Example Question - average rate of change
Here are examples of questions we've helped users solve.
Calculating Average Rate of Change for Exponential Function
The question asks for the average rate of change of the function f(x) = 100 * 2^x on the interval [0,4]. The average rate of change of a function over the interval [a, b] is given by the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] Here, our function f(x) = 100 * 2^x, our interval is [0, 4], so a = 0 and b = 4. We plug these into the function f to get f(0) and f(4): f(0) = 100 * 2^0 = 100 * 1 = 100 f(4) = 100 * 2^4 = 100 * 16 = 1600 Now plug f(0) and f(4) into the rate of change formula: \[ \text{Average Rate of Change} = \frac{f(4) - f(0)}{4 - 0} = \frac{1600 - 100}{4} = \frac{1500}{4} = 375 \] Hence, the average rate of change of f(x) on the interval [0,4] is 375.
Comparing Average Rates of Change of Functions
The question is asking to compare the average rates of change for the pair of functions over the given interval [1, 5]. The given functions are: - \( f(x) = 9x^2 \) - \( g(x) = 1 + 3x^2 \) The average rate of change of a function over an interval \([a, b]\) can be found using the formula: \[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \] Let's compute this for each function over the interval [1, 5]. For \( f(x) = 9x^2 \): - \( f(1) = 9(1)^2 = 9 \) - \( f(5) = 9(5)^2 = 225 \) - \( \text{Average rate of change of} \ f(x) \text{ over} \ [1, 5] = \frac{f(5) - f(1)}{5 - 1} = \frac{225 - 9}{4} = \frac{216}{4} = 54 \) For \( g(x) = 1 + 3x^2 \): - \( g(1) = 1 + 3(1)^2 = 4 \) - \( g(5) = 1 + 3(5)^2 = 1 + 75 = 76 \) - \( \text{Average rate of change of} \ g(x) \text{ over} \ [1, 5] = \frac{g(5) - g(1)}{5 - 1} = \frac{76 - 4}{4} = \frac{72}{4} = 18 \) Therefore: - The average rate of change of \( f(x) \) over [1, 5] is \( 54 \). - The average rate of change of \( g(x) \) over [1, 5] is \( 18 \). - The average rate of change of \( f(x) - g(x) \) over [1, 5] is \( 54 - 18 = 36 \). So, the average rate of change of \( f \) over [1, 5] is \( 54 \), of \( g \) over [1, 5] is \( 18 \), and of \( f - g \) over [1, 5] is \( 36 \).
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