Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his

Suppose that you travel 30 miles/hour for 2 hours, then 40 miles/hour for 1/2 hour, then 20 miles/hour for 4 hours. (a) What is the total distance you traveled? (b) Sketch a graph of the velocity function for this trip. (c) Represent the total distance traveled on your graph in part (b).

Figure 5.9 shows the velocity of a car for 0 t 12 and the rectangles used to estimate the distance traveled. (a) Do the rectangles represent a left or a right sum? (b) Do the rectangles lead to an upper or a lower estimate? (c) What is the value of n? (d) What is the value of t? (e) Give an approximate value for the estimate. 0 12 4 t Figure 5.9

Figure 5.10 shows the velocity of a runner for 0 t 15 and the rectangles used to estimate the distance traveled. (a) Do the rectangles represent a left or a right sum? (b) Do the rectangles lead to an upper or a lower estimate? (c) What is the value of n? (d) What is the value of t? (e) Give an approximate value for the estimate. 0 15 16 t Figure 5.10

A car comes to a stop six seconds after the driver applies the brakes. While the brakes are on, the velocities recorded are in Table 5.5. Table 5.5 Time since brakes applied (sec) 0 2 4 6 Velocity (ft/sec) 88 45 16 0 (a) Give lower and upper estimates for the distance the car traveled after the brakes were applied. (b) On a sketch of velocity against time, show the lower and upper estimates of part (a).

A car starts moving at time t = 0 and goes faster and faster. Its velocity is shown in the following table. Estimate how far the car travels during the 12 seconds. t (seconds) 0 3 6 9 12 Velocity (ft/sec) 0 10 25 45 75

Graph the rate of sales against time for the video game data in Example 2. Represent graphically the overestimate and the underestimate calculated in that example.

Figure 5.11 shows the velocity, v, of an object (in meters/ sec). Estimate the total distance the object traveled between t = 0 and t = 6. 1 2 3 4 5 6 10 20 30 40 t (sec) v (m/sec) Figure 5.11

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeffs data follow: Time since start (min) 0 15 30 45 60 75 90 Speed (mph) 12 11 10 10 8 7 0 (a) Assuming that Rogers speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour. (b) Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half.

The following table gives world oil consumption, in billions of barrels per year.1 Estimate total oil consumption during this 25-year period. Year 1985 1990 1995 2000 2005 2010 Oil (bn barrels/yr) 20.9 23.3 25.6 28.0 30.7 31.7

A car accelerates smoothly from 0 to 60 mph in 10 seconds with the velocity given in Figure 5.12. Estimate how far the car travels during the 10-second period. 5 10 20 40 60 t (sec) v (mph) Figure 5.12

The velocity of a car is f(t) = 5t meters/sec. Use a graph of f(t) to find the exact distance traveled by the car, in meters, from t = 0 to t = 10 seconds.

A village wishes to measure the quantity of water that is piped to a factory during a typical morning. A gauge on the water line gives the flow rate (in cubic meters per hour) at any instant. The flow rate is about 100 m3/hr at 6 amand increases steadily to about 280 m3/hr at 9 am. Using only this information, give your best estimate of the total volume of water used by the factory between 6 am and 9 am.

Filters at a water treatment plant become less effective over time. The rate at which pollution passes through the filters into a nearby lake is given in the following table. (a) Estimate the total quantity of pollution entering the lake during the 30-day period. (b) Your answer to part (a) is only an estimate. Give bounds (lower and upper estimates) between which the true quantity of pollution must lie. (Assume the rate of pollution is continually increasing.) Day 0 6 12 18 24 30 Rate (kg/day) 7 8 10 13 18 35

A car initially going 50 ft/sec brakes at a constant rate (constant negative acceleration), coming to a stop in 5 seconds. (a) Graph the velocity from t = 0 to t = 5. (b) How far does the car travel? (c) How far does the car travel if its initial velocity is doubled, but it brakes at the same constant rate?

Figure 5.13 shows the rate of change of a fish population. Estimate the total change in the population during this 12-month period. 4 8 12 5 10 15 20 25 time (months) rate (fish per month) Figure 5.13

Two cars travel in the same direction along a straight road. Figure 5.14 shows the velocity, v, of each car at time t. Car B starts 2 hours after car A and car B reaches a maximum velocity of 50 km/hr. (a) For approximately how long does each car travel? (b) Estimate car As maximum velocity. (c) Approximately how far does each car travel? Car A Car B t (hr) v (km/hr) Figure 5.14

Two cars start at the same time and travel in the same direction along a straight road. Figure 5.15 gives the velocity, v, of each car as a function of time, t. Which car: (a) Attains the larger maximum velocity? (b) Stops first? (c) Travels farther? Car A Car B t (hr) v (km/hr) Figure 5.15

Your velocity is given by v(t) = t2+1 in m/sec, with t in seconds. Estimate the distance, s, traveled between t = 0 and t = 5. Explain how you arrived at your estimate.

