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Section 5.6 Functions as Mathematical Models

Subsection 1. Properties of the basic functions

The eight basic functions are often used as models.

For these Examples, refer to the eight basic functions:

\begin{align*} f(x) \amp = x \amp f(x) \amp = \abs{x} \amp f(x) \amp = x^2 \amp f(x) \amp = x^3\\ f(x) \amp = \sqrt{x} \amp f(x) \amp = \sqrt[3]{x} \amp f(x) \amp = \dfrac{1}{x} \amp f(x) \amp = \dfrac{1}{x^2} \end{align*}

Subsubsection Examples

Example 5.41.

Which of the eight basic functions are always increasing?

Solution

\(f(x)=x,~~~f(x)=x^3,~~~f(x)=\sqrt{x},~~~f(x)=\sqrt[3]{x}\)

Example 5.42.

Which of the eight basic functions are concave up for positive \(x\text{?}\)

Solution

\(f(x)=x^2,~~~f(x)=x^3,~~~f(x)=\dfrac{1}{x},~~~f(x)=\dfrac{1}{x^2}\)

Subsubsection Exercises

Which of the eight basic functions are undefined at \(x=0\text{?}\)

Answer
\(f(x)=\dfrac{1}{x},~~~f(x)=\dfrac{1}{x^2}\)

Which of the eight basic functions are always non-negative?

Answer
\(f(x)=\abs{x},~~~f(x)=x^2,~~~f(x)=\sqrt{x},~~~f(x)=\dfrac{1}{x^2}\)

Subsection 2. Use familiar formulas

Here are some useful formulas often used in models.

Subsubsection Examples

Example 5.45.

Write a formula for the volume of a rectangular box, and identify the variables.

Solution

\(V=lwh~~\)

\(V\) stands for volume, and \(l,~w,\) and \(h\) stand for, respectively, the length, width, and height of the box.

Example 5.46.

Write a formula for the average of a number of scores, and identify the variables.

Solution

\(A=\dfrac{S}{n}~~\)

\(A\) stands for the average, \(S\) stands for the sum of the scores, and \(n\) stands for the number of scores.

Subsubsection Exercises

Choose the correct formula from the list below, and identify the variables.

  • \(\displaystyle I=Prt\)
  • \(\displaystyle P=R-C\)
  • \(\displaystyle d=rt\)
  • \(\displaystyle P=rW\)
  1. The distance traveled at a constant speed.
  2. The simple interest on an investment.
  3. The part specified by a percentage.
  4. The profit on sales of an item.
Answer

  1. \(d=rt\)

    \(~~d\) stands for the distance traveled at speed \(r\) for time \(t\text{.}\)

  2. \(I=Prt\)

    \(~~I\) stands for the interest earned on an investment \(P\) at interest rate \(r\) after a time period \(t\text{.}\)

  3. \(P=rW\)

    \(~~P\) stands for the quantity \(r\) percent of a whole amount \(W\text{.}\)

  4. \(P=R-C\)

    \(~~P\) stands for the profit left after the costs \(C\) are subtracted from the revenue \(R\text{.}\)

Choose the correct geometric formula from the list below, and identify the variables.

  • \(\displaystyle A=lw\)
  • \(\displaystyle P=2l+2w\)
  • \(\displaystyle A=\pi r^2\)
  • \(\displaystyle C=\pi d\)
  • \(\displaystyle V=\pi r^2h\)
  • \(\displaystyle V=\dfrac{4}{3}\pi r^3\)
  1. The volume of a cylinder.
  2. The area of a circle.
  3. The area of a rectangle.
  4. The perimeter of a rectangle.
  5. The volume of a sphere.
  6. The circumference of a circle.
Answer

  1. \(V=\pi r^2h\)

    \(~~V\) stands for the volume, \(r\) for the radius, and \(h\) for the height of the cylinder.

  2. \(A=\pi r^2\)

    \(~~A\) stands for the area and \(r\) for the radius of the circle.

  3. \(A=lw\)

    \(~~A\) stands for the area of the rectangle, \(l\) and \(w\) stand for its length and width.

  4. \(P=2l+2w\)

    \(~~P\) stands for the perimeter of the rectangle, \(l\) and \(w\) stand for its length and width.

  5. \(V=\dfrac{4}{3}\pi r^3\)

    \(~~V\) stands for the volume and \(r\) for the radius of the sphere.

  6. \(C=\pi d\)

    \(~~C\) stands for the circumference and \(d\) for the diameter of the circle.