More Applications of Definite Integrals 335- The change of temperature of a cup of
coffee measured in degrees Fahrenheit in a
certain room is represented by the function
f(t)=−cos
(
t
4)
for 0≤t≤5, wheretis
measured in minutes. If the temperature of
the coffee is initially 92◦F, find its
temperature after the first 5 minutes.- If thehalf-lifeof a radioactive element is
4500 years, and initially there are 100 grams
of this element, approximately how many
grams are left after 5000 years? - Find a solution of the differential equation:
dy
dx
=xcos (x^2 );y(0)=π. - If
d^2 y
dx^2
=x−5 and atx=0, y′=−2 and
y=1, find a solution of the differential
equation.
Part B Calculators are allowed.- Find the average value ofy=tanx
fromx=
π
4
tox=
π
3
.
- The acceleration function of a moving
particle on a straight line is given by
a(t)= 3 e^2 t, wheretis measured in seconds,
and the initial velocity is
1
2
. Find the
displacement and total distance traveled by
the particle in the first 3 seconds.
15. The sales of an item in a company follow an
exponential growth/decay model, wheretis
measured in months. If the sales drop from
5000 units in the first month to 4000 units
in the third month, how many units should
the company expect to sell during the
seventh month?
16. Find an equation of the curve that has a
slope of
2 y
x+ 1
at the point (x,y) and passes
through the point (0, 4).- The population in a city was approximately
750,000 in 1980, and grew at a rate of
3% per year. If the population growth
followed an exponential growth model, find
the city’s population in the year 2002. - Find a solution of the differential equation
4 ey=y′− 3 xeyandy(0)=0. - How much money should a person invest at
6.25% interest compounded continuously
so that the person will have $50,000 after
10 years? - The velocity function of a moving particle is
given asv(t)= 2 − 6 e−t,t≥0 andtis
measured in seconds. Find the total distance
traveled by the particle during the first
10 seconds. - Draw a slope field for differential equation
dy
dx
=x−y. - A rumor spreads through an office of
50 people at a model by
dP
dt
=. 65 P
(
1 −P
50
)
.On day zero, one
person knows the rumor. Find the model
for the population at timet, and use it to
predict when more than half the people in
the office will have heard the rumor.- A college dormitory that houses 200
students experiences an outbreak of
influenza. The illness is recognized when
two students are diagnosed on the same day.
The residents are quarantined to restrict the
infection to this one building. On the fifth
day of the outbreak, 12 students are ill. Use
a logistic model to describe the course of
infection and predict the number of
infected students on day 10. - Use Euler’s Method with a step size of
Δx= 0 .1 to computey(.5) ify(x)isthe
solution of the differential equation
dy
dx
=x^2 −y^3 with the conditiony(0)=1.