What is the average rate of change of the function f(x) = 42x + 1 over the interval from x = 0 to x = 5?

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The average rate of change of a function over a specified interval provides a measure of how much the function’s output value changes, on average, for each unit increase in the input value over that interval.

To find the average rate of change of the function f(x) = 42x + 1 from x = 0 to x = 5, we use the formula:

Average Rate of Change = \frac{f(b) - f(a)}{b - a}

Here, a = 0 and b = 5. First, we need to calculate the function values at these points:

  1. Calculate f(0): 
    f(0) = 42(0) + 1 = 1
  2. Calculate f(5): 
    f(5) = 42(5) + 1 = 210 + 1 = 211

Now, we can substitute these values back into our average rate of change formula:

Average Rate of Change = \frac{211 - 1}{5 - 0} = \frac{210}{5} = 42

Thus, the average rate of change of the function f(x) = 42x + 1 over the interval from x = 0 to x = 5 is 42.

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