Problem 49 - Calculus: Graphical, Numerical, Algebraic

Question

Sine Function Estimate the slope of the curve \(y=\sin x\) at \(x=1 .\) (Hint: See Exercises 41 and $42 . )

Step-by-Step Solution

Answer

The slope of the curve \(y = \sin(x)\) at \(x = 1\) is approximately \(\cos(1) ≈ 0.54\). This is an approximation because the cosine of 1 is not a simple, exact number.

Step 1: Identify the function

The function given is a sine function \(y = \sin(x)\). We are asked to find the slope of the curve that this function generates at the specific point where \(x = 1\).

Step 2: Find the derivative

The derivative of \(y = \sin(x)\) is \(y' = \cos(x)\). So, the derivative of our function is the cosine of \(x\).

Step 3: Substitute the given value

Substitute \(x = 1\) into the derivative: \(y'(1) = \cos(1)\).

Key Concepts

Derivative of Trigonometric Functions

In calculus, we often explore how functions change, which is what derivatives help us find. A derivative gives us the rate of change or the "slope" at any given point on a function's curve. When dealing with trigonometric functions like sine and cosine, knowing how to differentiate them is crucial.

For the sine function, denoted by \(y = \sin(x)\), the process of finding the derivative involves understanding its periodic nature. The derivative of \(\sin(x)\) is \(\cos(x)\). This means that at any point \(x\), the rate of change or slope of \(\sin(x)\) can be determined by the value of \(\cos(x)\).

• The derivative of \(\sin(x)\) is \(\cos(x)\).
• The derivative of \(\cos(x)\) is \(-\sin(x)\).
• This periodic pattern continues for other trigonometric functions.

Understanding these derivatives helps in multiple areas of calculus, especially in finding slopes and evaluating limits, providing a fundamental building block for more complex analysis.

Sine Function

The sine function, represented as \(y = \sin(x)\), is a fundamental mathematical function with a wave-like shape. This function is periodic, meaning it repeats its pattern every \(2\pi\) units. The key features of this function are its amplitude, period, and phase shift, which affect how the wave appears.

In the context of calculus, one of the sine function's core insights is its behavior over intervals:

• Amplitude: This is the height of the wave, showing how far the function rises or falls from its equilibrium, typically \(1\) for sine.
• Period: The interval it takes for the function to repeat. For \(\sin(x)\), the period is \(2\pi\).
• Phase Shift: Any horizontal shifts in the sine curve. For \(\sin(x)\), this shift is usually \(0\).

Knowing these attributes is essential when differentiating the sine function, especially when applying it to real-world situations like sound waves, light, and other periodic phenomena.

Slope of a Curve

The slope of a curve at a given point can provide valuable information about the function's behavior at that specific location. Calculus and derivatives make it possible to find this slope precisely. A slope is the "steepness" or "incline" of the curve, and is synonymous with the derivative.

When calculating the slope of a curve generated by a trigonometric function, such as \(y = \sin(x)\), we use the derivative of the function. For example:

• To find the slope of \(\sin(x)\) at \(x = 1\), we calculate \(y'(x) = \cos(x)\).
• Substituting \(x = 1\) gives \(y'(1) = \cos(1)\), which provides the precise slope at this point.
• This slope tells us the direction the curve is moving, and how steep the curve is, at \(x = 1\).

This slope calculation is fundamental in fields requiring precise directional movement understanding, such as physics, engineering, and economics, enabling a deeper analysis of dynamic changes within functions.