The value of a mutual fund increases at a rate of R = 500e0.04t dollars per year, with t in years since 2010. (a) Using t = 0, 2, 4, 6, 8, 10, make a table of values for R. (b) Use the table to estimate the total change in the value of the mutual fund between 2010 and 2020.

An old rowboat has sprung a leak. Water is flowing into the boat at a rate, r(t), given in the table. (a) Compute upper and lower estimates for the volume of water that has flowed into the boat during the 15 minutes. (b) Draw a graph to illustrate the lower estimate. t minutes 0 5 10 15 r(t) liters/min 12 20 24 16

The rate of change of the worlds population, in millions of people per year, is given in the following table. (a) Use this data to estimate the total change in the worlds population between 1950 and 2000. (b) The world population was 2555 million people in 1950 and 6085 million people in 2000. Calculate the true value of the total change in the population. How does this compare with your estimate in part (a)?

A car speeds up at a constant rate from 10 to 70 mph over a period of half an hour. Its fuel efficiency (in miles per gallon) increases with speed; values are in the table. Make lower and upper estimates of the quantity of fuel used during the half hour. Speed (mph) 10 20 30 40 50 60 70 Fuel efficiency (mpg) 15 18 21 23 24 25 26

Could the car travel half a mile in EV-only mode during the first 25 seconds of movement?

About how far, in feet, does the 2010 Prius Prototype travel in EV-only mode during the first 15 seconds of movement?

Assume two identical cars, one running in normal hybrid mode and one running in EV-only mode, accelerate together in a straight path from a stoplight. Approximately how far apart are the cars after 5 seconds?

Assume two identical cars, one running in normal hybrid mode and one running in EV-only mode, accelerate together in a straight path from a stoplight. Approximately how far apart are the cars after 15 seconds?

A car is traveling at 80 feet per second (approximately 55 miles per hour) and slows down as it passes through a busy intersection. The cars velocity is shown in the following table. Time since brakes applied (seconds) 0 3 6 9 12 velocity (ft/sec) 80 62 47 32 22 (a) Use a left sum to approximate the total distance the car has traveled over the 12 second interval. (b) Is your answer in part (a) a lower or upper estimate? How can you tell? (c) Use a right sum to approximate the total distance the car has traveled over the 12 second interval. (d) Is your answer in part (c) a lower or upper estimate? How can you tell? (e) Using your answers in parts (a) and (c), find a better estimate for the total distance the car has traveled.

The following table gives the total world emissions of CO2 from fossil fuels, in billions of tons per year.3 Year 1981 1986 1991 1996 2001 2006 2011 CO2 (bn tons per year) 20.1 21.3 21.9 23.4 24.2 29.1 34.7 (a) Use this data to estimate the total world CO2 emissions between 1981 and 2011 using a left sum with n = 6. (b) Is your answer in part (a) an upper or lower estimate? How can you tell? (c) Use this data to estimate the total world CO2 emissions between 1981 and 2011 using a right sum with n = 3. (d) Is your answer in part (c) an upper or lower estimate? How can you tell?

The rate of change of a quantity is given by f(t) = t2+1. Make an underestimate and an overestimate of the total change in the quantity between t = 0 and t = 8 using (a) t = 4 (b) t = 2 (c) t = 1 What is n in each case? Graph f(t) and shade rectangles to represent each of your six answers.

Use the expressions for left and right sums on page 245 and Table 5.6. (a) If n = 4, what is t? What are t0, t1, t2, t3, t4? What are f(t0), f(t1), f(t2), f(t3), f(t4)? (b) Find the left and right sums using n = 4. (c) If n = 2, what is t? What are t0, t1, t2? What are f(t0), f(t1), f(t2)? (d) Find the left and right sums using n = 2. Table 5.6 t 15 17 19 21 23 f(t) 10 13 18 20 30

Use the expressions for left and right sums on page 245 and Table 5.7. (a) If n = 4, what is t? What are t0, t1, t2, t3, t4? What are f(t0), f(t1), f(t2), f(t3), f(t4)? (b) Find the left and right sums using n = 4. (c) If n = 2, what is t? What are t0, t1, t2? What are f(t0), f(t1), f(t2)? (d) Find the left and right sums using n = 2.