IB Mathematics (SL)/Calculus

Introduction

Average Rate of Change (AROC) between x=a and x=b in f(X)

AROC = $\frac{f(a) - f(b)}{a-b}$

Instantaneous Rate of Change (IROC) at x=a is the slope of the line tangent at x=a:

IROC =$\lim_{x \to a} \frac{f(x) - f(a)}{x-a}$

Definition of the derivative f'(x) of a function f(x) (First principles of calculus):

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Basic Differentiation

The following formulas are shortcuts for finding the derivative

Derivative of a power function

$f(x) = ax^n \,\!$ , $f'(x) = anx^{n-1}\,\!$

Derivative of exponential function

$f(x) = e^x \,\!$ , $f'(x) = e^{x}\,\!$

Derivative of logarithmic function

$f(x) = \ln x \,\!$ , $f'(x) = \frac{1}{x} \,\!$

Derivative of trigonometric functions

$f(x) = \sin x \,\!$ , $f'(x) = \cos x \,\!$

$f(x) = \cos x \,\!$ , $f'(x) = -\sin x \,\!$

$f(x) = \tan x \,\!$ , $f'(x) = \frac{1}{\cos^2 x} = \sec^2 x \,\!$

Derivatives of sum of two functions

$f(x) = g(x) + h(x)\,\!$ , $f'(x) = g'(x) + h'(x)\,\!$

Chain Rule

$f(x) = g(h(x))\,\!$ , $f'(x) = g'(h(x)) \cdot h'(x)\,\!$

Product Rule

$f(x) = uv \,\!$ , $f'(x) = uv' + vu'\,\!$

Quotient Rule

$f(x) = \frac{u}{v} \,\!$ , $f'(x) = \frac{vu'-uv'}{v^2}\,\!$

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.