<< 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

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
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
anyone to explain some basic in calculus
Adegoke Reply
I can
Debdoot
A conical container of radius 10 ft and height 30 ft is filled with water to a depth of 15 ft. How much work is required to pump all the water out through a hole in the top of the container if the unit weight of the water is 62.4 lb/ft^3?
Milca Reply
hi am new here I really wants to know how the solve calculus
IBRAHIM
me too. I want to know calculation involved in calculus.
Katiba
evaluate triple integral xyz dx dy dz where the domain v is bounded by the plane x+y+z=a and the co-ordinate planes
BAGAM Reply
So how can this question be solved
Eddy
i m not sure but it could be xyz/2
Leo
someone should explain with a photo shot of the working pls
Adegoke
I think we should sort it out.
Eunice
Eunice Toe you can try it if you have the idea
Adegoke
how
Eunice
a^6÷8
Muzamil
i think a^6 ÷ 8
Muzamil
Practice Key Terms 3

Get the best Calculus volume 1 course in your pocket!





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