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Chapter 2: Problem 40

Find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval \([0,2 \pi] .\) What can you conclude about the slope of the sine function \(\sin a x\) at the origin? a.\(y=\sin 3 x\) b.\(y=\sin \frac{x}{2}\)

Short Answer

Expert verified
The slope of the tangent line at the origin for the sine function \(y=\sin 3x\) is 3, and for the function \(y=\sin x/2\), it is 1/2. This suggests that the slope of the tangent line at the origin for the function \(\sin ax\) is equal to the number of complete cycles the function completes in the interval [0, 2pi].

Step by step solution

01

Determine the general derivative

To begin with, the derivative of the basic sine function \(y=\sin x\) needs to be found. It is known from calculus that the derivative of \(\sin x\) is \(\cos x\).
02

Derive the specific functions and calculate slopes

Now, apply this rule to the specific functions given in the exercise. Take the derivative of \(y=\sin 3x\) with respect to \(x\). By using the chain rule, the derivative becomes \(3\cos(3x)\). Evaluate this at \(x=0\) to obtain the slope of the tangent at the origin, resulting in a value of 3. Repeat this process for the function \(y=\sin x/2\). Its derivative is \(\frac{1}{2}\cos (x/2)\) and the slope at the origin is 1/2.
03

Analyze the relationship

Now examine the relationship between the slopes obtained and the number of complete cycles in the interval [0, 2pi]. For \(y=\sin 3x\), there are 3 complete cycles and the slope is also 3. For \(y=\sin x/2\), the function completes half a cycle and the slope is 1/2. It could be concluded that the slope of the tangent line at the origin for the function \(\sin ax\) is equal to the number of complete cycles the function completes in the interval [0, 2pi].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative of sine function
Calculus provides us the tool to find the rate at which functions change, known as the derivative. For the basic sine function, represented as \(y = \sin x\), its derivative is \(\cos x\). This result means that the slope of the tangent line to the sine curve at any point \(x\) is given by the value of \(\cos x\):
  • If \(\cos x = 1\), the slope is 1.
  • If \(\cos x = 0\), the slope is 0.
  • If \(\cos x = -1\), the slope is -1.
Understanding the derivative of the sine function is crucial as it defines how rapidly the function is increasing or decreasing at any point. It's especially handy for analyzing the behavior of sine functions in various real-world phenomena.
Chain rule
The chain rule is an essential formula in calculus used when differentiating composite functions. It helps us to find the derivative of a function that is nested within another function. For example, in the function \(y = \sin(3x)\), the sine function contains another function \(3x\). The chain rule states: if we have \(y = f(g(x))\), then \(y' = f'(g(x)) \times g'(x)\).

Here's how the chain rule applies to our example:
  • Differentiate the outer function: Derivative of \(\sin\) is \(\cos\).
  • Multiply by the derivative of the inner function: Derivative of \(3x\) is 3.
  • Result is \(3 \cos(3x)\).
With this understanding, it's easier to handle more complicated functions involving trigonometric elements nested within each other.
Slope of tangent line
The concept of the slope of a tangent line is pivotal in calculus. This slope determines how steep the curve is at a particular point. In trigonometric functions, it's especially important. For the sine function, the slope at any point \(x\) is determined by its derivative, \(\cos x\).

The exercise gives:
  • For \(y = \sin 3x\), the derivative is \(3\cos(3x)\), and the slope at \(x = 0\) is 3.
  • For \(y = \sin \frac{x}{2}\), the derivative is \(\frac{1}{2} \cos(\frac{x}{2})\), and the slope at \(x = 0\) is \(\frac{1}{2}\).
Here, the slope of the tangent line at the origin gives us immediate insight into how fast or slow the function is changing right at the start.
Trigonometric functions
Trigonometric functions, such as sine, cosine, and tangent, are fundamental components of calculus dealing with angles and cycles. These functions are periodic and are commonly used in physics, engineering, and signal processing.

Key features of trigonometric functions include:
  • Periodic nature: They repeat their values over specific intervals.
  • Unit Circle: They can be visualized using the unit circle, showing relationships between angles and lengths.
  • Sine function: It oscillates between -1 and 1.
Understanding trigonometric functions helps in solving differential equations, modeling waves, and explaining phenomena such as sound and light.
Cycles in trigonometric functions
Cycles refer to how often a trigonometric function repeats itself within a given interval. For example, the standard sine function \(\sin x\) completes one full cycle in the interval \([0, 2\pi]\).

In our exercise:
  • \(y = \sin 3x\) completes 3 cycles in \([0, 2\pi]\).
  • \(y = \sin \frac{x}{2}\) completes half a cycle in \([0, 2\pi]\).
Interestingly, the number of cycles directly relates to the slope of the tangent line at the origin. The more cycles a function completes, the steeper the slope. This relationship aids in predicting complex wave behaviors and reinforces the connection between differential calculus and periodic functions.

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Most popular questions from this chapter

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. $$ x \cos y=1, \quad\left(2, \frac{\pi}{3}\right) $$

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