A mouse moves back and forth in a straight tunnel, attractedto bits of cheddar cheese

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 5.9. (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 student is speeding down Route 11 in his fancy red Porsche when his radar system warns him of an obstacle 400 feet ahead. He immediately applies the brakes, starts to slow down, and spots a skunk in the road directly ahead of him. The black box in the Porsche records the cars speed every two seconds, producing the following table. The speed decreases throughout the 10 seconds it takes to stop, although not necessarily at a constant rate. (a) What is your best estimate of the total distance the students car traveled before coming to rest? (b) Which one of the following statements can you justify from the information given? (i) The car stopped before getting to the skunk. (ii) The black box data is inconclusive. The skunk may or may not have been hit. (iii) The skunk was hit by the car.

Use Figure 5.76 to estimate - 3 0 f(x) dx.

Use Figure 5.77 to estimate - 20 15 f(x) dx.

Using the table, estimate - 100 0 f(t) dt.

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 am and increases steadily to about 280 m3/hr at 9 am. (a) Using only this information, give your best estimate of the total volume of water used by the factory between 6 am and 9 am. (b) How often should the flow rate gauge be read to obtain an estimate of this volume to within 6 m3?

You jump out of an airplane. Before your parachute opens you fall faster and faster, but your acceleration decreases as you fall because of air resistance. The table gives your acceleration, a (in m/sec2), after t seconds. (a) Give upper and lower estimates of your speed at t = 5. (b) Get a new estimate by taking the average of your upper and lower estimates. What does the concavity of the graph of acceleration tell you about your new estimate?

In Exercises 89, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t) = t 4, a = 1, b = 1

In Exercises 89, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t)=3t 4 5t 3 + 5t; a = 2, b = 1

Find the area of the regions in Exercises 1016. Between the parabola y = 4 x2 and the x-axis.

Find the area of the regions in Exercises 1016. Between y = x2 9 and the x-axis

Find the area of the regions in Exercises 1016. Under one arch of y = sin x and above the x-axis.

Find the area of the regions in Exercises 1016. Between the line y = 1 and one arch of y = sin

Find the area of the regions in Exercises 1016. Between y = x2 + 5x 4 and the x-axis, 0 x 3.

Find the area of the regions in Exercises 1016. Between y = cos x + 7 and y = ln(x 3), 5 x 7.

Find the area of the regions in Exercises 1016. Above the curve y = ex + e2(x1) and below the xaxis, for x 0.

Find - 1 1 |x| dx geometrically.

A car accelerates smoothly from 0 to 60 mph in 10 seconds with the velocity given in Figure 5.78. Estimate how far the car travels during the 10-second period.

A car going 80 ft/sec (about 55 mph) brakes to a stop in 8 seconds. Its velocity is recorded every 2 seconds and is given in the following table. (a) Give your best estimate of the distance traveled by the car during the 8 seconds. (b) To estimate the distance traveled accurate to within 20 feet, how often should you record the velocity?

Using the graph of 2 + cos x, for 0 x 4, list the following quantities in increasing order: the value of the integral - 4 0 (2 + cos x) dx, the left sum with n = 2 subdivisions, and the right sum with n = 2 subdivisions.

Your velocity is v(t) = sin(t 2) mph for t in hours, 0 t 1.1. Find the distance traveled during this time.

Your velocity is v(t) = ln(t 2+1) ft/sec for t in seconds, 0 t 3. Find the distance traveled during this time.

The following table gives the emissions, E, of nitrogen oxides in millions of metric tons per year in the US.9 Let t be the number of years since 1970 and E = f(t). (a) What are the units and meaning of - 30 0 f(t)dt? (b) Estimate - 30 0 f(t)dt.

After a spill of radioactive iodine, measurements showed the ambient radiation levels at the site of the spill to be four times the maximum acceptable limit. The level of radiation from an iodine source decreases according to the formula R(t) = R0e0.004t where R is the radiation level (in millirems/hour) at time t in hours and R0 is the initial radiation level (at t = 0). (a) How long will it take for the site to reach an acceptable level of radiation? (b) How much total radiation (in millirems) will have been emitted by that time, assuming the maximum acceptable limit is 0.6 millirems/hour?

Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas: (a) Make an overestimate and an underestimate of the total quantity of pollutants that escape during the first month. (b) Make an overestimate and an underestimate of the total quantity of pollutants that escape during the six months. (c) How often would measurements have to be made to find overestimates and underestimates which differ by less than 1 ton from the exact quantity of pollutants that escaped during the first six months?

Figure 5.79 shows the rate of change of the quantity of water in a water tower, in liters per day, during the month of April. If the tower had 12,000 liters of water in it on April 1, estimate the quantity of water in the tower on April 30.

For the function f in Figure 5.80, write an expression involving one or more definite integrals of f that denote: (a) The average value of f for 0 x 5. (b) The average value of |f| for 0 x 5.

For the even function f in Figure 5.80, consider the average value of f over the following intervals: I. 0 x 1 II. 0 x 2 III. 0 x 5 IV. 2 x 2 (a) For which interval is the average value of f least? (b) For which interval is the average value of f greatest? (c) For which pair of intervals are the average values equal?

(a) Suppose f (x) = sin(x2) and f(0) = 2. Use a graph of f (x) to decide which is larger: (i) f(0) or f(1) (ii) f(2) or f(2.5) (b) Estimate f(b) for b = 0, 1, 2, 3.

(a) If F(t) = t(ln t) t, find F (t). (b) Find , 12 10 ln t dt two ways: (i) Numerically. (ii) Using the Fundamental Theorem of Calculus.

Statisticians sometimes use values of the function F(b) = , b 0 ex2 dx. (a) What is F(0)? (b) Does the value of F increase or decrease as b increases? (Assume b 0.) (c) Find F(1), F(2), and F(3).

In Problems 3235, find - 5 2 f(x) dx. f(x) is odd and - 5 2 f(x) dx = 8

In Problems 3235, find - 5 2 f(x) dx. f(x) is even, - 2 2 f(x) dx = 6, and - 5 5 f(x) dx = 14

In Problems 3235, find - 5 2 f(x) dx. - 5 2 (3f(x) + 4) dx = 18

In Problems 3235, find - 5 2 f(x) dx. - 4 2 2f(x) dx = 8 and - 4 5 f(x) dx = 1

Without any computation, find the values of (a) , 2 2 sin x dx, (b) , x113 dx.

Suppose f is periodic, its graph repeating every p units. Given that - p 0 f(x) dx > 0, evaluate and simplify , 2p 0 f(x) dx + , 5p p f(x) dx , 7p 5p f(x) dx

The function f is even (that is, its graph is symmetric about the y-axis). Use this fact to evaluate and simplify - r r f(x) dx + - r 0 f(x) dx , 0 r f(x) dx , for r > 0. [Hint: Use the area interpretation of the integral.]

(a) Use Figure 5.81 to explain why - 3 3 xex2 dx = 0. (b) Find the left-hand sum approximation with n = 3 to - 3 0 xex2 dx. Give your answer to four decimal places. (c) Repeat part (b) for - 0 3 xex2 dx. (d) Do your answers to parts (b) and (c) add to 0? Explain.

Two trains travel along parallel tracks. The velocity, v, of the trains as functions of time t are shown in Figure 5.82. (a) Describe in words the trips taken by each train. (b) Estimate the ratio of the following quantities for Train A to Train B: (i) Maximum velocity (ii) Time traveled (iii) Distance traveled

Worldwide, wind energy generating capacity, W megawatts, t years after 2000, can be modeled10 by W = 21,000e0.22t . (a) What was wind energy generating capacity in 2000? in 2010? (b) What annual continuous percent increase is indicated by the model? (c) Estimate average wind energy generating capacity between 2000 and 2010.

