Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

For \(G(t)=t /(t+4)\), find and simplify \([G(a+h)-\) \(G(a)] / h .\)

Step-by-Step Solution

Answer

The simplified difference quotient is \( \frac{4}{(a+h+4)(a+4)} \).

Step 1: Substitute in the Function

We begin by evaluating the expression specifically for \(G(a+h)\) and \(G(a)\). Substitute \(t = a + h\) into the function \(G(t) = \frac{t}{t + 4}\). This gives:\[G(a+h) = \frac{a+h}{(a+h)+4} = \frac{a+h}{a+h+4}.\]Next, substitute \(t = a\) into the function to find \(G(a)\):\[G(a) = \frac{a}{a + 4}.\]

Step 2: Form the Difference Quotient

Now, we'll form the expression \( \frac{G(a+h) - G(a)}{h}\). Substitute the results from Step 1:\[ \frac{\frac{a+h}{a+h+4} - \frac{a}{a+4}}{h}.\]

Step 3: Find a Common Denominator

To simplify the expression from Step 2, find a common denominator for the two fractions in the numerator:The common denominator of \(a+h+4\) and \(a+4\) is \((a+h+4)(a+4)\). Rewrite each fraction:\[\frac{a+h}{a+h+4} = \frac{(a+h)(a+4)}{(a+h+4)(a+4)}\]\[\frac{a}{a+4} = \frac{a(a+h+4)}{(a+h+4)(a+4)}.\]

Step 4: Combine the Fractions

Now subtract the two fractions as they have a common denominator:\[\frac{(a+h)(a+4) - a(a+h+4)}{(a+h+4)(a+4)}.\]

Step 5: Simplify the Numerator

Expand and simplify the numerator from Step 4:\((a+h)(a+4) = a^2 + 4a + ah + 4h\)\(a(a+h+4) = a^2 + ah + 4a\)Subtracting these two gives:\[a^2 + 4a + ah + 4h - (a^2 + ah + 4a) = 4h.\]

Step 6: Complete the Division

Place the simplified numerator over the previous common denominator \((a+h+4)(a+4)\):\[\frac{4h}{h((a+h+4)(a+4))}.\]Cancel \(h\) in the numerator and the denominator:\[\frac{4}{(a+h+4)(a+4)}.\]

Step 7: Simplify Further if Necessary

The expression is as simplified as it can be without additional specific values for \(a\) or \(h\). The simplified difference quotient is:\[ \frac{4}{(a+h+4)(a+4)}.\]

Key Concepts

Limits

The concept of limits is fundamental in calculus. Limits help us understand the behavior of functions as they approach a particular point. They allow us to analyze the value that a function is heading towards, even if it's not explicitly defined at that point.

When finding the difference quotient, especially as seen in the exercise for the function \( G(t) = \frac{t}{t+4} \), limits play a crucial role. The difference quotient itself, \( \frac{G(a+h) - G(a)}{h} \), is often used to determine derivative values by examining its limit as \( h \) approaches zero. This helps determine the instantaneous rate of change of the function, which is a key concept in calculus.

An important detail is that as \( h \) becomes very small, the expression \( \frac{4}{(a+h+4)(a+4)} \) gets closer to representing the function's derivative at \( a \). This step is vital in understanding continuous, smooth behavior in calculus.

Calculus

Calculus is a branch of mathematics that investigates how things change. It deals primarily with concepts of derivatives and integrals. The derivative deals with rates of change, while the integral focuses on accumulating quantities.

In the context of the problem, the difference quotient \( \frac{G(a+h) - G(a)}{h} \) we calculated is an example of finding one approach to the derivative of a function using basic calculus principles. The derivative tells us how the function \( G(t) = \frac{t}{t+4} \) changes concerning \( t \).

• The derivative is important in a variety of contexts, including physics (motion analysis), economics (cost functions), and more.
• Understanding calculus opens up pathways for solving complex problems involving changes, optimization, and analysis.

By simplifying the difference quotient as done in the exercise, we are practicing a foundational calculus technique for finding derivatives.

Rational Functions

Rational functions are expressions where the numerator and the denominator are both polynomials. They can have interesting behaviors, such as vertical asymptotes and horizontal asymptotes, determined by their denominators.

For \( G(t) = \frac{t}{t+4} \), we deal with a simple rational function. In the original exercise, we analyzed it through substitution and simplification, revealing a deeper understanding of the behavior of rational functions.

• Rational functions often involve division by a polynomial, which can complicate calculations, especially when finding limits or integrations.
• As seen in the exercise, simplifying the expression \( \frac{G(a+h) - G(a)}{h} \) required combining and managing polynomial terms, which is a key skill in handling rational functions.

Working with rational functions requires careful attention to the operations included, particularly as they approach values that may lead to undefined or infinite behavior.