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Calculus Volume 2

Review Exercises

Calculus Volume 2Review Exercises

Review Exercises

True or False? Justify your answer with a proof or a counterexample.

262.

The differential equation y′=3x2y−cos(x)y″y′=3x2y−cos(x)y″ is linear.

263.

The differential equation y′=x−yy′=x−y is separable.

264.

You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

265.

You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.

For the following problems, find the general solution to the differential equations.

266.

y ′ = x 2 + 3 e x − 2 x y ′ = x 2 + 3 e x − 2 x

267.

y ′ = 2 x + cos −1 x y ′ = 2 x + cos −1 x

268.

y ′ = y ( x 2 + 1 ) y ′ = y ( x 2 + 1 )

269.

y ′ = e − y sin x y ′ = e − y sin x

270.

y ′ = 3 x − 2 y y ′ = 3 x − 2 y

271.

y ′ = y ln y y ′ = y ln y

For the following problems, find the solution to the initial value problem.

272.

y ′ = 8 x − ln x − 3 x 4 , y ( 1 ) = 5 y ′ = 8 x − ln x − 3 x 4 , y ( 1 ) = 5

273.

y ′ = 3 x − cos x + 2 , y ( 0 ) = 4 y ′ = 3 x − cos x + 2 , y ( 0 ) = 4

274.

x y ′ = y ( x − 2 ) , y ( 1 ) = 3 x y ′ = y ( x − 2 ) , y ( 1 ) = 3

275.

y ′ = 3 y 2 ( x + cos x ) , y ( 0 ) = −2 y ′ = 3 y 2 ( x + cos x ) , y ( 0 ) = −2

276.

( x − 1 ) y ′ = y − 2 , y ( 0 ) = 0 ( x − 1 ) y ′ = y − 2 , y ( 0 ) = 0

277.

y ′ = 3 y − x + 6 x 2 , y ( 0 ) = −1 y ′ = 3 y − x + 6 x 2 , y ( 0 ) = −1

For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field.

278.

y ′ = 2 y − y 2 y ′ = 2 y − y 2

279.

y′=1x+lnx−y,y′=1x+lnx−y, for x>0x>0

For the following problems, use Euler’s Method with n=5n=5 steps over the interval t=[0,1].t=[0,1]. Then solve the initial-value problem exactly. How close is your Euler’s Method estimate?

280.

y ′ = −4 y x , y ( 0 ) = 1 y ′ = −4 y x , y ( 0 ) = 1

281.

y ′ = 3 x − 2 y , y ( 0 ) = 0 y ′ = 3 x − 2 y , y ( 0 ) = 0

For the following problems, set up and solve the differential equations.

282.

A car drives along a freeway, accelerating according to a=5sin(πt),a=5sin(πt), where tt represents time in minutes. Find the velocity at any time t,t, assuming the car starts with an initial speed of 6060 mph.

283.

You throw a ball of mass 22 kilograms into the air with an upward velocity of 88 m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by g=9.8m/s2.g=9.8m/s2.

284.

You drop a ball with a mass of 55 kilograms out an airplane window at a height of 50005000 m. How long does it take for the ball to reach the ground?

285.

You drop the same ball of mass 55 kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a proportionality constant of 33 and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?

286.

A drug is administered to a patient every 2424 hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant 0.2.0.2. If the patient needs a baseline level of 55 mg to be in the bloodstream at all times, how large should the dose be?

287.

A 10001000-liter tank contains pure water and a solution of 0.20.2 kg salt/L is pumped into the tank at a rate of 11 L/min and is drained at the same rate. Solve for total amount of salt in the tank at time t.t.

288.

You boil water to make tea. When you pour the water into your teapot, the temperature is 100°C.100°C. After 55 minutes in your 15°C15°C room, the temperature of the tea is 85°C.85°C. Solve the equation to determine the temperatures of the tea at time t.t. How long must you wait until the tea is at a drinkable temperature (72°C)?(72°C)?

289.

The human population (in thousands) of Nevada in 19501950 was roughly 160.160. If the carrying capacity is estimated at 1010 million individuals, and assuming a growth rate of 2%2% per year, develop a logistic growth model and solve for the population in Nevada at any time (use 19501950 as time = 0). What population does your model predict for 2000?2000? How close is your prediction to the true value of 1,998,257?1,998,257?

290.

Repeat the previous problem but use Gompertz growth model. Which is more accurate?

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