5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:5

More Applications of Definite Integrals 337

followed an exponential growth model, find

the city’s population in the year 2002.

18. Find a solution of the differential equation
    4 ey=y′− 3 xeyandy(0)=0.


19. How much money should a person invest at
    6.25% interest compounded continuously

so that the person will have $50,000 after

10 years?

20. 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.

14.8 Cumulative Review Problems

(Calculator) indicates that calculators are
permitted.

21. If 3ey=x^2 y, find
    dy
    dx

.

22. Evaluate

∫ 1

0

x^2

x^3 + 1

dx.

23. The graph of a continuous functionf,
    which consists of three line segments on
    [−2, 4], is shown in Figure 14.8-1. If
    F(x)=

∫x

− 2

f(t)dtfor− 2 ≤x≤4,

7 6 5 4 3 2 1

• 2 – 1123450

y

f

t

Figure 14.8-1

(a) FindF(−2) andF(0).

(b) FindF′(0) andF′(2).

(c) Find the value ofxsuch thatFhas a

maximum on [−2, 4].

(d) On which interval is the graph ofF

concave upward?

24. (Calculator) The slope of a function
    y=f(x) at any point (x,y)is

y

2 x+ 1

and

f(0)=2.

(a) Write an equation of the line tangent

to the graph offatx=0.

(b) Use the tangent in part (a) to find the

approximate value off(0.1).

(c) Find a solutiony=f(x) for the

differential equation.

(d) Using the result in part (c), find

f(0.1).

25. (Calculator) LetRbe the region in the
    first quadrant bounded byf(x)=ex− 1
    andg(x)=3 sinx.

(a) Find the area of regionR.

(b) Find the volume of the solid obtained

by revolvingRabout thex-axis.

(c) Find the volume of the solid havingR

as its base and semicircular cross

sections perpendicular to thex-axis.