Differential Calculus
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I Functions and Models
II Limits and Derivatives
2.2 The Limit of a Function
The Idea of a Limit
Left and Right Limits
Limits That Do Not Exist
Infinite Limits and Vertical Asymptotes
2.3 Calculating Limits Using the Limit Laws
The Limit Laws
Direct Substitution
Limits of Trigonometric Functions
The Squeeze Theorem
2.4 The Precise Definition of a Limit
Proof of Limit Laws
Proofs of Trigonometric Limits
Limits at Infinity by Definition
Proof of Direct Substitution Property
Proof of The Squeeze Theorem
2.5 Continuity
Limits and Continuity
Types of Discontinuities
Properties of Continuous Functions
Proof of Continuity Laws
The Intermediate Value Theorem
Proof of The IVT
Continuity of Trigonometric Functions
2.6 Limits at Infinity; Horizontal Asymptotes
Limits At Infinity
Limits at Infinity by Definition
2.7 Derivatives and Rates of Change
Application: Tangent Lines
2.8 The Derivative as a Function
Differentiability
Differentiability & Continuity
Notation
III Differentiation Rules
3.1 Derivatives of Polynomials and Exponential Functions
Power Functions
Proof of Differentiation Rules
Derivatives of Exponential Functions
3.2 The Product and the Quotient Rule
The Product Rule
Proof of the Product Rule
The Quotient Rule
Proof of the Quotient Rule
3.3 Derivatives of Trigonometric Functions
Derivative of
\(\sin(x)\)
and
\(\cos(x)\)
Other Trigonometric Derivatives
Proofs of Trigonometric Derivatives
Higher Derivatives
3.4 The Chain Rule
Proof of The Chain Rule
3.5 Implicit Differentiation
Derivatives of Inverse Trigonometric Functions
Proof of Inverse Trig Derivative Laws
3.6 Derivatives of Logarithmic and Inverse TrigonometricFunctions
Logarithmic Derivatives
Proof of Logarithmic Derivative Laws
Logarithmic Differentiation
Proof of the Power Rule
3.9 Related Rates
3.10 Linear Approximations and Differentials
Linear Approximation
Differentials
IV Applications of Differentiation
4.1 Maximum and Minimum Values
Critical Points
Maximum and Minimum Values
Proof of the Extreme Value Theorem
Finding Absolute Extrema
Calculus I
Calculus I
Differential Calculus
About This Book