Question

the graph of $f(x) = 2^x + 1$ represents the growth of a population of rabbits on a farm over $x$ years. complete the sentence about the graph. the average rate of change for $1 \leq x \leq 5$ is ... show your work here

Explanation:

Step1: Recall average rate formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 1 \), \( b = 5 \), and \( f(x)=2^x + 1 \).

Step2: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) \): \( f(1)=2^1 + 1 = 2 + 1 = 3 \).

Step3: Calculate \( f(5) \)

Substitute \( x = 5 \) into \( f(x) \): \( f(5)=2^5 + 1 = 32 + 1 = 33 \).

Step4: Compute average rate

Using the formula, \(\frac{f(5)-f(1)}{5 - 1}=\frac{33 - 3}{4}=\frac{30}{4}=7.5\).

Answer:

\( 7.5 \)