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MATH 1070: Mathematics of Change, part I
Calculus Group
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Front Matter
Preface: approximations and bounds
1
Where should I put my ladder?
2
Bounding
3
Concavity
0
Variables, functions and graphs
0.1
Notation and terminology
0.2
Some useful functions
0.3
Properties of functions
0.4
Graphing
1
Units, proportionality and mathematical modeling
1.1
Physical units and formulas
1.2
Modeling
1.3
Inverse functions
1.4
Exponential and logarithmic relationships
2
Limits
2.1
Definitions of limit
2.2
Variations
2.3
Continuity
2.4
Computing limits
2.4.1
The Most Important Fact About Limits
2.4.2
Limit rules and
\(\infty\)
2.4.3
Two algebraic techniques for computing limits
2.4.4
the log trick
3
Derivatives
3.1
Concept of the derivative
3.2
Definitions
3.3
First and second derivatives, and sketching
4
Computing derivatives
4.1
Tools for computing derivatives
4.2
Arguments and proofs
5
Some Applications of Derivatives
5.1
Differentiating inverse functions
5.2
Related rates
5.3
Exponentials revisited
5.3.1
Differential equations
5.3.2
Time constants
5.4
Tangent line estimates and bounds using calculus
5.4.1
The mean value theorem
6
Asymptotic analysis and L’Hôpital’s Rule
6.1
L’Hôpital’s Rule
6.2
Orders of growth at infinity
6.3
Comparisons elsewhere and orders of closeness
7
Optimization
7.1
Definitions of Minima and Maxima, and their existence
7.2
The first derivative and extrema
7.3
Some example applications
8
Applying the optimization procedure
8.1
Optimization in geometry
8.2
Optimization in economics and business
9
Summation
9.1
Sequences
9.2
Finite series
9.3
Some series you can explicitly sum
9.4
Infinite series
9.5
Financial applications
10
Integrals
10.1
Area
10.2
Riemann sums and the definite integral
10.3
Trapezoidal approximation
10.4
Interpretations of the integral
10.5
The fundamental theorem of calculus
10.6
Estimating sums via integrals
11
Computing integrals
11.1
Remembering and guessing
11.2
Integration by parts
11.3
Substitution
12
Integrals over the whole real line
12.1
Definitions
12.2
Convergence
13
Integration and Probability
13.1
Probability densities
13.2
Summary statistics
13.3
Some common probability densities
14
Taylor approximations
14.1
Taylor polynomials
14.2
Taylor and Maclaurin polynomials in graphing
14.3
Computing Taylor Polynomials
14.3.1
Simple Compositions
14.3.2
Products
14.3.3
More Complicated Compositions
14.3.4
Putting it together: Taylor series
14.4
The Mean Value Theorem and Taylor’s Theorem
Chapter
5
Some Applications of Derivatives
Calculus has been around for 300 years. The applications and techniques don’t all fit nicely into chapter length categories. Here, we tie up some loose ends and mention a few things we think you shouldn’t miss.
🔗
5.1
Differentiating inverse functions
5.2
Related rates
5.3
Exponentials revisited
5.4
Tangent line estimates and bounds using calculus
🔗
