AboutSocrates Expertise I can answer any questions from the standard four semester Calulus sequence. Derivatives, partial derivatives, chain rule, single and multiple integrals, change of variable, sequences and series, vector integration (Green`s Theorem, Stokes, and Gauss) and applications.
Pre-Calculus, Linear Algebra and Finite Math questions are also welcome.
Experience Ph.D. in Mathematics and many years teaching undergraduate courses at three state universities.
Question Hi Socrates,
I am working on this practice problem
Water usage rate for a city in a given 24-hour time period is represented in the table below (NOTE: t = 0 corresponds to midnight). Time is measured in hours, and the rate of water usage is measured in millions of gallons per hour.
t R(t)
0 0.5
2 0.6
4 0.9
6 1.3
8 1.7
10 1.5
12 1.3
14 1.4
16 1.5
18 1.4
20 1.1
22 0.7
24 0.5
a) Make an estimate of the average water usage over the 24-hour period.
b) Find the average rate of change of R(t) over the 24-hour period.
c) Estimate R'(12). Explain the physical meaning of this value. Be sure to use proper units in your answer.
could you tell me how to proceed - Thanks!
Answer a) 27.8 million gallons
From t=0 to t=2 , the average rate of water use is (.5+.6)/2 =.55 million gallons per hour.
For the two hours, this means approximately 2 x .55 = 1.1 million gallons will be used
From t=2 to t=4 , the average rate of water use is (.6+.9)/2 = .75
For the two hours, this means approximately 2 x .75 = 1.5 million
gallons will be used
From t=4 to t=8 , the average rate of water use is (.9+ 1.3)/2 = 1.1
For the two hours, this means approximately 2 x 1.1 = 2.2 million gallons will be used.
Keep going like this until you get the last estimate for amount of water used from t=23 to t=24. Then add up the estimates.
1.1 + 1.5 + 2.2 +.......= 27.8 million gallons used in 24 hours
c) -.025 million gallons per hour per hour. The rate of water usage is decreasing at approximately .025 million gallons per hour per hour at t=12
You really need to see a graph for this part.
Use the slope of the line joining (10,R(10)) and (14,R(14))to estimate the slope of the tangent line when t=12 . The slope of the tangent line to the graph when t=12 will be R'(12).
