Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Calculate \(g(3.141)\) if \(g(u)=\frac{\sqrt{u^{3}+2 u}}{2+u}\).

Step-by-Step Solution

Answer

The value of \( g(3.141) \) is approximately 1.188.

Step 1: Understand the function

First, we note that the function is expressed as \( g(u) = \frac{\sqrt{u^3 + 2u}}{2+u} \). We need to substitute \( u = 3.141 \) into this expression.

Step 2: Substitute the value

Replace \( u \) with \( 3.141 \) in the function: \( g(3.141) = \frac{\sqrt{(3.141)^3 + 2 \times 3.141}}{2 + 3.141} \).

Step 3: Calculate the expression inside the square root

Compute \( (3.141)^3 + 2 \times 3.141 \). \[ (3.141)^3 = 31.003593621 \] and \[ 2 \times 3.141 = 6.282 \] so the expression becomes \[ 31.003593621 + 6.282 = 37.285593621 \].

Step 4: Take the square root

Calculate the square root of \( 37.285593621 \), which is approximately \( 6.104 \).

Step 5: Calculate the denominator

Compute the denominator \( 2 + 3.141 = 5.141 \).

Step 6: Perform the division

Divide the result from the square root by the denominator: \( \frac{6.104}{5.141} \approx 1.188 \).

Step 7: Result

Conclude that \( g(3.141) \approx 1.188 \).

Key Concepts

Substitution Method

In mathematics, the substitution method is a technique used to find the value of a function by replacing its variables with specific values. In our problem, we have the function \( g(u) = \frac{\sqrt{u^3 + 2u}}{2+u} \). To evaluate the function at \( u = 3.141 \), we substitute \( 3.141 \) into the function in place of \( u \). This means everywhere you see the letter \( u \), you literally replace it with the number 3.141. By doing this, we convert the function into a numerical expression, allowing us to perform further calculations. Through substitution, complex functions become simpler numbers that can be handled using basic arithmetic operations.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform on numbers, such as addition, subtraction, multiplication, and division. They form the foundation for solving many math problems. In the exercise, after substituting \( u = 3.141 \), we need to compute \( (3.141)^3 + 2 \times 3.141 \). Here:

• We first calculate the cube of 3.141, resulting in 31.003593621.
• Next, multiply 3.141 by 2, which gives 6.282.
• Finally, add these results together to get 37.285593621.

These steps show how arithmetic operations simplify our substitutions into solvable parts. Understanding how these simple operations work lays the groundwork for evaluating more complex expressions like this one.

Square Root Calculation

Taking the square root of a number is a process that finds a value that, when multiplied by itself, equals the original number. In this exercise, once we calculate \( (3.141)^3 + 2 \times 3.141 \) to be approximately 37.285593621, we need to find its square root. Calculating \( \sqrt{37.285593621} \), we discover that it equals approximately 6.104. Square root calculations are crucial for simplifying expressions that contain quadratic terms. They play a significant role in mathematics, bridging the gap between linear and quadratic equations. It's important to note that square root values can often be irrational, meaning they cannot be expressed as a finite decimal or fraction, which is why approximations are often used.

Numerical Approximation

Numerical approximation involves finding close estimates for mathematical expressions, especially when exact values are difficult or impossible to determine. In this problem, after performing the arithmetic and square root calculations, we continue by dividing the square root result by the simplified denominator. The denominator \( 2 + 3.141 \) is computed as 5.141. Dividing the approximate square root 6.104 by 5.141, we get approximately 1.188. Numerical approximations simplify complex or irrational numbers into manageable forms for easier interpretation and use in subsequent calculations. This process allows us to work with approximate solutions in real-world settings where precision is necessary, but absolute exactness may not be feasible.