Differential Calculus (50%) Differential Calculus
50%
- 2.1The Derivative
Definitions of the derivative; Derivatives of elementary functions; Derivatives of sums, products and quotients; Derivative of a composite function (chain rule); Implicit differentiation; Derivative of the inverse of a function; Higher order derivatives; Corresponding characteristics of graphs of f, f', and f''; Statement of the Mean Value Theorem; applications and graphical illustrations; Relation between differentiability and continuity; Use of L'Hospital's Rule (quotient and indeterminate forms)
- 2.2Applications of the Derivative
Slope of a curve at a point; Tangent lines and linear approximation; Curve sketching: increasing and decreasing functions; relative and absolute maximum and minimum points; concavity; points of inflection; Extreme value problems; Velocity and acceleration of a particle moving along a line; Average and instantaneous rates of change; Related rates of change
Integral Calculus (40%) Integral Calculus
40%
- 3.1Antiderivatives and Techniques of Integration
Concept of antiderivatives; Basic integration formulas; Integration by substitution (use of identities and change of variable)
- 3.2Applications of Antiderivatives
Distance and velocity from acceleration with initial conditions; Solutions of y' = ky and applications to growth and decay
- 3.3The Definite Integral
Definition of the definite integral as the limit of a sequence of Riemann sums and approximations of the definite integral using areas of rectangles; Properties of the definite integral; The Fundamental Theorem
- 3.4Applications of the Definite Integral
Average value of a function on an interval; Area, including area between curves; Other (e.g., accumulated change from a rate of change)