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 1.1877.

Step 1: Identify the function and input

We are asked to calculate the value of the function \(g(u)\) for the input value \(u = 3.141\). The function is given as \(g(u) = \frac{\sqrt{u^3 + 2u}}{2 + u}\). This means we need to substitute \(u = 3.141\) into this expression.

Step 2: Substitute the value into the function

Substitute \(u = 3.141\) into the expression for \(g(u)\): \[ g(3.141) = \frac{\sqrt{(3.141)^3 + 2 \times 3.141}}{2 + 3.141} \]

Step 3: Calculate the numerator

Calculate the numerator of the expression. This involves computing \((3.141)^3 + 2 \times 3.141\) and then taking the square root. 1. Calculate \((3.141)^3 = 3.141 \times 3.141 \times 3.141 = 31.003357181\).2. Calculate \(2 \times 3.141 = 6.282\).3. Sum the results: \(31.003357181 + 6.282 = 37.285357181\).4. Take the square root: \(\sqrt{37.285357181} \approx 6.1051\).

Step 4: Calculate the denominator

Calculate the denominator of the expression: \(2 + 3.141 = 5.141\).

Step 5: Compute the final value

Substitute the computed numerator and denominator into the expression to find \(g(3.141)\):\[ g(3.141) = \frac{6.1051}{5.141} \approx 1.1877 \]

Step 6: Conclusion of the calculation

The value of the function \(g(u)\) at \(u = 3.141\) is approximately \(1.1877\).

Key Concepts

Evaluating Functions

Evaluating functions involves finding the output of a function for a particular input. In this context, we have the function \(g(u) = \frac{\sqrt{u^3 + 2u}}{2 + u}\) and are tasked to find \(g(3.141)\). This process helps us understand how functions behave and what they represent in mathematical terms. To evaluate a function, follow these steps:

• Identify the function and the specific input value. Here, the function is \(g(u)\) and the input is \(3.141\).


• Substitute the input value into the function. Replace \(u\) with \(3.141\) in the function's expression.


• Calculate the expression using mathematical operations, keeping the order of operations in mind.

Understanding evaluation is crucial for solving many mathematical problems, especially in calculus, where functions often model real-world situations.

Substitution

Substitution is a foundational technique in calculus that involves replacing a variable with a given value or another expression. It is an essential part of solving equations, computing derivatives, or integrating functions. In our problem, substitution means taking the function \(g(u) = \frac{\sqrt{u^3 + 2u}}{2 + u}\) and replacing \(u\) with \(3.141\). The substitution process includes:

• Identifying the variable to be replaced—in this case, \(u\).
• Rewriting the original equation with the specified value or expression—in this example, \(u = 3.141\). After substitution, the function becomes \(g(3.141) = \frac{\sqrt{(3.141)^3 + 2 \times 3.141}}{2 + 3.141}\).


• Proceeding with the calculations as per the substituted expression to find the desired output value.

Substitution simplifies complex equations into more manageable ones, and it's extensively used in calculus to transform problems into solvable formats.

Numerical Calculation

Numerical calculation involves computing numerical results from mathematical expressions, particularly using arithmetic operations and following the order of operations: parenthesis, exponents, multiplication, division, addition, and subtraction. In our problem with \(g(3.141) = \frac{\sqrt{(3.141)^3 + 2 \times 3.141}}{2 + 3.141}\), numerical calculation steps are precise:

• First, compute the exponent: \((3.141)^3\), which yields approximately \(31.003\).


• Then, calculate \(2 \times 3.141\), obtaining \(6.282\).


• Add these results: \(31.003 + 6.282\) gives \(37.285\).


• Take the square root of \(37.285\), getting approximately \(6.1051\).


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


• Lastly, divide the numerator \(6.1051\) by the denominator \(5.141\) to get the final value of about \(1.1877\).

Numerical calculations are crucial in verifying step-by-step solutions and ensuring accuracy in mathematical problems.