IB Mathematics SL/Calculus
Average Rate of Change (AROC) between x=a and x=b in f(X)
Instantaneous Rate of Change (IROC) at x=a is the slope of the line tangent at x=a:
Definition of the derivative f'(x) of a function f(x) (First principles of calculus):
The following formulas are shortcuts for finding the derivative
Derivative of a power function
Derivative of exponential function
Derivative of logarithmic function
Derivative of trigonometric functions
Derivatives of sum of two functions
Applications to the derivative
The derivative is the slope at one point in a function. The slope is the rate of change. Ergo, with the derivative you can determine the rate of change at a given point. Given a displacement graph, where time is represented by x and position is represented by y, the derivative of any point on any function graphed will say the rate of change at that position; this is known as the velocity. The derivative of a velocity graph shows the acceleration.
Introduction to Integrals
Integrals find the area under the curve and are also known as the anti-derivative. This means that if the integral of the derivative is found, the original equation will be given but with an arbitrary constant c. A documented method of integration is the use of u-substitution.
Applications to integration
-Total Distance traveled -Area under a curve -Volume of revolution