature of calculus limits.

In essence, learning this approximation via graphing helps solidify the concept of derivatives and their practical applications in mathematical modelling.