I need help!! Describe how to determine the average rate of change between x = 1 and x = 3 for the function f(x) = 3x^3 + 1. Inc
lude the average rate of change in your answer.
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Answer: If we define 2:00pm as our 0 in time; then:
at t= 0. the velocity is 30 mi/h.
then at t = 10m (or 1/6 hours) the velocity is 50mi/h
Then, if we think in the "mean acceleration" as the slope between the two velocities, we can find the slope as:
a= (y2 - y1)/(x2 - x1) = (50 mi/h - 30 mi/h)/(1/6h - 0h) = 20*6mi/(h*h) = 120mi/
Now, this is the slope of the mean acceleration between t= 0h and t = 1/6h, then we can use the mean value theorem; who says that if F is a differentiable function on the interval (a,b), then exist at least one point c between a and b where F'(c) = (F(b) - F(a))/(b - a)
So if v is differentiable, then there is a time T between 0h and 1/6h where v(T) = 120mi/
Since you haven't provided the graph, I'll explain each one and you choose the one suiting your given.
<u>The parent modulus function is:</u>
g(x) = |x|
It is centered at the origin and opens upwards.
A coefficient inside the modulus |x+k| means that the function is shifted along the x-axis
If "k" is <u>positive</u>, the shift will be to the <u>left</u>. If "k" is <u>negative</u>, the shift will be to the <u>right</u>.
A coefficient outside the modulus |x| + h means that the function is shifted along the y-axis
If "h" is <u>positive</u>, the shift will be <u>upwards</u>. If "h" is <u>negative,</u> the shift will be <u>downwards</u>.
<u>Now, let's check each of the options:</u>
<u>g(x) = |x+4| - 2 :</u>
This function is shifted 4 units to the left and 2 units down. It will be centered at (-4,-2). Check the blue graph in the attachment.
<u>g(x) = |x-4| - 2 :</u>
This function is shifted 4 units to the right and 2 units down. It will be centered at (4,-2). Check the black graph in the attachment.
<u>g(x) = |x-2| - 4 :</u>
This function is shifted 2 units to the right and 4 units down. It will be centered at (2,-4). Check the red graph in the attachment.
<u>g(x) = |x-2| + 4 :</u>
This function is shifted 2 units to the right and 4 units up. It will be centered at (2,4). Check the green graph in the attachment.
All 4 graphs are shown in the attached picture.
Hope this helps :)