In Problems 4245, let r(t) be the rate, in kg/sec, at which a spaceship burns fuel at time t sec. Assume that r(t) > 0 and r (t) < 0 for 0 t T , where T is the time at which all the fuel has burned. Let Th be the time at which half the fuel has burned and Q be the initial amount of fuel in kg. State which of the following two expressions is larger, or that they are equal, or that there is not enough information to decide. , T 0 r(t) dt and Q

In Problems 4245, let r(t) be the rate, in kg/sec, at which a spaceship burns fuel at time t sec. Assume that r(t) > 0 and r (t) < 0 for 0 t T , where T is the time at which all the fuel has burned. Let Th be the time at which half the fuel has burned and Q be the initial amount of fuel in kg. State which of the following two expressions is larger, or that they are equal, or that there is not enough information to decide. , 0.5T 0 r(t) dt and , T 0.5T r(t) dt

In Problems 4245, let r(t) be the rate, in kg/sec, at which a spaceship burns fuel at time t sec. Assume that r(t) > 0 and r (t) < 0 for 0 t T , where T is the time at which all the fuel has burned. Let Th be the time at which half the fuel has burned and Q be the initial amount of fuel in kg. State which of the following two expressions is larger, or that they are equal, or that there is not enough information to decide. , T/3 0 r(t) dt and Q/3

In Problems 4245, let r(t) be the rate, in kg/sec, at which a spaceship burns fuel at time t sec. Assume that r(t) > 0 and r (t) < 0 for 0 t T , where T is the time at which all the fuel has burned. Let Th be the time at which half the fuel has burned and Q be the initial amount of fuel in kg. State which of the following two expressions is larger, or that they are equal, or that there is not enough information to decide. , Th 0 r(t) dt and , 0.5T 0 r(t) dt

Use the property - a b f(x) dx = - b a f(x) dx to show that - a a f(x) dx = 0.

The average value of y = v(x) equals 4 for 1 x 6, and equals 5 for 6 x 8. What is the average value of v(x) for 1 x 8?

(a) For any continuous function f, is - 2 1 f(x) dx + - 3 2 f(x) dx = - 3 1 f(x) dx? (b) For any function f, add the left-hand sum approximation with 10 subdivisions to - 2 1 f(x) dx to the left-hand sum approximation with 10 subdivisions to - 3 2 f(x) dx. Do you get the left-sum approximations with 10 subdivisions to - 3 1 f(x) dx? If not, interpret the result as a different Riemann Sum.

Using Figure 5.83, list the following integrals in increasing order (from smallest to largest). Which integrals are negative, which are positive? Give reasons. I. - b a f(x) dx II. - c a f(x) dx III. - e a f(x) dx IV. - e b f(x) dx V. - c b f(x) dx

In Problems 5052, the function F(t) gives the thickness, in inches, of ice on a roof t hours after midnight, where f(t) = F (t), f(2) = 0.5, F(7) = 4, F(9) = 5, and , 3.5 2 f (t) dt = 0.75 and , 7 4 f(t) dt = 1.5. Evaluate the expressions and explain what your answers tell you about the ice on the roof. , 9 7 f(t) d

In Problems 5052, the function F(t) gives the thickness, in inches, of ice on a roof t hours after midnight, where f(t) = F (t), f(2) = 0.5, F(7) = 4, F(9) = 5, and , 3.5 2 f (t) dt = 0.75 and , 7 4 f(t) dt = 1.5. Evaluate the expressions and explain what your answers tell you about the ice on the roof. F(4)

In Problems 5052, the function F(t) gives the thickness, in inches, of ice on a roof t hours after midnight, where f(t) = F (t), f(2) = 0.5, F(7) = 4, F(9) = 5, and , 3.5 2 f (t) dt = 0.75 and , 7 4 f(t) dt = 1.5. Evaluate the expressions and explain what your answers tell you about the ice on the roof. F (3.5)

For the even function f graphed in Figure 5.84: (a) Suppose you know - 2 0 f(x) dx. What is - 2 2 f(x) dx? (b) Suppose you know - 5 0 f(x) dx and - 5 2 f(x) dx. What is - 2 0 f(x) dx? (c) Suppose you know - 5 2 f(x) dx and - 2 2 f(x) dx. What is - 5 0 f(x) dx?

For the even function f graphed in Figure 5.84: (a) Suppose you know - 2 2 f(x) dx and - 5 0 f(x) dx. What is - 5 2 f(x) dx? (b) Suppose you know - 5 2 f(x) dx and - 0 2 f(x) dx. What is - 5 2 f(x) dx? (c) Suppose you know - 5 2 f(x) dx and - 5 2 f(x) dx. What is - 2 0 f(x) dx?

