Calculating the Average Rate of Change of a Function Over an Interval | A Step-by-Step Guide with Example

The function f is given by f(x)=1+3cosx. What is the average rate of change of f over the interval [0,π] ?

To find the average rate of change of a function f(x) over a given interval [a, b], you need to find the slope of the secant line that passes through the points (a, f(a)) and (b, f(b))

To find the average rate of change of a function f(x) over a given interval [a, b], you need to find the slope of the secant line that passes through the points (a, f(a)) and (b, f(b)).

In this case, the interval is [0, π], so a = 0 and b = π.

To find f(a), you substitute a = 0 into the function f(x):

f(0) = 1 + 3cos(0) = 1 + 3(1) = 1 + 3 = 4.

Similarly, to find f(b), you substitute b = π into the function f(x):

f(π) = 1 + 3cos(π) = 1 + 3(-1) = 1 – 3 = -2.

So, the points we need are (0, 4) and (π, -2).

The slope of the secant line passing through these two points is given by:

Average rate of change = (f(b) – f(a))/(b – a).

Plugging in the values, we get:

Average rate of change = (-2 – 4)/(π – 0).

Simplifying, we have:

Average rate of change = (-6)/(π) = -6/π.

Therefore, the average rate of change of f over the interval [0, π] is -6/π.

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