Calculus Overview
Purpose
Calculate area under a curve or volume
Integral
Integral on Wikipedia
Wikipedia: Assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data
Definite Integral
Definite integral notation: integral Subscript a Superscript b Baseline f left-parenthesis x right-parenthesis d x
Wikipedia: Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.
Example calculation
Procedure to calculate the formula: let the term be calculated by increasing the value of the exponent by one and then dividing by that number. Using this term, subtract the value of the term with the lower limit x value substituted into it from the value of the term with the upper limit x value substituted
integral Subscript 2 Superscript 5 Baseline x squared d x equals StartFraction x Subscript b Superscript 3 Baseline Over 3 EndFraction minus StartFraction x Subscript a Superscript 3 Baseline Over 3 EndFraction equals StartFraction 5 cubed Over 3 EndFraction minus StartFraction 2 cubed Over 3 EndFraction
Calculate the Slope / rate of change / steepness of the curve
Slope (estimated without calculus) between two points in time is the rise (change in y) divided by the run (change in x)
Derivative
The derivative is the calculation of the exact slope / rate of change at one point on the curve or one precise moment in time - the rate at which the value y of the function changes with respect to the change of the variable x.
First [order] derivative
First [order] derivative notations: f prime left-parenthesis x right-parenthesis equals StartFraction d x Over d y EndFraction equals y prime
Procedure to calculate the formula: multiply each coefficient by the value of the exponent, and decrement the exponent
First [order] derivative f left-parenthesis x right-parenthesis equals 2 x squared plus 2 x plus 1 f prime left-parenthesis x right-parenthesis equals 4 x plus 2