Problem 68 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum slope of the graph of \(y=\sin (b x)\) is \(b\).

Step-by-Step Solution

Answer

The statement is true. The maximum slope of the graph of \(y=\sin (b x)\) is indeed \(b\), as the derivative of \(y=\sin (b x)\) reaches its maximum value \(b\), when \(bx = 2n\pi\), where \(n\) is any integer, causing the cos function to be 1.

Step 1: Understanding the function and its derivative

The function given is \(y=\sin (b x)\). The maximum slope of this function can be determined by computing its first derivative. Using chain rule, the derivative of \(y=\sin (b x)\) is \(y' = b \cos (b x)\).

Step 2: Finding the maximum slope

The cos function ranges between -1 and +1, therefore the maximum value of \(\cos(bx)\) is 1. Therefore, the maximum value for the slope \(y'\) is \(b\). This happens when the value inside the cos function is an even multiple of \(\pi\), \(bx = 2n\pi\), where \(n\) is any integer.

Step 3: Validate the statement

Since the maximum value of the derivative is indeed \(b\), as it is the result from Step 2, the statement 'The maximum slope of the graph of \(y=\sin (b x)\) is \(b\)' is true.

Key Concepts

Calculating Derivatives

The process of calculating derivatives is essential in understanding the behavior of functions, particularly when you're trying to find rates of change or slopes of tangents to curves. The derivative of a function at a point is the slope of the tangent line to the function's graph at that point. To calculate the derivative of a basic sine function, let's take, for example, the function
\[ y = \text{sin}(x). \]
Its derivative is found using the foundational rules of differentiation:
\[ y' = \text{cos}(x). \]
This simple example shows that the slope of the tangent to the sine curve at any point x is given by the cosine of that point. When we introduce a coefficient, such as b in y = sin(bx), we have to account for this scaling factor in our derivative. Using the chain rule, we differentiate the outer function, sine, as we would normally, then multiply by the derivative of the inner function, bx, which is just b. This gives us a slope formula dependent on the coefficient b:
\[ y' = b \text{cos}(bx). \]
To understand how the slope behaves across different values of x, one must consider the properties of the cosine function.

Chain Rule in Calculus

The chain rule is a powerful tool in calculus that allows us to differentiate composite functions. A composite function is one which has another function nested inside it, like f(g(x)). In such cases, we cannot apply the standard rules of differentiation directly. That's when the chain rule comes into play. The formal statement of the chain rule is:
\[ (f\circ g)'(x) = f'(g(x))g'(x). \]
In the context of our exercise, we applied the chain rule to find the derivative of the sine function with an argument that's not just x, but bx. Even with its simplicity, the chain rule can be difficult for some students to wrap their heads around. Think of it like peeling an onion, where we differentiate from the outside to the inside layers. For the function
\[ y = \text{sin}(bx), \]
we start by differentiating the outer layer, sin, yielding cos, and then we multiply by the derivative of the inner layer, which is b. It's this multiplication by the inner derivative that adjusts the slope correctly for scaled arguments.

Sine Function Properties

The sine function has a rich set of properties that are essential to understand in trigonometry and calculus. Some of its key characteristics include:

• It is periodic with a period of 2π, meaning that sin(x) = sin(x + 2nπ) for any integer n.
• The function oscillates between -1 and 1. Thus, its maximum and minimum values are 1 and -1, respectively.
• It starts at 0, rises to 1 at π/2, falls back to 0 at π, descends to -1 at 3π/2, and returns again to 0 at 2π.

These properties have direct implications for the slopes of sine functions. Like the sine function, its derivative, the cosine function, is also periodic and oscillates between -1 and 1. In the context of our exercise, since the amplitude of the cosine function is 1, the absolute maximum value of b cos(bx) is b, when cos(bx) = 1. This occurs at certain intervals, specifically at the multiples of 2π. At these points, the slope of the sine function, the value of its derivative, reaches its peak. Understanding these properties enables us to confidently predict the behavior of sine functions and, by extension, their derivatives.