Problem 64 - University Calculus: Early Transcendentals

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

The graphs show that as \( h \rightarrow 0 \), \( \frac{\cos(x+h)-\cos x}{h} \) approaches \(-\sin x\), illustrating derivative calculation.

Step 1: Understand the Exercise

The exercise requires us to graph two functions. The first is \( y = -\sin x \) over \( -\pi \leq x \leq 2\pi \). The second involves an expression that approximates a derivative: \( y = \frac{\cos(x + h) - \cos x}{h} \). We need to graph the latter expression for various values of \( h \) and observe what happens as \( h \) approaches zero from the right (\( h \rightarrow 0^{+} \)) and from the left (\( h \rightarrow 0^{-} \)). The goal is to identify the phenomenon being illustrated.

Step 2: Graph \( y = -\sin x \)

Graph the function \( y = -\sin x \) for \( -\pi \leq x \leq 2\pi \). This is a standard sine function reflected across the x-axis, with a period of \(2\pi\), starting at \(0\), reaching \(-1\) at \(\pi/2\), back to \(0\) at \(\pi\), \(1\) at \(3\pi/2\), and \(0\) again at \(2\pi\).

Step 3: Graphing the Difference Quotient with Positive h

Graph the function \( y = \frac{\cos(x+h) - \cos x}{h} \) for \( h = 1, 0.5, 0.3, \) and \( 0.1 \) using the range \( -\pi \leq x \leq 2\pi \). As \( h \) decreases, observe how the graph approaches the graph of \( y = \sin x \), indicating the derivative of the cosine function, which is \(-\sin x\).

Step 4: Graphing the Difference Quotient with Negative h

Next, graph the same function \( y = \frac{\cos(x+h) - \cos x}{h} \) for \( h = -1, -0.5, \) and \( -0.3 \). Observe how as \( h \rightarrow 0^{-} \), the graph continues to approach the graph of \( y = \sin x \).

Step 5: Observing the Phenomenon as h Approaches Zero

As \( h \rightarrow 0^{+} \) and \( h \rightarrow 0^{-} \), the graph of \( y = \frac{\cos(x+h) - \cos x}{h} \) approaches \(-\sin x\). This illustrates the definition of the derivative. Here, the derivative of \( \cos x \) is \(-\sin x \). The graphs serve as a visual demonstration of this limit process.

Step 6: Conclusion: Identify the Phenomenon

The phenomenon being illustrated is the concept of differentiation and the use of limits to find the derivative of a function. Specifically, this exercise demonstrates that the derivative of \( \cos x \) is \(-\sin x \), using a graphical approach and the difference quotient method as \( h \) approaches zero from both sides.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that refers to finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point. In the context of this exercise, differentiation helps us visually grasp how the function \( \cos x \) changes as we adjust the value of \( h \).

• The derivative of a function at a point indicates the slope of the tangent line to the function's graph at that point.
• When you calculate the derivative of a function, you are essentially finding a function that gives the slope at each point of the original function.
• For this exercise, the derivative of \( \cos x \) is \( -\sin x \).

Understanding differentiation helps us predict the behavior of a function without having to constantly plot points on a graph. By approaching this graphically, you can see how the difference quotient becomes the derivative when \( h \) approaches zero.

Limits

Limits play a crucial role when we talk about derivatives. They help in understanding the behavior of functions as the input approaches a certain value. In this exercise, using the function \( y = \frac{\cos(x+h) - \cos x}{h} \), you get to observe what happens as \( h \) gets closer and closer to zero.

• A limit helps us find the value that a function or sequence "approaches" as the input or index approaches some value.
• The expression \( \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h} \) graphically shows that it approaches \( -\sin x \) as \( h \) gets smaller.
• This visualization is key in comfortably understanding how derivatives are defined as limits.

By employing limits, differentiation becomes straightforward, showcasing that as \( h \rightarrow 0 \), the expression \( \frac{\cos(x+h) - \cos x}{h} \) settles into its derivative form \( -\sin x \).

Difference Quotient

The difference quotient is a central technique in calculus for understanding derivatives. It is essentially an approximation of the derivative and is represented by the formula \( \frac{f(x+h) - f(x)}{h} \).

• This quotient gives us the average rate of change of a function over the interval from \( x \) to \( x+h \).
• As \( h \) gets smaller, the difference quotient approaches the actual derivative if it exists.
• The graphing in this exercise shows the transition from the difference quotient to the derivative as \( h \) approaches zero both positively and negatively.

For \( \cos x \), the difference quotient \( \frac{\cos(x+h) - \cos x}{h} \) clearly approaches \( -\sin x \), visually confirming the derivative. This method is not only pivotal for theoretical understanding but also serves as groundwork for numerical approximations in advanced mathematics and practical applications.