334 STEP 4. Review the Knowledge You Need to Score High
- Use Euler’s Method with a step size ofΔx= 0 .5 to computey(2), ify(x)isthe
solution of the differential equation
dy
dx
=y+xywith initial conditiony(0)=1.
Answer: y(0)=1;y(0.5)= 1 + 0 .5(1+ 0 .1)= 1 .5;
y(1)= 1. 5 + 0. 5 [ 1. 5 +(0.5)(1.5)]= 1. 5 + 0 .5[2.25]= 2. 625
y(1.5)= 2. 625 + 0. 5 [ 2. 625 +(1)(2.625)]= 2. 625 + 2. 625 = 5. 25
y(2)= 5. 25 + 0. 5 [ 5. 25 +(1.5)(5.25)]= 5. 25 + 0 .5[13.125]= 11. 8125
13.9 Practice Problems
Part A The use of a calculator is not
allowed.
- Find the value ofcas stated in the Mean
Value Theorem for Integrals forf(x)=x^3
on [2, 4]. - The graph offis shown in Figure 13.9-1.
Find the average value off on [0, 8].
(4,4)
1
012345678
2
3
4
x
y
f
Figure 13.9-1
- The position function of a particle moving
on a coordinate line is given ass(t)=t^2 −
6 t−7, 0≤t≤10. Find the displacement
and total distance traveled by the particle
from 1≤t≤4. - The velocity function of a moving particle
on a coordinate line isv(t)= 2 t+ 1
for 0≤t≤8. Att=1, its position is−4.
Find the position of the particle att=5.
5.The rate of depreciation for a new piece of
equipment at a factory is given asp(t)=
50 t−600 for 0≤t≤10, wheretis
measured in years. Find the total loss of
value of the equipment over the first 5 years.
6.If the acceleration of a moving particle on a
coordinate line isa(t)=−2 for 0≤t≤4,
and the initial velocityv 0 =10, find the
total distance traveled by the particle during
0 ≤t≤4.
7.The graph of the velocity function of a
moving particle is shown in Figure 13.9-2.
What is the total distance traveled by the
particle during 0≤t≤12?
v(t)
v
t
20
10
–10
(^0) 1 2 3 4 5 6 7 8 9 10 11 12
Figure 13.9-2
8.If oil is leaking from a tanker at the rate of
f(t)= 10 e^0.^2 tgallons per hour, wheretis
measured in hours, how many gallons of oil
will have leaked from the tanker after the
first 3 hours?