A mouse moves back and forth in a straight tunnel, attracted to bits of cheddar cheese alternately introduced to and removed from the ends (right and left) of the tunnel. The graph of the mouses velocity, v, is given in Figure 5.85, with positive velocity corresponding to motion toward the right end. Assuming that the mouse starts (t = 0) at the center of the tunnel, use the graph to estimate the time(s) at which: (a) The mouse changes direction. (b) The mouse is moving most rapidly to the right; to the left. (c) The mouse is farthest to the right of center; farthest to the left. (d) The mouses speed (i.e., the magnitude of its velocity) is decreasing. (e) The mouse is at the center of the tunnel.

Problems 5657 concern hybrid cars such as the Toyota Prius that are powered by a gas-engine, electric-motor combination, but can also function in Electric-Vehicle (EV) only mode, without power from the engine. Figure 5.8611 shows the velocity, v, of a 2010 Prius Plug-in Hybrid Prototype operating in normal hybrid mode and EV-only mode, respectively, while accelerating from a stoplight. About how far, in feet, does the 2010 Prius Prototype travel in EV-only mode during the first 15 seconds of movement?

Problems 5657 concern hybrid cars such as the Toyota Prius that are powered by a gas-engine, electric-motor combination, but can also function in Electric-Vehicle (EV) only mode, without power from the engine. Figure 5.8611 shows the velocity, v, of a 2010 Prius Plug-in Hybrid Prototype operating in normal hybrid mode and EV-only mode, respectively, while accelerating from a stoplight. 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?

The Montgolfier brothers (Joseph and Etienne) were eighteenth-century pioneers in the field of hot-air ballooning. Had they had the appropriate instruments, they might have left us a record, like that shown in Figure 5.87, of one of their early experiments. The graph shows their vertical velocity, v, with upward as positive. (a) Over what intervals was the acceleration positive? Negative? (b) What was the greatest altitude achieved, and at what time? (c) At what time was the upward acceleration greatest? (d) At what time was the deceleration greatest? (e) What might have happened during this flight to explain the answer to part (d)? (f) This particular flight ended on top of a hill. How do you know that it did, and what was the height of the hill above the starting point?

The Glen Canyon Dam at the top of the Grand Canyon prevents natural flooding. In 1996, scientists decided an artificial flood was necessary to restore the environmental balance. Water was released through the dam at a controlled rate12 shown in Figure 5.88. The figure also shows the rate of flow of the last natural flood in 1957. (a) At what rate was water passing through the dam in 1996 before the artificial flood? (b) At what rate was water passing down the river in the pre-flood season in 1957? (c) Estimate the maximum rates of discharge for the 1996 and 1957 floods. (d) Approximately how long did the 1996 flood last? How long did the 1957 flood last? (e) Estimate how much additional water passed down the river in 1996 as a result of the artificial flood. (f) Estimate how much additional water passed down the river in 1957 as a result of the flood.

Using Figure 5.89, list from least to greatest, (a) f (1). (b) The average value of f(x) on 0 x a. (c) The average value of the rate of change of f(x), for 0 x a. (d) - a 0 f(x) dx

In Problems 6164, let f(x) = F (x) where, for all x, f (x) > 0 and f(x) < 0. Explain whether the expression gives the exact value of f(3.1), an underestimate of f(3.1), or an overestimate of f(3.1), or if there is not enough information to decide. f(3) + 0.1f (3)

In Problems 6164, let f(x) = F (x) where, for all x, f (x) > 0 and f(x) < 0. Explain whether the expression gives the exact value of f(3.1), an underestimate of f(3.1), or an overestimate of f(3.1), or if there is not enough information to decide. f(3) + , 3.1 3 f (t) dt

In Problems 6164, let f(x) = F (x) where, for all x, f (x) > 0 and f(x) < 0. Explain whether the expression gives the exact value of f(3.1), an underestimate of f(3.1), or an overestimate of f(3.1), or if there is not enough information to decide. F(3.11) F(3.1) 0.01

In Problems 6164, let f(x) = F (x) where, for all x, f (x) > 0 and f(x) < 0. Explain whether the expression gives the exact value of f(3.1), an underestimate of f(3.1), or an overestimate of f(3.1), or if there is not enough information to decide. f(3) + +n i=1 f (xi) x, where n > 0, x = 0.1/n, x0 = 3, xn = 3.1, with x0, x1,...,xn evenly spaced.

