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Section 1.4 Projects: Functions and Notation

Subsection Exercises

1

Let \(I=f(T)\) and \(H=g(T)\) be two functions that return \(I\text{,}\) the number of ice cream bars you sell at your food cart in one day, and \(H\text{,}\) the number of hot coffee drinks you sell at your food cart in one day. Let \(T\) be the average temperature outside in degrees Fahrenheit.

  1. Do you expect the function \(f\) should be increasing or decreasing? Explain your answer.

  2. Do you expect the function \(g\) should be increasing or decreasing? Explain your answer.

  3. If you charge \($2.50\) for an ice cream bar and \($1.75\) for a hot coffee drink, write an expression (using function notation) that represents the total amount of money you will make from coffee and ice cream in one day, if the average temperature outside is \(80\) degrees.

2

Let \(f(t)=78.4t-9.8t^2\)

  1. Setch a graph of the function. Label the horizontal and vertical intercepts

  2. Identify the intervals on which the function is increasing/decreasing.

  3. Identify the intervals on which \(f(t)>0\)

  4. Identify the intervals on which the function is concave down.

  5. Calcuate \(\frac{\Delta f}{\Delta t}\)on the interval \(0 \leq t \leq 2\)

  6. Suppose \(f(t)\) describes the height from the ground of a model rocket in meters and \(t\) is time measured in seconds.

    1. Based on the sketch of your graph and the situation described, what is a reasonable interval of time for this function to make sense?

    2. The entire expression “\(\frac{\Delta f}{\Delta t}=-29.4\) on the interval \(5 \leq t \leq 6\)” actually means something in English. Interpret the meaning of the expression in terms of the situation. What is the importance of the inequality as part of the expression?