Problem 42 - Calculus - AP Edition

Question

Find formulas for \(d y\) and \(\Delta y\). $$y=\sin x$$

Step-by-Step Solution

Answer

Differentials: \(dy = \cos x \, dx\); Change in function: \(\Delta y = \sin x \, (\cos(\Delta x) - 1) + \cos x \, \sin(\Delta x)\).

Step 1: Differentiate the Function

To find the differential, we need to differentiate the function with respect to \(x\). Given the function \(y = \sin x\), the derivative is obtained as follows:\[ \frac{dy}{dx} = \cos x \]

Step 2: Write the Formula for dy

The differential \(dy\) can be expressed by multiplying the derivative by \(dx\):\[ dy = \cos x \, dx \]

Step 3: Use the Definition for Δy

The change in \(y\), represented by \(\Delta y\), is given by the change in the function value \(\Delta y = y(x + \Delta x) - y(x)\). For \(y = \sin x\), this becomes:\[ \Delta y = \sin(x + \Delta x) - \sin x \]

Step 4: Apply the Sine Addition Formula

Use the sine addition formula to simplify \(\sin(x + \Delta x)\):\[ \sin(x + \Delta x) = \sin x \, \cos(\Delta x) + \cos x \, \sin(\Delta x) \]This makes \(\Delta y\):\[ \Delta y = (\sin x \, \cos(\Delta x) + \cos x \, \sin(\Delta x)) - \sin x \]

Step 5: Simplify Δy Expression

Simplify the expression for \(\Delta y\):\[ \Delta y = \sin x \, (\cos(\Delta x) - 1) + \cos x \, \sin(\Delta x) \]

Key Concepts

Differential of a Function

In calculus, the differential of a function helps us understand how a function changes as its inputs change. It's like looking at a map and figuring out the direction you need to go to keep on track. The differential, denoted as \( dy \), gives a linear approximation of a small change in \( y \) when there is a small change in \( x \), called \( dx \).
For a function like \( y = \sin x \), we find the derivative first. The derivative tells us how \( y \) changes with respect to \( x \). In this case, the derivative is \( \frac{dy}{dx} = \cos x \).
This derivative multiplied by \( dx \) helps us write the differential as:

• \( dy = \cos x \cdot dx \)

So, \( dy \) represents a small change in \( y \) as \( x \) changes slightly.

Sine Addition Formula

The sine addition formula is a handy tool for solving problems involving angles. It allows us to break down the sine of a sum of two angles into simpler components. The formula is written as:

• \( \sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b \)

This formula is applied when analyzing changes in the sine function as we assess \( \Delta y \), the change in \( y \), for small variations in \( x \).
In the original problem, we use the sine addition formula for a small angle \( \Delta x \) added to \( x \). This makes the expression \( \sin(x + \Delta x) = \sin x \cdot \cos(\Delta x) + \cos x \cdot \sin(\Delta x) \).
Breaking down complex expressions into manageable parts helps simplify the calculation of changes in functions.

Derivative of Sine Function

Understanding the derivative of the sine function is essential for working with changes in trigonometric functions. The derivative of \( \sin x \), found using differential calculus, describes the rate at which the sine function changes as \( x \) changes.
Given \( y = \sin x \), when we take the derivative with respect to \( x \), we find:

• \( \frac{dy}{dx} = \cos x \)

This result tells us that the slope of the tangent to the curve \( y = \sin x \) at any point \( x \) is \( \cos x \).
Knowing the derivative helps us compute differentials, like \( dy \), and predict how the function behaves when \( x \) is adjusted. Moreover, it's central in understanding how trigonometric functions are interconnected and in solving complex calculus problems.