1220-006 CALCULUS II
Review sheet for chapters 8, 9 and 10
Alan M. Watson
November 27, 2015
1
Sections 8.1 & 8.2: Indeterminate forms and l’Hppital’s rule
Things you should know:
(i) L’Hppital’s rule states that if a limit limx→a
f (x)
g(x)
is of the form
0
0
or
∞
∞,
then
f 0 (x)
f (x)
= lim 0
x→a g (x)
x→a g(x)
lim
provided that the latter exists. If it doesn’t, l’Hppital’s theorem cannot be applied.
(ii) There are other indeterminate forms 1∞ , 00 , ∞0 that can be reduced to the forms
by taking logarithms.
Sample questions:
• Compute
Rx√
lim
0
x→0
1 + sin t dt
x
• Convince yourself that you can’t use l’Hppital’s rule to compute the limit
x2 sin(1/x)
x→0
tan x
lim
and find this limit using standard techniques.
• Consider the function
f (x) =
ln x
x−1
c
x 6= 1
x=1
What value of c makes f (x) continuous at x = 1?
Trickier Determine the constants a, b, c so that
ax4 + bx3 + 1
=c
x→1 (x − 1) sin(πx)
lim
1
0
0
or
∞
∞
• Compute the limit
lim (tan x)cos x
x→π/2−
• Compute the limits
1/x
lim x
x→0
2
,
lim
x→∞
1
1+
x
x
,
lim
x→1
1
x
−
x − 1 ln x
Sections 8.3 & 8.4: Improper integrals
Things you should know:
(i) Improper integrals of the form
∞
Z
R∞
a
Z
f (x) dx = lim
b→∞ a
a
f (x) dx or
b
Rb
−∞ f (x) dx
Z
can be computed as
b
f (x) dx,
Z
f (x) dx = lim
−∞
a→−∞ a
b
f (x) dx
If these limits are finite, the improper integrals are said to converge.
(ii) The integral
R∞
1
1
xp
dx converges if p > 1 and diverges otherwise.
(iii) In order to establish the convergence of an improper integral, we may use a comparison
test: if 0 ≤ f (x) ≤ g(x) over [a, ∞) then:
• If
• If
R∞
a g(x) dx
R∞
a f (x) dx
R∞
converges, then a f (x) dx converges too.
R∞
diverges, then a g(x) dx diverges too.
Sample questions:
• Show that the integral
R∞
1
1
xp
dx converges if p > 1 and diverges otherwise.
• Use a comparison test to decide whether
Z
2
∞
1
dx
x2 ln(x + 1)
converges.
• Does the integral
• Does
Re
• Does
R 3π/4
1
1 x ln x
π/4
R∞
2
√ 1
x+1−1
dx converge or not?
dx converge?
tan x dx converge?
2
3
Sections 9.2, 9.3, 9.4 & 9.5: Convergence tests for series
Things you should know:
(i) A geometric series
P
n≥0 r
n
converges to
1
1−r
if |r| < 1 and diverges otherwise.
P
(ii) A harmonic series n≥1 n1 diverges.
P
(iii) A p-series n≥1 n1p converges if p > 1.
(iv) You should be able to compute sums of collapsing series (e.g.
1
n≥1 n(n+1) ,
P
k
n≥1 ln k+1 ).
P
(v) You should be able to determine whether a series converges or not. You’ll be able to use
a table of convergence tests.
Sample questions:
Check the quizzes and Midterm 2.
4
Sections 9.6 & 9.7: Power series and operations on them
Things you should know:
P
(i) Given a power series n≥0 an (x − a)n , you should be able to determine the convergence
set (the values of x that make the series convergent). This is accomplished by imposing
an+1 xn+1 <1
lim
n→∞ an xn (ii) You should be able to compute derivatives, integrals, products and quotients of power
series.
Sample questions:
• Find the values of x for which the power series
5
n (x−2)
n≥1 (−1)
n
P
• Find a power series expansion in x for f (x) =
ex
1+ln(1+x) .
