Calculus Volume 2

# Chapter Review Exercises

Calculus Volume 2Chapter 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.

$y′=x2+3ex−2xy′=x2+3ex−2x$

267.

$y′=2x+cos−1xy′=2x+cos−1x$

268.

$y′=y(x2+1)y′=y(x2+1)$

269.

$y′=e−ysinxy′=e−ysinx$

270.

$y′=3x−2yy′=3x−2y$

271.

$y′=ylnyy′=ylny$

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

272.

$y′=8x−lnx−3x4,y(1)=5y′=8x−lnx−3x4,y(1)=5$

273.

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

274.

$xy′=y(x−2),y(1)=3xy′=y(x−2),y(1)=3$

275.

$y′=3y2(x+cosx),y(0)=−2y′=3y2(x+cosx),y(0)=−2$

276.

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

277.

$y′=3y−x+6x2,y(0)=−1y′=3y−x+6x2,y(0)=−1$

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.

$y′=2y−y2y′=2y−y2$

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.

$y′=−4yx,y(0)=1y′=−4yx,y(0)=1$

281.

$y′=3x−2y,y(0)=0y′=3x−2y,y(0)=0$

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?