2WBB0 – calculus
Summary fi nal exam
Contents
P.1 Real numbers and the real line........................................................................................4
Rules for inequalities........................................................................................................... 4
Intervals.............................................................................................................................. 4
Absolute value.................................................................................................................... 4
Properties of absolute values..............................................................................................4
Equations and inequalities involving absolute values..........................................................4
P.2 Cartesian coordinates in the plane...................................................................................5
Increments and distances...................................................................................................5
Distance D between P(x1, y1) and Q(x2, y2).........................................................................5
Straight lines....................................................................................................................... 5
Equations of lines................................................................................................................5
P.3 Graphs of quadratic equations.........................................................................................6
Circles and disks.................................................................................................................6
Equations of parabolas.......................................................................................................6
Shifting a graph...................................................................................................................6
P.4 functions and their graphs................................................................................................6
P.5 Combining functions to make new functions....................................................................7
Sums, differences, products, quotients and multiples.........................................................7
Composite functions...........................................................................................................7
Piecewise defined functions................................................................................................7
P.6 Polynomials and rational functions...................................................................................7
The factor theorem..............................................................................................................7
Roots and factors of quadratic polynomials.........................................................................8
P.7 the trigonometric functions...............................................................................................8
Some useful identities.........................................................................................................8
Other trigonometric functions..............................................................................................9
Sine law.............................................................................................................................. 9
Cosine law.......................................................................................................................... 9
1.1 Examples of velocity, growth rate, and area...................................................................10
The area of a circle........................................................................................................... 10
Average velocity................................................................................................................ 10
1.2 Limits of functions........................................................................................................... 10
One-sided limits................................................................................................................ 10
, The squeeze theorem.......................................................................................................10
1.3 Limits at infinity and infinite limits....................................................................................10
Limits at infinity and negative infinity.................................................................................10
Limits at infinity for rational functions................................................................................10
Infinite limits...................................................................................................................... 11
1.4 Continuity....................................................................................................................... 11
Continuity at an interior point............................................................................................11
Right and left continuity.....................................................................................................11
2.1 Tangent lines and their slopes........................................................................................12
The slope of a curve......................................................................................................... 12
Normals............................................................................................................................ 12
2.2 The derivative................................................................................................................. 12
Right derivative................................................................................................................. 12
Left derivative................................................................................................................... 12
2.3 Differentiation rules.........................................................................................................13
Differentiation rules........................................................................................................... 13
The reciprocal rule............................................................................................................ 13
The quotient rule............................................................................................................... 13
2.4 The chain rule................................................................................................................. 13
2.5 Derivatives of trigonometric functions.............................................................................13
An important trigonometric limit.........................................................................................14
Derivative of sine function.................................................................................................14
Derivative of cosine function.............................................................................................14
Derivatives of other trigonometric functions......................................................................14
2.8 The mean-value theorem................................................................................................14
3.1 Inverse functions............................................................................................................ 15
3.2 Exponential and logarithmic functions............................................................................15
Laws of logarithms............................................................................................................ 15
3.5 The inverse trigonometric functions................................................................................15
4.3 Indeterminate forms........................................................................................................16
4.4 Linear approximation......................................................................................................17
4.10 Taylor polynomials........................................................................................................17
5.4 Properties of the definite integral....................................................................................18
Mean-value theorem for integrals......................................................................................18
5.5 The fundamental theorem of calculus.............................................................................18
Part I................................................................................................................................. 18
Part II................................................................................................................................ 19
, 5.6 The method of substitution.............................................................................................19
Integrals of tangent, cotangent, secant, and cosecant......................................................19
6.1 Integration by parts.........................................................................................................20
6.2 Techniques of integration...............................................................................................20
6.5 Improper integrals........................................................................................................... 20
Improper integrals of type I...............................................................................................20
Improper integrals of type II..............................................................................................20
7.9 First-order differential equations.....................................................................................21
Separable equations......................................................................................................... 21
First-order linear equations...............................................................................................21
