Problem 51 - Thomas Calculus

Question

Graph \(y=\cos x\) 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 \(h \rightarrow 0^{-7} ?\) What phenomenon is being illustrated here?

Step-by-Step Solution

Answer

The difference quotient converges to \(-\sin x\) as \(h \to 0\). This illustrates the derivative of \(\cos x\).

Step 1: Understanding the Problem

We need to graph the function \( y = \cos x \) on the interval \( -\pi \leq x \leq 2\pi \) and compare it with the graph of the difference quotient \( y = \frac{\cos(x+h) - \cos x}{h} \) for different values of \( h \). Then, we observe the behavior as \( h \) approaches zero from both positive and negative directions.

Step 2: Graphing \( y = \cos x \)

The function \( y = \cos x \) is a standard cosine curve with a period of \( 2\pi \). It will start at the maximum value at \( x = -\pi \), touch zero at \( x = -\frac{\pi}{2} \), reach a minimum at \( x = 0 \), and complete one full wave by \( x = \pi \) before continuing to another max at \( x = 2\pi \). Draw this curve from \( x = -\pi \) to \( x = 2\pi \).

Step 3: Graphing the Difference Quotient for Positive h Values

Graph \( y = \frac{\cos(x+h) - \cos x}{h} \) for \( h = 1, 0.5, 0.3, \text{ and } 0.1 \). These graphs represent the slopes of secant lines as \( h \) changes. Plot these lines on the same graph as \( y = \cos x \) to show how the secant lines become tangent to the curve as \( h \) approaches 0 from the positive side.

Step 4: Graphing the Difference Quotient for Negative h Values

Open a new graph window and plot \( y = \frac{\cos(x+h) - \cos x}{h} \) for \( h = -1, -0.5, \text{ and } -0.3 \). Observe that these secant lines also start to resemble tangents as \( h \) approaches 0 from the negative side. This illustrates the approach of the derivative as \( h \) approaches zero.

Step 5: Understanding the Limit Behavior

Observe the behavior of the difference quotients as \( h \to 0^{+} \) and \( h \to 0^{-} \). In both cases, the difference quotients converge to the derivative of the cosine function at that point, which is \( -\sin x \). This phenomenon illustrates the definition of the derivative as the limit of the difference quotient as \( h \to 0 \).

Key Concepts

Difference quotient

The difference quotient is a fundamental concept in calculus. It's used to approximate the derivative of a function, which essentially gives us the function's "rate of change." For a function \( f(x) \), the difference quotient is expressed as: \[ \frac{f(x+h) - f(x)}{h} \] This formula calculates the average slope of the line between two points on the graph of \( f(x) \). As \( h \) becomes smaller, the line becomes closer to the tangent at \( x \), providing a more accurate estimate of the derivative. In our specific example, the function being examined is the cosine function, thus applying the difference quotient helps to understand its change over a small interval.

Cosine function

The cosine function is a trigonometric function often represented as \( y = \cos x \). It's periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units. Some key features include:

• A maximum value of 1 and a minimum value of -1.
• Starts at 1 when \( x = 0 \), decreases to -1 by \( x = \pi \), and returns to 1 by \( x = 2\pi \).
• Symmetrical around the y-axis, making it an even function, meaning \( \cos(-x) = \cos x \).

The behavior and shape of the cosine graph are essential when considering the effects of the difference quotient, as the slope of secant lines calculated in this context relates directly to the cosine curve's characteristics.

Graphical analysis

In graphical analysis, visually interpreting the behavior of functions or expressions on a graph is critical. For this exercise, we graph both \( y = \cos x \) and the difference quotient \( y = \frac{\cos(x+h) - \cos(x)}{h} \) across a range of \( h \) values. The purpose of this is to observe how secant lines, the average rate of change of the cosine function over the interval \( h \), change as \( h \) gets smaller.

• As \( h \) approaches zero, the secant lines become closer to the cosine curve, reflecting the tangent line at a particular point.
• Both positive and negative \( h \) values are examined to show convergence from both sides.

Through graphical analysis, one gains a visual understanding of how the derivative (or instantaneous rate of change) represents the slope of the tangent line, providing deeper insight into calculus's fundamental concepts.

Limit behavior

Limit behavior in calculus describes how a function behaves as its input approaches some value. When dealing with derivatives, we study the limit of the difference quotient as \( h \) approaches zero to define derivative precisely. For the cosine function, this limit is: \[ \lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h} = -\sin x \] This equation shows that as \( h \) gets smaller (from both positive and negative sides), the difference quotient becomes the derivative of \( \cos x \), which is \( -\sin x \). Understanding this behavior is crucial:

• It helps confirm the underlying rules of calculus - that the derivative at a point tells us about the instantaneous rate of change.
• By approaching zero from both directions, we ensure that the limit truly exists, illustrating smooth transition and consistency.

This foundational approach paves the way for solving real-world problems involving changing rates and velocities.