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Chapter 0: Problem 7

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

Short Answer

Expert verified
g(3.141) \approx 1.188.

Step by step solution

01

Substitute the Value of u

To find the value of \(g(3.141)\), substitute \(u = 3.141\) into the function \(g(u)\). The given function is \(g(u) = \frac{\sqrt{u^3 + 2u}}{2 + u}\).
02

Calculate the Numerator

Calculate \(u^3 + 2u\) for \(u = 3.141\). This is calculated as: \(3.141^3 + 2 \times 3.141\). Evaluate the power and the product: \(3.141^3 = 31.006 \) and \(2 \times 3.141 = 6.282\). Adding these values together, the result is: \(31.006 + 6.282 = 37.288\).
03

Take the Square Root of the Numerator

Take the square root of \(37.288\), which results from the previous step. Thus, \(\sqrt{37.288} \approx 6.108\).
04

Calculate the Denominator

Calculate \(2 + u\) for \(u = 3.141\), which gives: \(2 + 3.141 = 5.141\).
05

Evaluate the Expression

Now evaluate the expression \(\frac{\sqrt{37.288}}{5.141}\) using the results from previous steps. Substitute the values: \(\frac{6.108}{5.141} \approx 1.188\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution is a fundamental concept in mathematics and involves replacing a variable with a given value in an expression or function.
  • When dealing with functions like our example, substituting a value allows you to evaluate the function at a specific point.
  • In the exercise above, we substitute the value of the variable "u" with "3.141" into the function \( g(u) = \frac{\sqrt{u^3 + 2u}}{2 + u} \).
  • This step is crucial because it converts the general formula into a specific calculation tailored to the given value.
Making these substitutions correctly is the key to moving forward with accurate calculations. Substitution may seem straightforward, but it lays the groundwork for deeper numerical processing.
Numerical Calculation
Numerical calculation involves computing numerical expressions using arithmetic operations such as addition, multiplication, and exponentiation.
To evaluate the function after substitution, you'll often deal with powers and products.
  • In the given exercise, once the substitution was made, we computed \(u^3 + 2u\) with \(u = 3.141\).
  • The first part, \(3.141^3\), involves raising 3.141 to the power of 3, resulting in \(31.006\).
  • The second part involves multiplying 3.141 by 2, which yields 6.282.
Adding these results, you get \(31.006 + 6.282 = 37.288\).Doing these calculations accurately is essential for finding precise results in mathematical problems.
Square Root
The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number.
The symbol for square root is \(\sqrt{}\).
  • In our exercise, after calculating the expression \(u^3 + 2u\) to be 37.288, the next step was to find \(\sqrt{37.288}\).
  • To determine this, you calculate or estimate the square root to find the approximate value of 6.108.
  • This step is all about using the result from a previous calculation and transforming it through a new operation.
Finding square roots is crucial in many areas of mathematics, including geometry and algebra, because it simplifies expressions and helps in solving equations.
Division
Division is one of the basic arithmetic operations, where you split a number into equal parts. It is represented by the symbol "/" or sometimes the division line in fractions.
It is often the final step when dealing with fractions in functions.
  • In the example problem, division is used to combine the results from previous steps to evaluate \(g(3.141)\).
  • With the square root value as the numerator \(6.108\) and the computed 5.141 as the denominator, you form the expression \(\frac{6.108}{5.141}\).
  • Carrying out this division gives the approximate result of 1.188.
Understanding division is crucial for working with ratios, proportions, and more complex mathematical expressions. It helps in finding the relative values of different quantities, making it a powerful tool in problem-solving.

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Most popular questions from this chapter

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x=-y^{2}+1\)

The circular frequency \(v\) of oscillation of a point is given by \(v=\frac{2 \pi}{\text { period }}\). What happens when you add two motions that have the same frequency or period? To investigate, we can graph the functions \(y(t)=2 \sin (\pi t / 5)\) and \(y(t)=\sin (\pi t / 5)+\) \(\cos (\pi t / 5)\) and look for similarities. Armed with this information, we can investigate by graphing the following functions over the interval [-5,5] (a) \(y(t)=3 \sin (\pi t / 5)-5 \cos (\pi t / 5)+2 \sin ((\pi t / 5)-3)\) (b) \(y(t)=3 \cos (\pi t / 5-2)+\cos (\pi t / 5)+\cos ((\pi t / 5)-3)\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(y=\frac{x}{x^{2}+1}\)

Circular motion can be modeled by using the parametric representations of the form \(x(t)=\sin t\) and \(y(t)=\cos t\) (A parametric representation means that a variable, \(t\) in this case, determines both \(x(t)\) and \(y(t) .\) This will give the full circle for \(0 \leq t \leq 2 \pi .\) If we consider a 4 -foot-diameter wheel making one complete rotation clockwise once every 10 seconds, show that the motion of a point on the rim of the wheel can be represented by \(x(t)=2 \sin (\pi t / 5)\) and \(y(t)=2 \cos (\pi t / 5)\) (a) Find the positions of the point on the rim of the wheel when \(t=2\) seconds, 6 seconds, and 10 seconds. Where was this point when the wheel started to rotate at \(t=0 ?\) (b) How will the formulas giving the motion of the point change if the wheel is rotating counterclockwise. (c) At what value of \(t\) is the point at (2,0) for the first time?

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(|x|+|y|=4\)
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