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

\(t^2\)

Step 1: Substitute Functions into F

First, identify the expressions for the functions. Here, \(f(t) = t \cos t\) and \(g(t) = \sec^2 t\). We are asked to find \(F(f(t), g(t))\). Substitute these into the function \(F(x, y) = x^2 y\). This means we replace \(x\) with \(f(t)\) and \(y\) with \(g(t)\) to obtain \(F(t \cos t, \sec^2 t)\).

Step 2: Calculate the Substituted Function

Using the substitution from Step 1, calculate the expression. Substitute \(f(t) = t \cos t\) for \(x\) and \(g(t) = \sec^2 t\) for \(y\) in the formula \(x^2 y\). This gives \((t \cos t)^2 \sec^2 t = (t^2 \cos^2 t) \sec^2 t\).

Step 3: Simplify the Expression

Simplify the expression \((t^2 \cos^2 t) \sec^2 t\). Recall that \(\sec t = \frac{1}{\cos t}\), so \(\sec^2 t = \frac{1}{\cos^2 t}\). Therefore, \(t^2 \cos^2 t \sec^2 t = t^2 \cos^2 t \times \frac{1}{\cos^2 t} = t^2\). The \(\cos^2 t\) terms cancel out.

Key Concepts

Function Composition

Function composition is an essential concept in mathematics, particularly in multivariable calculus. When we compose functions, we're essentially building new functions by applying one function to the outcome of another. This is similar to how a computer program might call one function with the result of another.

Here, we're working with two functions:

• The function \( F(x, y) = x^2 y \), which takes two variables and uncovers how their interaction results in a new value.
• Two single-variable functions \( f(t) = t \cos t \) and \( g(t) = \sec^2 t \), each of which manipulates the input \( t \) to produce its respective value.

The goal is to find \( F(f(t), g(t)) \), so we replace \( x \) and \( y \) with \( f(t) \) and \( g(t) \) respectively and compute: \( F(t \cos t, \sec^2 t) \). This composition of functions allows us to evaluate how the changes in \( t \) influence the overall output.

Trigonometric Functions

Trigonometric functions are cornerstones of calculus, encapsulating relationships in angles and sides of triangles. These come in handy when dealing with oscillations, rotations, and waves. Let's dissect the trigonometric components in our exercise.

- The function \( f(t) = t \cos t \) uses \( \cos t \), which is the cosine function. Cosine relates the angle \( t \) of a right-angled triangle to the adjacent side over the hypotenuse.- The function \( g(t) = \sec^2 t \) involves the secant function, known as \( \sec t = \frac{1}{\cos t} \). Thus, \( \sec^2 t = \left(\frac{1}{\cos t}\right)^2 \), making it the square of the reciprocal of the cosine function.

Understanding these functions is essential for manipulating and simplifying expressions involving trigonometric identities. Recognizing that \( \sec^2 t \) can transform into simpler forms using algebraic rules clarifies the solving process.

Algebraic Manipulation

Algebraic manipulation is a mathematical technique to simplify or alter expressions using algebraic rules and properties. It's crucial for making complex expressions more manageable, especially when dealing with multiple variables or intricate trigonometric functions.

In the given solution, after substituting the functions into the expression, we arrive at: \((t \cos t)^2 \sec^2 t \). Breaking this down algebraically:

• First, compute \((t \cos t)^2\). This results in \(t^2 \cos^2 t\).
• Next, incorporate \(\sec^2 t\) which is \(\frac{1}{\cos^2 t}\).

When multiplying these, \(t^2 \cos^2 t \times \frac{1}{\cos^2 t} = t^2\), the \(\cos^2 t\) terms cancel, leaving a simplified form \(t^2\). By using these algebraic simplifications, the expression becomes easier to interpret and work with, demonstrating how strategic manipulation can distill complex expressions into their essence.