Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.

c. Find the distance traveled by the object on the interval 0≤t≤6.​

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.

b. Find the displacement of the object on the interval 0≤t≤6.

Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.

d. How long does it take the racer to travel 300 ft?

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.

e. How far has the racer traveled when it reaches a speed of 178 ft/s?

Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.

b. How far does the car travel in the time it takes to come to rest?

40–43. Population growth

S​​​​tarting with an initial value of P(0)=55, the population of a prairie dog community grows at a rate of P′(t)=20−t/5 (prairie dogs/month), for 0≤t≤200, where t is measured in months.

b. Find the population P(t), for 0≤t≤200.

40–43. Population growth

When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.

b. F​ind the populati​on P(t) at an​y tim​e t≥0.

Oil production An oil refinery produces oil at a variable rate given by​​ Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels. 

a. How many barrels are produced in the first 35 days?

Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 

a. Find Q(t), the total amount of oil extracted by the nation after t years.​​

Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 

c. Find the minimum decay constant k for which the total oil reserves will last forever.

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.

a. How much water flows into the cistern in 1 hour?

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.

b. Find the function that gives the amount of water in the tank at any time t≥0.

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.

c. When will the tank be full?

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.

a. Verify that the amount of blood pumped over a one-second interval is 70 mL.

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.

b. Find the function that gives the total blood pumped between t=0 and a future time t>0.

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.

c. What is the cardiac output over a period of 1 min? (Use calculus; then check your answer with algebra.)

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.

​​a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.

​​b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.

​​c. Suppose P=10, A=50, and r=5. If the initial population is N(0)=10, does the population ever become extinct? Explain.

Power and energy ​The terms power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶ J, or 250 Cal. On the other hand, power is the rate at which energy is used and is measured in watts (W; 1W=1 J/s). Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for 1 hr, the total amount of energy used is 1 kilowatt-hour (kWh), which is 3.6×10⁶ J. Suppose the power function of a large city over a 24-hr period is given by P(t) = E'(t) = 300 - 200 sin πt/12, where P is measured in megawatts and t=0 corresponds ​to 6:00 P.M​. (see figure).

b. Burning 1 kg of coal produces about 450 kWh of energy. How many kilograms of coal are required to meet the energy needs of the city for 1 day? For 1 year? 

55–58. Marginal cost Consider the following marginal cost functions.

a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.

C′(x)=200−0.05x

55–58. Marginal cost Consider the following marginal cost functions.

b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.

C′(x)=200−0.05x

55–58. Marginal cost Consider the following marginal cost functions.

a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.

C′(x) = 300+10x−0.01x²

55–58. Marginal cost Consider the following marginal cost functions.

b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.

C′(x) = 300+10x−0.01x²

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The distance traveled by an object moving along a line is the same as the displacement of the object.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from A units to 2A units is greater than the cost of increasing production from 2A units to 3A units.

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.

v(t)=2 sin t, for 0≤t≤π

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.

v(t) = t(25−t²)^1/2, for 0≤t≤5