5 Steps to a 5 AP Calculus BC 2019

336 STEP 4. Review the Knowledge You Need to Score High

25. Use Euler’s Method with a step size of
    Δx= 0 .5 to computey(3) ify(x)isthe

solution of the differential equation

dy

dx

=y− 2 xwith initial conditiony(0)=1.

13.10 Cumulative Review Problems

(Calculator) indicates that calculators are
permitted.

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

.

27. Evaluate

∫ 1

0

x^2

x^3 + 1

dx.

28. The graph of a continuous functionfthat
    consists of three line segments on [−2, 4] is
    shown in Figure 13.10-1. If
    F(x)=

∫x

− 2

f(t)dtfor− 2 ≤x≤4,

7 6 5 4 3 2 1

–2 0–1 1 2 3 4 5

y

f

t

Figure 13.10-1

(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?

29. (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).

30. (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.

31. An object traveling on a path defined by
    〈x(θ),y(θ)〉has an acceleration vector of
    〈sinθ,−cosθ〉. If the velocity of the object
    at timeθ=
    π
    3
    is

〈

−1, 0

〉

and the initial

position of the object is the origin, find the

position whenθ=π.

32.

∫

x^2 e^5 x−^2 dx

33. A projectile follows a path defined by
    x=t−2,y=sin^2 ton the interval
    0 ≤t≤π. Find the point at which the
    object reaches its maximumy-value.