40 select the correct answer. jane planted a ...
40 select the correct answer. jane planted a marigold sapling in her garden and recorded its growth every week. the plants height, in inches, is modeled by the function h(x), where x is the number of weeks since jane planted the sapling. x h(x)=3√7x + 1 0 3 1 8.49 2 11.62 3 14.07 4 16.16 5 18 what is true of the function as the x - values increase? a. the function is decreasing, but the average rate of change is increasing. b. the function is decreasing, and the average rate of change is decreasing. c. the function is increasing, but the average rate of change is decreasing. d. the function is increasing, and the average rate of change is increasing.
Answer
# Explanation:
## Step1: Analyze function values
As \(x\) increases from \(0\) to \(5\) (from \(x = 0,h(0)=3\) to \(x = 5,h(5)=18\)), the values of \(h(x)\) are \(3,8.49,11.62,14.07,16.16,18\). Since these values are getting larger, the function \(h(x)\) is increasing.
## Step2: Calculate average - rate of change
The average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).
Over the interval \([0,1]\): \(\frac{h(1)-h(0)}{1 - 0}=\frac{8.49 - 3}{1}=5.49\).
Over the interval \([1,2]\): \(\frac{h(2)-h(1)}{2 - 1}=\frac{11.62 - 8.49}{1}=3.13\).
Over the interval \([2,3]\): \(\frac{h(3)-h(2)}{3 - 2}=\frac{14.07 - 11.62}{1}=2.45\).
Over the interval \([3,4]\): \(\frac{h(4)-h(3)}{4 - 3}=\frac{16.16 - 14.07}{1}=2.09\).
Over the interval \([4,5]\): \(\frac{h(5)-h(4)}{5 - 4}=\frac{18 - 16.16}{1}=1.84\).
The average - rate of change is decreasing as \(x\) increases.
# Answer:
C. The function is increasing, but the average rate of change is decreasing