336 STEP 4. Review the Knowledge You Need to Score High
- 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.
- If 3ey=x^2 y, find
dy
dx
.
- Evaluate
∫ 1
0
x^2
x^3 + 1
dx.
- 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?
- (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).
- (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.
- 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
- 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.