The number of days a cold-blooded organism, such as an insect, takes to mature depends on the surrounding temperature, H. Each organism has a minimum temperature Hmin below which no development takes place.13 For an interval of time, t, on which the temperature is constant, the increase in maturity of the organism can be measured by the number of degree-days, S, where t is in days and S = (H Hmin)t. (a) If H varies with time, so H = f(t), write an integral that represents the total number of degree-days, S, required if development to maturity takes T days. (b) An organism which has Hmin = 15C requires 125 degree-days to develop to maturity. Estimate the development time if the temperature, HC, at time t days is in Table 5.10.

Let (x) be the angle of the tangent line at x of the graph of f with the horizontal, as in Figure 5.90. Show that (a) tan = f (x) (b) 1+(f (x))2 = 1 cos2 (c) d dx = cos2 d dx(tan ) = f(x) 1+(f (x))2 (d) (b) (a) = , b a f(x) 1+(f(x))2 dx

Consider the definite integral - 1 0 x4dx. (a) Write an expression for a right-hand Riemann sum approximation for this integral using n subdivisions. Express each xi, i = 1,2,. . . ,n, in terms of i. (b) Use a computer algebra system to obtain a formula for the sum you wrote in part (a) in terms of n. (c) Take the limit of this expression for the sum as n , thereby finding the exact value of this integral.

Repeat Problem 67, using the definite integral - 1 0 x5dx.

For Problems 6971, you will write a Riemann sum approximating a definite integral and use a computer algebra system to find a formula for the Riemann sum. By evaluating the limit of this sum as the number of subdivisions approaches infinity, you will obtain the definite integral. (a) Using summation notation, write the left-hand Rie mann sum with n subdivisions for - 2 1 t dt. (b) Use a computer algebra system to find a formula for the Riemann sum. (c) Evaluate the limit of the sum as n approaches infinity. (d) Calculate directly the area under the graph of y = t between t = 1 and t = 2, and compare it with your answer to part (c).

For Problems 6971, you will write a Riemann sum approximating a definite integral and use a computer algebra system to find a formula for the Riemann sum. By evaluating the limit of this sum as the number of subdivisions approaches infinity, you will obtain the definite integral. (a) Using summation notation, write the left-hand Riemann sum with n subdivisions for - 2 1 t 2 dt. (b) Use a computer algebra system to find a formula for the Riemann sum. (c) Evaluate the limit of the sum as n approaches infinity. (d) What is the area under the graph of y = t 2 between t = 1 and t = 2?

For Problems 6971, you will write a Riemann sum approximating a definite integral and use a computer algebra system to find a formula for the Riemann sum. By evaluating the limit of this sum as the number of subdivisions approaches infinity, you will obtain the definite integral. (a) Using summation notation, write the right-hand Riemann sum with n subdivisions for - 0 sin x dx. (b) Use a computer algebra system to find a formula for the Riemann sum. [Note: Not all computer algebra systems can evaluate this sum.] (c) Use a computer algebra system to evaluate the limit of the sum as n approaches infinity. (d) Confirm your answer to part (c) by calculating the definite integral with the computer algebra system.

In Problems 7273: (a) Use a computer algebra system to compute the given definite integral. (b) From your answer to part (a) and the Fundamental Theorem of Calculus, guess a function whose derivative is the integrand. Check your guess using the computer algebra system. [Hint: Make sure that the constants a, b, and c do not have previously assigned values in your computer algebra system.] , b a sin(cx) dx

In Problems 7273: (a) Use a computer algebra system to compute the given definite integral. (b) From your answer to part (a) and the Fundamental Theorem of Calculus, guess a function whose derivative is the integrand. Check your guess using the computer algebra system. [Hint: Make sure that the constants a, b, and c do not have previously assigned values in your computer algebra system.] , c a x 1 + bx2 dx, b > 0