A. Functions, Graphs and Limits 

A.1 Analysis of Graphs 


A.1.1 
Geometric and Analytic Analysis of Graphs of Different Functions 


A.1.2 
Prediction and Explanation of Local and Global Behaviour of a Function by using both Geometric and Analytic and Calculus 

A.2 Limits of Functions [Including OneSided Limits] 


A.2.1 
Solving the Limits of a function using Algebra 


A.2.2 
Estimating Limits from Graphs 

A.3 Asymptotic and Unbounded Behaviour of Graphs 


A.3.1 
Concept of Asymptotes as a Graphical Behaviour 


A.3.2 
Describing Asymptotic Behavior in terms of Limits Involving Infinity 


A.3.3 
Comparing Relative Magnitudes of Functions and their Rates of Change 

A.4 Continuity 


A.4.1 
Continuous Function and Concept of Continuity 


A.4.2 
Continuity in terms of Limits [Right Hand Limit and Left Hand Limit] 


A.4.3 
Graphical Representation of Continuous Functions 


A.4.4 
Intermediate Value Theorem and Extreme Value Theorem 

A.5 Parametric, Polar and Vector Functions 


A.5.1 
Analysis of Planar Curves given in Parametric, Polar and Vector. 
B. Derivatives 

B.1 Concept of the Derivative 


B.1.1 
Graphical, Numerical and Analytical concept of Differentiation 


B.1.2 
Instantaneous Rate of Change 


B.1.3 
Derivative as the Limit of the Difference Quotient [First Principle Method] 


B.1.4 
Relationship between Differentiability and Continuity 

B.2 Derivative at a Point 


B.2.1 
Slope of a Curve at a Point [Points having Vertical Tangents and No Tangents] 


B.2.2 
Tangent Line to a Curve at a Point and Local Linear Approximation 


B.2.3 
Instantaneous Rate of Change as the Limit of Average Rate of Change 


B.2.4 
Approximate Rate of Change from Graphs 

B.3 Derivative as a Function 


B.3.1 
Corresponding Characteristics of Graphs of ƒ and ƒ' 


B.3.2 
Relationship between the Increasing and Decreasing behavior of ƒ and the sign of ƒ' 


B.3.3 
The Mean Value Theorem and its Geometric Interpretation 


B.3.4 
Equations Involving Derivatives 


B.3.5 
Verbal Descriptions Translation into Equations Involving Derivatives and vice versa 

B.4 Second Derivatives 


B.4.1 
Corresponding Characteristics of the graphs of ƒ , ƒ' and ƒ'' 


B.4.2 
Relationship between the Concavity of ƒ and the Sign of ƒ'' 


B.4.3 
Points of Inflection as Places where Concavity Changes 

B.5 Applications of Derivatives 


B.5.1 
Analysis of Curves 


B.5.2 
Planar Curves Analysis [in Parametric, Polar and Vector Form including Velocity and Acceleration] 


B.5.3 
Optimization both Absolute/Global and Relative/Local Extrema 


B.5.4 
Modelling Rtes of Change [including related rates problems] 


B.5.5 
Implicit Differentiation for differentiating an inverse function 


B.5.6 
Derivation as a rate of change [velocity and acceleration] 


B.5.7 
Geometric Interpretation of Differential Equations [including curves] 


B.5.8 
Differential Equations Solution using Euler's Method 


B.5.9 
L'Hospital's Rule [usage in determining Limits and Convergence of Improper Integrals and Series] 

B.6 Computation of Derivatives 


B.6.1 
Differentiation of functions[for example basic functions, including power, exponential, logarithmic, trigonometric and inverse trigonometric functions ] 


B.6.2 
Basic Differentiating Rules [Sums, Products and Quotients Rules] 


B.6.3 
Chain Rule and Implicit Differentiation 


B.6.4 
Derivative of Parametric, Polar and Vector functions 