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AP Precalculus - Average Rate of Change

Formula: f(b) - f(a) / (b - a)

Key Points:

  • The formula calculates the average rate of change of a function over a given interval.

  • The difference in function values (F(b) - F(a)) is divided by the difference in input values (b - a).

Example 1:

  • Function: f(x) = 3x^2

  • Interval: [1, 3] [a, b] a = 1, b = 3

  • Average Rate of Change = (F(3) - F(1)) / (3 - 1)

  • Substituting into the function: (27 - 3) / 2 = 24 / 2 = 12

In Example 1, the function f(x) = 3x^2 has an average rate of change of 12 over the interval [1, 3].

Example 2:

  • Function: f(x) = x^2 + 2

  • Interval: [2, 4] [a, b] a = 2, b = 4

  • Average Rate of Change = (F(4) - F(2)) / (4 - 2)

  • Substituting into the function: (18 - 6) / 2 = 12 / 2 = 6

In Example 2, the function f(x) = x^2 + 2 has an average rate of change of 6 over the interval [2, 4].

Conclusion:

  • Overall, the Average rate of change provides insight into the trend of the function within the specified interval.

SB

AP Precalculus - Average Rate of Change

Formula: f(b) - f(a) / (b - a)

Key Points:

  • The formula calculates the average rate of change of a function over a given interval.

  • The difference in function values (F(b) - F(a)) is divided by the difference in input values (b - a).

Example 1:

  • Function: f(x) = 3x^2

  • Interval: [1, 3] [a, b] a = 1, b = 3

  • Average Rate of Change = (F(3) - F(1)) / (3 - 1)

  • Substituting into the function: (27 - 3) / 2 = 24 / 2 = 12

In Example 1, the function f(x) = 3x^2 has an average rate of change of 12 over the interval [1, 3].

Example 2:

  • Function: f(x) = x^2 + 2

  • Interval: [2, 4] [a, b] a = 2, b = 4

  • Average Rate of Change = (F(4) - F(2)) / (4 - 2)

  • Substituting into the function: (18 - 6) / 2 = 12 / 2 = 6

In Example 2, the function f(x) = x^2 + 2 has an average rate of change of 6 over the interval [2, 4].

Conclusion:

  • Overall, the Average rate of change provides insight into the trend of the function within the specified interval.

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