Honors Calculus The Mean (average) Value Theorem page 4-1 solutions
The mean value theorem relates the average rate of change over a time interval to the instantaneous rate at a particular moment. The jist of it is this: Suppose it took Ray Ray 30 minutes to drive the 20 miles to work. Did the speedometer ever read 40 mph?
1. The number of gallons of water in a bucket over a
time interval is shown in the graph below.
a) What is the average rate of change between time 3 and time 8? Draw the corresponding line segment and label the slope.
b) Mark points on the curve where the instantaneous rate of change is clearly greater than the average rate, clearly less than average, and about equal to the average rate.
c) Analytically find the coordinates of a point on the curve where the instantaneous rate of change is equal to the average rate of change.
d) What we have done could
be summarized as solving: 
. Explain!
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1a)
.5 gallons/min
b) see graph 
c) x = 5.5 (solve y’ =
.5 
x – 5 = .5
)
d) The right side of the equality is the average rate of change over the interval. The left side of the equality is the slope of the curve at c. We solved for the value c that makes the equation true.