Summary fi nal exam
Contents
P.1 Real numbers and the real line........................................................................................4
Rules for inequalities........................................................................................................... 4
Intervals.............................................................................................................................. 4
Absolute value.................................................................................................................... 4
Properties of absolute values..............................................................................................4
Equations and inequalities involving absolute values..........................................................4
P.2 Cartesian coordinates in the plane...................................................................................5
Increments and distances...................................................................................................5
Distance D between P(x1, y1) and Q(x2, y2).........................................................................5
Straight lines....................................................................................................................... 5
Equations of lines................................................................................................................5
P.3 Graphs of quadratic equations.........................................................................................6
Circles and disks.................................................................................................................6
Equations of parabolas.......................................................................................................6
Shifting a graph...................................................................................................................6
P.4 functions and their graphs................................................................................................6
P.5 Combining functions to make new functions....................................................................7
Sums, differences, products, quotients and multiples.........................................................7
Composite functions...........................................................................................................7
Piecewise defined functions................................................................................................7
P.6 Polynomials and rational functions...................................................................................7
The factor theorem..............................................................................................................7
Roots and factors of quadratic polynomials.........................................................................8
P.7 the trigonometric functions...............................................................................................8
Some useful identities.........................................................................................................8
Other trigonometric functions..............................................................................................9
Sine law.............................................................................................................................. 9
Cosine law.......................................................................................................................... 9
1.1 Examples of velocity, growth rate, and area...................................................................10
The area of a circle........................................................................................................... 10
Average velocity................................................................................................................ 10
1.2 Limits of functions........................................................................................................... 10
One-sided limits................................................................................................................ 10
, The squeeze theorem.......................................................................................................10
1.3 Limits at infinity and infinite limits....................................................................................10
Limits at infinity and negative infinity.................................................................................10
Limits at infinity for rational functions................................................................................10
Infinite limits...................................................................................................................... 11
1.4 Continuity....................................................................................................................... 11
Continuity at an interior point............................................................................................11
Right and left continuity.....................................................................................................11
2.1 Tangent lines and their slopes........................................................................................12
The slope of a curve......................................................................................................... 12
Normals............................................................................................................................ 12
2.2 The derivative................................................................................................................. 12
Right derivative................................................................................................................. 12
Left derivative................................................................................................................... 12
2.3 Differentiation rules.........................................................................................................13
Differentiation rules........................................................................................................... 13
The reciprocal rule............................................................................................................ 13
The quotient rule............................................................................................................... 13
2.4 The chain rule................................................................................................................. 13
2.5 Derivatives of trigonometric functions.............................................................................13
An important trigonometric limit.........................................................................................14
Derivative of sine function.................................................................................................14
Derivative of cosine function.............................................................................................14
Derivatives of other trigonometric functions......................................................................14
2.8 The mean-value theorem................................................................................................14
3.1 Inverse functions............................................................................................................ 15
3.2 Exponential and logarithmic functions............................................................................15
Laws of logarithms............................................................................................................ 15
3.5 The inverse trigonometric functions................................................................................15
4.3 Indeterminate forms........................................................................................................16
4.4 Linear approximation......................................................................................................17
4.10 Taylor polynomials........................................................................................................17
5.4 Properties of the definite integral....................................................................................18
Mean-value theorem for integrals......................................................................................18
5.5 The fundamental theorem of calculus.............................................................................18
Part I................................................................................................................................. 18
Part II................................................................................................................................ 19
, 5.6 The method of substitution.............................................................................................19
Integrals of tangent, cotangent, secant, and cosecant......................................................19
6.1 Integration by parts.........................................................................................................20
6.2 Techniques of integration...............................................................................................20
6.5 Improper integrals........................................................................................................... 20
Improper integrals of type I...............................................................................................20
Improper integrals of type II..............................................................................................20
7.9 First-order differential equations.....................................................................................21
Separable equations......................................................................................................... 21
First-order linear equations...............................................................................................21
