Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(F(f(t), g(t))\) if \(F(x, y)=x^{2} y\) and \(f(t)=t \cos t\), \(g(t)=\sec ^{2} t .\)

Step-by-Step Solution

Answer

\( F(f(t), g(t)) = t^2 \)

Step 1: Identify Functions

First, identify the given functions: - Function \( F(x, y) = x^2 y \). - Function \( f(t) = t \cos t \). - Function \( g(t) = \sec^2 t \).

Step 2: Substitute Functions Into F

Substitute \( f(t) \) and \( g(t) \) into the variables \( x \) and \( y \) of the function \( F \). That means replacing \( x \) with \( t \cos t \) and \( y \) with \( \sec^2 t \) in \( F(x, y) = x^2 y \).

Step 3: Evaluate Substituted Expression

Now evaluate the expression by substituting: \[ F(f(t), g(t)) = (t \cos t)^2 \cdot \sec^2 t \]. So, the function becomes: \[ F(f(t), g(t)) = t^2 \cos^2 t \cdot \sec^2 t \].

Step 4: Simplify the Expression

We know \( \sec^2 t = \frac{1}{\cos^2 t} \), so use this identity to simplify:\[ F(f(t), g(t)) = t^2 \cos^2 t \times \frac{1}{\cos^2 t} \].The \( \cos^2 t \) terms cancel out, leaving:\[ F(f(t), g(t)) = t^2 \].

Key Concepts

Function Composition

Function composition is a crucial concept in calculus and plays a significant role in forming complex functions from simpler ones. When composing functions, we take one function and substitute its output as the input of another function. This helps in simplifying complex equations and analyzing the behavior of combined functions.

• The notation for function composition with two functions \( f \) and \( g \) is \( (f \circ g)(x) = f(g(x)) \).
• In our exercise, we need to understand how to input functions \( f(t) \) and \( g(t) \) into a given function \( F(x, y) \).
• By substituting \( f(t) = t \cos t \) and \( g(t) = \sec^2 t \) into \( F(x, y) = x^2 y \), we are essentially performing function composition.

This process helps in simplifying the problem by allowing us to break down complex expressions into manageable parts. It is a strategy that can be applied to various mathematical problems involving multiple functions.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions and are true for all values of the variables involved. These identities help simplify complex trigonometric expressions and are invaluable in calculus.

• One fundamental identity is \( \sec t = \frac{1}{\cos t} \), which was used in our solution to simplify the expression.
• When dealing with trigonometric functions like \( \cos t \) and \( \sec t \), recognizing such identities helps reduce complex functions to simpler forms.
• For example, using \( \sec^2 t = \frac{1}{\cos^2 t} \), we can transform more complex trigonometric forms into manageable algebraic expressions.

By using these identities, understanding and simplifying expressions involving trigonometric functions become more straightforward. They also establish the relationship between angles and lengths, crucial in solving real-world application problems.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to find solutions more easily. This is a fundamental skill in calculus as it enables us to handle and simplify complex equations.

• In this exercise, after substituting the functions into the expression \( F(x, y) = x^2 y \), we performed algebraic manipulation to simplify it.
• By recognizing and canceling the \( \cos^2 t \) terms due to \( \sec^2 t \), we simplified the expression to \( t^2 \).
• This simplification process is essential to reduce expressions, making calculations easier and solutions clear.

Mastering algebraic manipulation involves practice and familiarity with algebraic rules and identities. It allows students to approach complex calculus problems methodically, enhancing both understanding and problem-solving abilities.