University:
Virtual High School
Course:
MCV4U | Calculus and Vectors
Academic year:

2023
Views:

15

Pages:

2
Author:

anon

MCV4U – Grade 12 Calculus & Vectors – Rates of Change Rates of Change and the slope of a curve ● Average rate of change: refers to the rate at which a function changes over a period of time. It relates to the angle of the secant between the interval's two endpoints. ● Instantaneous rate of change: speaks of the rate of change at a particular time. It is equivalent to the slope of a tangent passing through a single point, also known as a tangent point, on a function graph. ○ An estimate of the instantaneous rate of change can be obtained by calculating the average rate of change over the smallest interval for which data are available. ○ It can also be determined by sketching a tangent on a graph Rates of Change using Equations ● For a given function y = f(x), the instantaneous rate of change at x = a is estimated by calculating the slope of a secant over a very small interval, a <= x <= a + h such that h is a very small number ● The expression f(a+h) – f(a) / h, where h not = 0, is called the difference quotient ○ It is used to calculate the slope of the secant between (a, f(a)) and (a + h, f(a + h)). It will let you have an estimate for the slope of the tangent as h approaches 0. Limits ● A sequence is a function, f(n) = tn , whose domain is the set of natural numbers N ● limit as x approaches a of f(x) exists if the following are true: ○ limit as x approaches a- of f(x) exists (left hand limit exists) ○ limit as x approaches a+ of f(x) exists (right hand limit exists) ○ right side limit and left side limit are equal ● If they aren’t equal and they criteria aren’t met, then the limit does not exists ● Limits can be a real number of infinity ● A function is continuous at a value if these are true: ○ f(a) exists ○ limit as x approaches a exists ○ The limit as x approaches a and f(a) are the same Limits and Continuity ● The limit of a function at x = a may exist even though the function is discontinuous at x = a. ● In short, the graph of a discontinuous function cannot be drawn without lifting your pencil ● There are 3 kinds if discontinuity: Jump, Removable, and Infinite ● When direct substitution of x = a results in a limit of an indeterminate , 0/0, determine an equivalent function that represents f(x). ● Discontinuity may be removed by factoring, rationalizing the numerator or denominator, expanding, and simplifying. Introduction to derivatives ● The derivative of y = f(x) is a new function y = f’(x), which represents the slope of the tangent, or instantaneous rate of change, at any point on the curve of y = f(x). ● The derivative function is defined by the first principles definition for the derivative, f’(x) = lim h -> 0 f(x+h) – f(x) / h, if the limit exists ● Different notations for the derivative of y = f(x) are y = f’(x), y’, dy/dx, and d/dx f(x). ● If the derivative does not exist on a point on the curve, the function is nondifferentiable at that x-value. This can occur at points where the function is discontinuous or in cases where the function has an abrupt change, which is represented by a cusp or corner on the graph.