Lieve Göbbels
2WBB0 Calculus 2
Semester 1, 2020-2021
Calculus 2
Recapitulation Basic Calculus 3
Powers 3
Simplifying expressions 3
Fractions 4
Trigonometric identities 4
Trigonometry 5
Rationalizing denominators 6
Partial fraction decomposition 6
Graphing functions 7
Differentiation 9
Antiderivatives 10
Equations 11
Special functions 13
Inequalities 13
Sine and cosine rule 15
Vectors 16
Line of intersection of two points 16
Distance between a point and a plane 16
Area of a triangle or parallelogram in 16
Volume of parallelepiped 17
Angles 17
Limits and Continuity 18
Limits 18
Squeeze theorem 19
Limits at in nity and in nite limits 19
Limits for exponential and logarithmic functions 19
Growth and decay 19
Indeterminate forms (l’Hôpital rules) 20
Continuity 20
Max-Min theorem 21
Intermediate value theorem 21
Differentiation 2.0 22
Tangent lines 22
The derivative 22
Trigonometric differentiation 22
Mean-Value theorem (MVT) 23
Inverse functions 24
General properties 24
Inverse trigonometric functions 25
Extreme Values, Approximation, Taylor Polynomials and The Sum Notation 26
Extreme values 26
, Linear approximations 26
Taylor polynomials 27
Sums and Sigma notation 27
More on Integrals 28
The de nite integral 28
The fundamental theorem 29
Methods 30
Improper integrals 31
First-Order (differential) equations 32
, Recapitulation Basic Calculus
In short:
• Powers
• Simplifying expressions
• Fractions
• Trigonometric identities
• Trigonometry
• Rationalizing denominators
• Partial fraction decomposition
• Graphing functions
• Differentiation
• Antiderivatives
• Equations
• Special functions
• Inequalities
• Sine and Cosine rule
Powers
1 ap
a p ⋅ a q = a p+q an
= a −n aq
= a p−q (a p )q = a pq
q
n
(a b) p = a p b p an = p
aq = a p ab = a⋅ b
a2
Simplifying expressions
(a + b)(c + d ) = a c + a d + bc + bd
๏ Special forms:
(a + b)2 = a 2 + 2a b + b 2
(a − b)2 = a 2 − 2a b + b 2
(a − b)(a + b) = a 2 − b 2
๏ Important:
(x + a)(x + b) = x 2 + (a + b)x + a b
(x 3 − a 3) = (x − a)(x 2 + a x + a 2 )
2WBB0 Calculus 2
Semester 1, 2020-2021
Calculus 2
Recapitulation Basic Calculus 3
Powers 3
Simplifying expressions 3
Fractions 4
Trigonometric identities 4
Trigonometry 5
Rationalizing denominators 6
Partial fraction decomposition 6
Graphing functions 7
Differentiation 9
Antiderivatives 10
Equations 11
Special functions 13
Inequalities 13
Sine and cosine rule 15
Vectors 16
Line of intersection of two points 16
Distance between a point and a plane 16
Area of a triangle or parallelogram in 16
Volume of parallelepiped 17
Angles 17
Limits and Continuity 18
Limits 18
Squeeze theorem 19
Limits at in nity and in nite limits 19
Limits for exponential and logarithmic functions 19
Growth and decay 19
Indeterminate forms (l’Hôpital rules) 20
Continuity 20
Max-Min theorem 21
Intermediate value theorem 21
Differentiation 2.0 22
Tangent lines 22
The derivative 22
Trigonometric differentiation 22
Mean-Value theorem (MVT) 23
Inverse functions 24
General properties 24
Inverse trigonometric functions 25
Extreme Values, Approximation, Taylor Polynomials and The Sum Notation 26
Extreme values 26
, Linear approximations 26
Taylor polynomials 27
Sums and Sigma notation 27
More on Integrals 28
The de nite integral 28
The fundamental theorem 29
Methods 30
Improper integrals 31
First-Order (differential) equations 32
, Recapitulation Basic Calculus
In short:
• Powers
• Simplifying expressions
• Fractions
• Trigonometric identities
• Trigonometry
• Rationalizing denominators
• Partial fraction decomposition
• Graphing functions
• Differentiation
• Antiderivatives
• Equations
• Special functions
• Inequalities
• Sine and Cosine rule
Powers
1 ap
a p ⋅ a q = a p+q an
= a −n aq
= a p−q (a p )q = a pq
q
n
(a b) p = a p b p an = p
aq = a p ab = a⋅ b
a2
Simplifying expressions
(a + b)(c + d ) = a c + a d + bc + bd
๏ Special forms:
(a + b)2 = a 2 + 2a b + b 2
(a − b)2 = a 2 − 2a b + b 2
(a − b)(a + b) = a 2 − b 2
๏ Important:
(x + a)(x + b) = x 2 + (a + b)x + a b
(x 3 − a 3) = (x − a)(x 2 + a x + a 2 )