• Find a power series expansion in x for f (x) =
Rx
0
tan−1 t
t
n
converges.
dt
Section 9.8 & 9.9 Taylor and Maclaurin series
Things you should know:
(i) You should be able to find the Taylor series of a function
X f n) (a)
n≥0
n!
3
(x − a)n
(ii) Find an upper bound for the error that is made when approximating a function by a
Taylor polynomial. Establish the order of the Taylor polynomial that is required in order
to approximate the value of a function with a specified accuracy.
Sample questions:
• Find the Taylor series through terms of order 4 for the function
Z x
1
−1
√
sin (x) =
dt
1 − t2
0
• Represent
√
1 + x as a Maclaurin series and use it to approximate
√
1.1 to 5 decimal places.
• Find the Taylor polynomial of order 2 for the function f (x) = ln[cos2 (x)] around the point
π
6 and give an upper bound for the error that we would make if we used this polynomial
to approximate f ( π3 ).
• Determine the order of the Maclaurin polynomial for ex that is required to approximate e
to 5 decimal places.
• Calculate cos 63 accurate to 3 decimal places.
6
Section 10.1, 10.2 & 10.3: Translation and rotation of axes
Things you should know:
(i) Given a fixed line ` and a fixed point F , a conic is the locus of points P satisfying the
relation
P F = eP `, e = constant
The constant e is called eccentricity and its value determines the shape of the conic.
• When e = 1, the conic is a parabola, which has standard equation y =
• When 0 < e < 1, the conic is an ellipse, which has standard equation
• When 1 < e, the conic is a hyperbola, which has standard equation
1 2
2p x .
x2
a2
x2
a2
+
−
y2
b2
y2
b2
= 1.
= 1.
(ii) In general, a polynomial equation of degree 2 in two variables x, y determines a conic
Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0
In order to reduce this equation to its standard form it suffices to perform a rotation
followed by a translation of the coordinate axes.
• First perform a rotation
x = u cos θ − v sin θ
y = u sin θ + v cos θ
and determine the angle tan(2θ) =
B
A−C
4
that results in an equation with no uv-term.
• Then complete squares in order to reduce the (u, v)-equation to its standard form.
Sample questions:
• Name the conic represented by the following equations and sketch its graph.
(a) x2 + y 2 − 2x + 2y + 1 = 0.
(b) 9x2 + 4y 2 + 72x − 16y + 160 = 0.
(c) y 2 − 4y − 5x − 6 = 0.
(d) x2 + xy + y 2 = 6.
(e) 4xy − 3y 2 = 64.
(f) 4x2 − 3xy = 18.
√
√
• Consider the conic 4x2 + 2 3xy + 2y 2 + 10 3x + 10y = 5
(a) What rotation do we need to perform in order to eliminate the cross-product term?
(b) Suppose that this rotation results in an equation 5u2 + v 2 + 20u = 5. Find the
standard form of this conic and sketch its graph in (x, y)-coordinates.
• Find the points of x2 + 14xy + 49y 2 = 100 that are closest to the origin.
• For what values of B is the graph of x2 + Bxy + y 2 = 1
(a) an ellipse.
(b) a circle.
(c) a hyperbola.
(d) two parallel lines.
7
Section 10.5: Polar coordinates
Things you should know:
(i) You should be able to convert points and equations written in cartesian coordinates to
polar coordinates and vice-versa.
(ii) You should be able to use the polar expressions
r=
d
,
cos(θ − θ0 )
r = 2a cos(θ − θ0 ),
r=
ed
1 + e cos(θ − θ0 )
to quickly identify conics.
Sample questions:
For each of the following polar equations, name the curve that is being represented. If it’s a
conic, give the eccentricity.
5
• r = 6.
8
• θ=
2π
3 .
• r=
4
1+cos θ ,
• r=
6
4−cos θ
• r=
4
2+2 cos(θ−π/3)
• r=
4
3 cos(θ−π/3)
• r=
4
1/2+cos(θ−π)
r=
4
1+2 cos θ
Section 10.7: Calculus in polar coordinates
Things you should know:
Sample questions:
6