Grade 12 – Calculus and Vectors
Rates of Change Test
Rates of Change and the slope of a curve
- Average rate of change: refers to the rate of change of a function over an interval. It corresponds to the slope of the secant connecting the two endpoints of the interval
- Instantaneous rate of change: refers to the rate of change at a specific point. It corresponds to the slope of a tangent passing through a single point, or tangent point, on the graph of a function.
- 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.
- 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 continues 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 derivatives 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 non-differentiable 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.