MATH 18.01 - FINAL EXAM REVIEW: SUMMARY OF SOME KEY
CONCEPTS
18.01 Calculus, Fall 2014
Professor: Jared Speck
a. Parametric curves
(a) Are curves in the (x, y) plane expressed as
x = F (t),
y = G(t),
a ≤ t ≤ b, where t is called the parameter.
b. Arc length of a curve
Rb
(a) Arc length is equal to a ds.
(b) a is the parameter starting point, b is the parameter end
q point.
p
p
2
dx 2
(c) For curves in parametric form, ds = (dx)2 + (dy)2 =
+ dy
dt = (F 0 (t))2 + (G0 (t))2 dt
dt
dt
(Pythagorean theorem).
q
p
dy 2
(d) For curves y = f (x), the formula reduces to ds = 1 + dx
dx = 1 + (f 0 (x))2 dx
(and x is the parameter).
c. Surface area of a solid formed by revolving a curve around the x−axis (for revolution around
the y−axis, interchange the roles of x and y in everything that follows)
(a) Divide the surface into small strips that are portions of cones (the cone strip radii are
parallel to the y−axis, and the cone strip axes of symmetry are parallel to the x−axis).
(b) RSurface area is given by
Rconical strip circumference × slant edge length
= 2π conical strip radius × ds
q
2
2
+( dy
( dx
dt )
dt )
y
}|
{
R t=b z}|{ zp
0
2
0
= t=a 2π G(t) (F (t)) + (G (t))2 dt.
(c) a is the parameter starting point, b is the parameter end point.
q
dy 2
1+( dx
)
y
z
}|
{
R x=b z}|{ p
(d) For curves y = f (x), the formula reduces to x=a 2π f (x) 1 + (f 0 (x))2 dx.
d. Polar coordinates
(a) x = r cos θ, y = r sin θ
p
(b) In the standard formulation, r = x2 + y 2 , θ is the polar angle, and 0 ≤ θ < 2π
1
Final Exam - Review Sheet
2
(c) Area in polar coordinates: Area under the curve r = f (θ) in between the angles θ1 and
Rθ
Rθ
θ2 is given by Area = 12 θ12 r2 dθ = 12 θ12 [f (θ)]2 dθ
e. L’Hôpital’s rule
(a) Sometimes allows one to evaluate limits of the form 00 , ∞
, 0 × ∞, 00 , 1∞
∞
0
(b) Many of the above limits can be massaged into the form 0 or ∞
, where L’Hôpital’s rule
∞
can sometimes directly be applied. For example, the 00 case can be massaged into the
0
case with the help of ln .
0
(c) In the “ 00 ” case: If f, g are differentiable functions, a is a finite number, f (a) = g(a) = 0,
0 (x)
0 (x)
(x)
= L, then limx→a fg(x)
= limx→a fg0 (x)
= L. Furthermore, it is sometimes
and limx→a fg0 (x)
0
(a)
true that L = fg0 (a)
(for example, when f 0 (x) and g 0 (x) are continuous at x = a and
g 0 (a) 6= 0).
∞
(d) In the “ ∞
” case: If f, g are differentiable functions, a is a finite number, limx→a f (x) =
0 (x)
0 (x)
(x)
exists, then limx→a fg(x)
= limx→a fg0 (x)
.
limx→a g(x) = ∞, and limx→a fg0 (x)
(e) Analogous statements hold if we replace limx→a with limx→∞ or limx→−∞ .
f. Improper integrals
R∞
RM
(a) If f (x) is continuous for 0 ≤ x < ∞, then by definition, 0 f (x) dx = limM →∞ 0 f (x) dx
(i) If the limit exists, we say the improper integral converges. Otherwise, we say it
diverges.
(b) If f (x) is continuous for a < x ≤ b but is not continuous at x = a, then by definition,
Rb
Rb
f (x) dx = limx0 →a+ x0 f (x) dx
a
(i) If the limit exists, we say the improper integral converges. Otherwise, we say it
diverges.
g. Infinite series
P
···
(a) Are series of the form ∞
k=0 ak = a0 + a1 + a2 + a3 +
PM
P∞
(b) By definition, k=0 ak = limM →∞ SM , where SM = k=0 ak = a0 + a1 + a2 + · · · + aM
is the M th partial sum.
(i) If limM →∞ SP
converges. Otherwise, we say it diverges.
M exists, we say the seriesP
∞
1
k
k
(c) Geometric series: k=0 x = 1−x if |x| < 1. ∞
if |x| ≥ 1.
k=0 x diverges
P∞
P
(d) Comparison: If 0 ≤ ak ≤ bk for all large k, and if k=0 ak diverges, then ∞
k=0 bk
diverges too (divergence
of
smaller
=⇒
divergence
of
bigger).
If
0
≤
a
≤
b
for
all
k
k
P∞
P∞
large k, and if k=0 bk converges, then k=0 ak converges too (convergence of bigger
=⇒ convergence of smaller).
P∞
(e) Limit comparison test: If
a
≥
0,
b
≥
0
for
all
large
k
and
a
∼
b
,
then
k
k
k
k
k=0 ak
P
converges if and only if ∞
b
converges.
Here,
a
∼
b
means
that
there
exists
a
k
k
k
k=0
ak
non-zero number L such that limk→∞ bk = L.
(f) Integral comparison:
for
R∞
P∞ If f (x) is continuous, f (x) ≥ 0 for all x, and f (x) is decreasing
all large x, then k=0 f (k) converges if and only if the improper integral x=0 f (x) dx
converges.
h. Taylor’s series with base point b = 0
Final Exam - Review Sheet
3
For x near 0 : f (x) = a0 + a1 x + a2 x2 + a3 x3 + · · ·
(n)
an = f n!(0) , where f (n) is the nth derivative of f
P
xk
x2
x3
x4
ex = ∞
k=0 k! = 1 + x + 2! + 3! + 4! + · · ·
P
2
3
4
k+1 k
ln(1 + x) = ∞
x = x − x2 + x3 − x4 ± · · ·
k=1 (−1)
P
2
4
(−1)k x2k
cos x = ∞
= 1 − x2! + x4! ∓ · · ·
k=0
(2k)!
P
3
5
(−1)k x2k+1
= x − x3! + x5! ∓ · · ·
(f) sin x = ∞
k=0 (2k+1)!
i. Taylor’s series with base point b
(a) For x near b : f (x) = a0 + a1 (x − b) + a2 (x − b)2 + a3 (x − b)3 + · · ·
(n)
(b) an = f n!(b)
A−2
A(A−1)(A−2)bA−3
2
(c) xA = bA + AbA−1 (x − b) + A(A−1)b
(x
−
b)
+
(x − b)3 + · · ·
2!
3!
(a)
(b)
(c)
(d)
(e)