WTW 158 Calculus Exam Proofs
Sipho
Theme 1: Functions
1. Properties of Absolute Values
The absolute value of x is defined as:
(
x
if x ≥ 0
|x| =
−x if x < 0
|xy| = |x||y|
x
|x|
=
,
y
|y|
y ̸= 0
[Triangle Inequality]
|x + y| ≤ |x| + |y|
[Reverse Triangle Inequality]
|x − y| ≥ ||x| − |y||
Theme 2: Limits, Continuity, and Derivatives
2. Limit Laws
[Sum]
lim [f (x) + g(x)] = lim f (x) + lim g(x)
x→c
x→c
x→c
[Difference]
lim [f (x) − g(x)] = lim f (x) − lim g(x)
x→c
x→c
x→c
[Constant Multiple]
lim [k · f (x)] = k · lim f (x)
x→c
x→c
[Product]
lim [f (x) · g(x)] = lim f (x) · lim g(x)
x→c
x→c
x→c
[Quotient]
f (x)
limx→c f (x)
lim
=
,
x→c g(x)
limx→c g(x)
1
if lim g(x) ̸= 0
x→c
3. Squeeze Theorem
If f (x) ≤ g(x) ≤ h(x) for all x in some interval around c (except possibly at c)
and
lim f (x) = lim h(x) = L,
x→c
x→c
then
lim g(x) = L.
x→c
4. Fundamental Theorem of Calculus
[Part 1] If f is continuous on [a, b] and F (x) =
Rx
a
f (t) dt, then F ′ exists and
F ′ (x) = f (x).
[Part 2] If f is continuous on [a, b], then
Z b
f (x) dx = F (b) − F (a),
a
where F is any antiderivative of f , i.e., F ′ = f .
Theme 3: Differentiation
5. Derivatives of Trigonometric Functions
d
sin x = cos x
dx
Proof.
sin(x + h) − sin x
d
sin x = lim
h→0
dx
h
Using the sum-to-product identities,
sin x cos h + cos x sin h − sin x
h→0
h
cos h − 1
sin h
= lim sin x
+ cos x
h→0
h
h
= sin x · 0 + cos x · 1 = cos x
= lim
Theme 4: Applications of Differentiation
6. Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f attains its absolute maximum
and minimum on [a, b]. That is, there exist numbers c and d in [a, b] such that
f (c) ≥ f (x)
and f (d) ≤ f (x)
2
for all x ∈ [a, b].
7. Rolle’s Theorem
If f is continuous on the closed interval [a, b], differentiable on the open interval
(a, b), and f (a) = f (b), then there exists a number c ∈ (a, b) such that
f ′ (c) = 0.
8. Mean Value Theorem
If f is continuous on the closed interval [a, b] and differentiable on the open
interval (a, b), then there exists a number c ∈ (a, b) such that
f ′ (c) =
f (b) − f (a)
.
b−a
9. L’Hôpital’s Rule
′
(x)
If limx→c f (x) = limx→c g(x) = 0 or ±∞ and limx→c fg′ (x)
exists, then
f ′ (x)
f (x)
= lim ′
.
x→c g (x)
x→c g(x)
lim
Theme 5: Integration
10. Antiderivative Definition
A function F is an antiderivative of f on an interval I if
F ′ (x) = f (x)
for all x ∈ I.
Theme 6: Vector Algebra
11. Dot Product
The dot product of two vectors a = ⟨a1 , a2 , a3 ⟩ and b = ⟨b1 , b2 , b3 ⟩ is
a · b = a 1 b1 + a 2 b2 + a 3 b3 .
12. Cross Product
The cross product of two vectors a = ⟨a1 , a2 , a3 ⟩ and b = ⟨b1 , b2 , b3 ⟩ is
a × b = ⟨a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ⟩ .
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