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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