Math Problem Statement

What is the average rate of change on the interval from x = 0 to x = 4?

Solution

To calculate the average rate of change of the function f(x)f(x) on the interval from x=0x = 0 to x=4x = 4, we will use the formula for the average rate of change:

Average rate of change=f(x2)f(x1)x2x1\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

In this case:

  • x1=0x_1 = 0 and x2=4x_2 = 4
  • We need to find f(0)f(0) and f(4)f(4) from the graph.

From the graph:

  • f(0)=10f(0) = 10 (the height at time x=0x = 0)
  • f(4)=4f(4) = 4 (the height at time x=4x = 4)

Now we substitute these values into the formula:

Average rate of change=f(4)f(0)40=4104=64=32feet per second\text{Average rate of change} = \frac{f(4) - f(0)}{4 - 0} = \frac{4 - 10}{4} = \frac{-6}{4} = -\frac{3}{2} \, \text{feet per second}

Thus, the average rate of change is 32-\frac{3}{2} feet per second.

Would you like further details or have any questions?

Here are 5 related questions:

  1. How is the average rate of change related to the slope of a line?
  2. What does the negative sign in the rate of change signify in this context?
  3. How can you determine the rate of change from a table of values?
  4. What happens to the rate of change when the slope of the graph is horizontal?
  5. How would you calculate the rate of change over a different interval, say from x=4x = 4 to x=8x = 8?

Tip: The rate of change in a linear graph is constant, so the slope remains the same across the entire line.

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Math Problem Analysis

Mathematical Concepts

Average Rate of Change
Linear Functions
Slope

Formulas

Average rate of change = (f(x2) - f(x1)) / (x2 - x1)

Theorems

Slope of a Linear Function

Suitable Grade Level

Grades 8-10