Water usage rate for Seattle in a given 24-hour period is represented in the table and...

 

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Water usage rate for Seattle in a given 24-hour period is represented in the table and graph below (Note: t = 0 corresponds to midnight). Time is measure in hours, and the rate of water usage is measured in millions of gallons per hour. Rate (millions of gallons/hour) Time 0 Rate 0.5 1.70 1.55- 1.40- 1.25- 1.10- 0.95- 0.80- 0.65- 0.50- 0.00 2 0.6 4 0.9 4.00 6 8 10 12 14 16 18 20 22 24 1.3 1.7 1.5 1.3 1.4 1.5 1.4 1.1 0.7 0.5 Water Usage Rate for Seattle, WA, Over 24 Hours 8.00 12.00 Time (hours) 24 16.00 20.00 24.00 A. Using three trapezoids of equal width, estimate frate(t) dt. Explain the physical meaning of this definite Integral. Frate(r) di B. Does the estimate in part A overestimate or underestimate the value of frate(t)dt on the Interval 0 t 8? Explain your reasoning. C. Using your answer from part A, estimate the average water usage over the 24-hour period. D. Estimate the slope of the usage rate curve at t = 12. Describe the method that you use. 4. The function f(x) has the value f(-1) = 1. The slope of the curve y= f(x) at any point is given by dy the expression -= (2x + 1)(y + 1). (2x dx A. Write an equation for the line tangent to the curve y=f(x) at x=-1. B. Use the tangent line from part A to estimate f(-0.9). C. Use separation of variables to find an explicit or implicit formula for y=f(x), with no integrals remaining. D. Find lim f(x). X- Water usage rate for Seattle in a given 24-hour period is represented in the table and graph below (Note: t = 0 corresponds to midnight). Time is measure in hours, and the rate of water usage is measured in millions of gallons per hour. Rate (millions of gallons/hour) Time 0 Rate 0.5 1.70 1.55- 1.40- 1.25- 1.10- 0.95- 0.80- 0.65- 0.50- 0.00 2 0.6 4 0.9 4.00 6 8 10 12 14 16 18 20 22 24 1.3 1.7 1.5 1.3 1.4 1.5 1.4 1.1 0.7 0.5 Water Usage Rate for Seattle, WA, Over 24 Hours 8.00 12.00 Time (hours) 24 16.00 20.00 24.00 A. Using three trapezoids of equal width, estimate frate(t) dt. Explain the physical meaning of this definite Integral. Frate(r) di B. Does the estimate in part A overestimate or underestimate the value of frate(t)dt on the Interval 0 t 8? Explain your reasoning. C. Using your answer from part A, estimate the average water usage over the 24-hour period. D. Estimate the slope of the usage rate curve at t = 12. Describe the method that you use. 4. The function f(x) has the value f(-1) = 1. The slope of the curve y= f(x) at any point is given by dy the expression -= (2x + 1)(y + 1). (2x dx A. Write an equation for the line tangent to the curve y=f(x) at x=-1. B. Use the tangent line from part A to estimate f(-0.9). C. Use separation of variables to find an explicit or implicit formula for y=f(x), with no integrals remaining. D. Find lim f(x). X-