Problem 51 - University Calculus: Early Transcendentals

Question

In Exercises \(51-70,\) find \(d y / d t\). $$y=\sin ^{2}(\pi t-2)$$

Step-by-Step Solution

Answer

\( \frac{dy}{dt} = \pi \cdot \sin(2\pi t - 4) \)

Step 1: Identify the Functions

The given function is a composition of a trigonometric function and its square, i.e., \( y = [ ext{sin}( heta)]^2 \) where \( \theta = \pi t - 2 \). Here, we have an inner function \( \theta \) and an outer function \( ext{sin}^2 \theta \).

Step 2: Use the Chain Rule

To differentiate \( y = ext{sin}^2(\theta) \) with respect to \( t \), apply the chain rule. First differentiate the outer function \( ext{sin}^2(\theta) \) with respect to \( \theta \), then multiply it by the derivative of the inner function \( \theta \) with respect to \( t \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( ext{sin}^2(\theta) \) with respect to \( \theta \). The derivative is \( 2 ext{sin}(\theta) ext{cos}(\theta) = ext{sin}(2\theta) \) using the double angle identity: sin(2θ) = 2sin(θ)cos(θ).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( \theta = \pi t - 2 \) with respect to \( t \). The derivative is \( \pi \).

Step 5: Apply the Chain Rule

Combine the results from Steps 3 and 4 using the chain rule: \[ \frac{dy}{dt} = \frac{d}{d\theta}\left(\text{sin}^2(\theta)\right) \cdot \frac{d\theta}{dt} = \text{sin}(2(\pi t - 2)) \cdot \pi \].

Step 6: Simplify the Expression

This simplifies to the derivative of the function: \[ \frac{dy}{dt} = \pi \cdot \text{sin}(2\pi t - 4) \].

Key Concepts

Trigonometric Differentiation

The process of finding the derivative of a function involving trigonometric functions is known as trigonometric differentiation. In the original problem, the function \( y = \sin^2(\pi t - 2) \) involves the sine function. Trigonometric rules can help in the differentiation process, with common derivatives being:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \(-\sin(x) \).

When dealing with functions like \( \sin^2(\theta) \), the differentiation requires a bit more work. Using identities like the double angle formula, \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \), helps simplify the derivative process. Breaking down trigonometric functions with these identities makes differentiation easier and less prone to error.

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions, where one function is nested inside another. It states that if a function \( y \) depends on \( u \) and \( u \) depends on \( x \), then \( y \) depends on \( x \). The chain rule is expressed as:\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]In the given problem, this is demonstrated in the differentiation of \( y = \sin^2(\theta) \) with respect to \( t \). Here:

• The outer function is \( \sin^2(\theta) \).
• The inner function is \( \theta = \pi t - 2 \).

Differentiating \( \sin^2(\theta) \) involves first processing the outer function. The derivative, according to the chain rule, requires you to multiply the derivative of the outer function by the derivative of the inner function. This ensures all parts of the composite function are properly accounted for. The chain rule is essential when working through layers of functions.

Composite Functions

Composite functions are functions made up of two or more simpler functions. In such cases, one function is applied to the result of another. The format can be written as \( f(g(x)) \), where \( g(x) \) is the inner function and \( f \) is the outer function. This concept is crucial in understanding the trigonometric differentiation and chain rule discussed earlier.In the original exercise, the composite function is \( y = \sin^2(\pi t - 2) \). Here:

• The inner function \( g(t) \) is \( \pi t - 2 \).
• The outer function \( f(x) \) is \( \sin^2(x) \).

Composite functions require careful analysis to differentiate. By tackling the inner function first and then applying the outer function, students can understand the layered structure and how differentiation applies to each component. Composite functions often require the use of the chain rule to break down the differentiation process into simpler steps.