Rates of Change
The algebra formula used to find the average rate of change between two coordinates (x1, y1) and (x2, y2)
Slope formula: m=(y2-y1)/(x2-x1)
This mathematical concept asks what value a function approaches as the input gets closer and closer to a target number.
limit
The absolute first step you should always try when evaluating any analytical limit problem.
direct substitution
This type of straight line cuts through a curve at two distinct points to show an average rate of change.
secant line
Translate this symbol into a plain English sentence: limx-->3f(x) = 7
What is "The limit of f(x) as x approaches 3 equals 7"
Evaluate limx--5 (2x - 3)
What is 7
(Calculation: 2(5) - 3 = 7)
The average rate of change for the function f(x) = x2 on the closed interval [1, 4]
5, (Calculation: (f(4)-f(1))/(4-1) = (16-1)/(3) = (15)/(3) = 5)
When tracing a graph with your fingers from both the left and right sides, this is what happens if your fingers don't meet at the same height.
the limit does not exist" (DNE)
What you must do with your target x-value immediately after successfully canceling out a troublesome denominator term
substitute the target value back in (or try direct substitution again)
This ideal line grazes a curve at exactly one single point, representing an instantaneous rate of change.
tangent line
For a general two-sided limit to exist, these two separate pathways must approach the exact same value.
the left-hand and right-hand limits
The three specific mathematical conditions that must all be true for a function to be considered perfectly continuous at a point \(x = a\)
The formal function notation setup representing the average rate of change from x = a to x = b
(f(b)-f(a))/(b-a)
Limits do not care about what happens exactly at a specific coordinate point; they only care about this.
Only what happens near (or around) the point
Evaluate limx-->-3 (x2 + 5x + 6)/(x + 3).
What is -1?
(Calculation: Factor numerator to (x+3)(x+2), cancel out (x+3), then substitute: -3 + 2 = -1)