<< Chapter < Page Chapter >> Page >

Solve the initial value problem d y d x = 3 x −2 , y ( 1 ) = 2 .

y = 3 x + 5

Got questions? Get instant answers now!

Initial-value problems arise in many applications. Next we consider a problem in which a driver applies the brakes in a car. We are interested in how long it takes for the car to stop. Recall that the velocity function v ( t ) is the derivative of a position function s ( t ) , and the acceleration a ( t ) is the derivative of the velocity function. In earlier examples in the text, we could calculate the velocity from the position and then compute the acceleration from the velocity. In the next example we work the other way around. Given an acceleration function, we calculate the velocity function. We then use the velocity function to determine the position function.

Decelerating car

A car is traveling at the rate of 88 ft/sec ( 60 mph) when the brakes are applied. The car begins decelerating at a constant rate of 15 ft/sec 2 .

  1. How many seconds elapse before the car stops?
  2. How far does the car travel during that time?
  1. First we introduce variables for this problem. Let t be the time (in seconds) after the brakes are first applied. Let a ( t ) be the acceleration of the car (in feet per seconds squared) at time t . Let v ( t ) be the velocity of the car (in feet per second) at time t . Let s ( t ) be the car’s position (in feet) beyond the point where the brakes are applied at time t .
    The car is traveling at a rate of 88 ft/sec . Therefore, the initial velocity is v ( 0 ) = 88 ft/sec. Since the car is decelerating, the acceleration is
    a ( t ) = −15 ft/s 2 .

    The acceleration is the derivative of the velocity,
    v ( t ) = 15 .

    Therefore, we have an initial-value problem to solve:
    v ( t ) = −15 , v ( 0 ) = 88 .

    Integrating, we find that
    v ( t ) = −15 t + C .

    Since v ( 0 ) = 88 , C = 88 . Thus, the velocity function is
    v ( t ) = −15 t + 88 .

    To find how long it takes for the car to stop, we need to find the time t such that the velocity is zero. Solving −15 t + 88 = 0 , we obtain t = 88 15 sec.
  2. To find how far the car travels during this time, we need to find the position of the car after 88 15 sec. We know the velocity v ( t ) is the derivative of the position s ( t ) . Consider the initial position to be s ( 0 ) = 0 . Therefore, we need to solve the initial-value problem
    s ( t ) = −15 t + 88 , s ( 0 ) = 0 .

    Integrating, we have
    s ( t ) = 15 2 t 2 + 88 t + C .

    Since s ( 0 ) = 0 , the constant is C = 0 . Therefore, the position function is
    s ( t ) = 15 2 t 2 + 88 t .

    After t = 88 15 sec, the position is s ( 88 15 ) 258.133 ft.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Suppose the car is traveling at the rate of 44 ft/sec. How long does it take for the car to stop? How far will the car travel?

2.93 sec , 64.5 ft

Got questions? Get instant answers now!

Key concepts

  • If F is an antiderivative of f , then every antiderivative of f is of the form F ( x ) + C for some constant C .
  • Solving the initial-value problem
    d y d x = f ( x ) , y ( x 0 ) = y 0

    requires us first to find the set of antiderivatives of f and then to look for the particular antiderivative that also satisfies the initial condition.

For the following exercises, show that F ( x ) are antiderivatives of f ( x ) .

F ( x ) = 5 x 3 + 2 x 2 + 3 x + 1 , f ( x ) = 15 x 2 + 4 x + 3

F ( x ) = 15 x 2 + 4 x + 3

Got questions? Get instant answers now!

F ( x ) = x 2 + 4 x + 1 , f ( x ) = 2 x + 4

Got questions? Get instant answers now!

F ( x ) = x 2 e x , f ( x ) = e x ( x 2 + 2 x )

F ( x ) = 2 x e x + x 2 e x

Got questions? Get instant answers now!

F ( x ) = cos x , f ( x ) = sin x

Got questions? Get instant answers now!

F ( x ) = e x , f ( x ) = e x

F ( x ) = e x

Got questions? Get instant answers now!

For the following exercises, find the antiderivative of the function.

Questions & Answers

any genius online ? I need help!!
Guzorochi Reply
how can i help you?
Pina
need to learn polynomial
Zakariya
i will teach...
nandu
I'm waiting
Zakariya
evaluate the following computation (x³-8/x-2)
Murtala Reply
teach me how to solve the first law of calculus.
Uncle Reply
what is differentiation
Ibrahim Reply
f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
explain for me
Usman
okay I have such documents
Fitzgerald
please share it
Hamza
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask