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First Derivative test
If the first derivative changes from negative to positive at c then f has a relative minimum at the point (c,f(c)) If it goes from positive to negative it is a relative maximum
concavity
a graph is concave up is the tangent lines are below the graph; a graph is concave down if the tangent lines are above the graph
Limits at infinity
The dependent variable approaches a finite number as the independent variable becomes arbitrarily large
Second derivative test
If the second derivative is greater than zero then the function has a relative minimum
Asymptotes
y=±b/a(x-h)+k
Non differentiable points
Result of change in concavity and Increasing or decreasing. Example: Increasing Concave down- Decreasing Concave Up
Decreasing-Increasing
Relative Minimum
Increasing-Decreasing
Relative Maximum
point of inflection
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
Process for sketching
1.Solve for intercepts
Rolle's Theorem
If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists a point c∈(a,b), where f '(c) = 0.
Y'=0 or is undefined
Y has a critical point
Y is increasing
Y' > 0
Y is decreasing
Y' < 0
Y has a local minimum
Y' changes from - to +
Y has a local maximum
Y' changes from + to -
Y is concave up
Y' is increasing or Y'' > 0
Y is concave down
Y' is decreasing or Y'' < 0
Y has a point of inflection
Y' switches from increasing to decreasing or decreasing to increasing. Y''switches from + to - or - to +
Y has absolute max or min
Check all critical points and endpoints
Asymptote
An imaginary line on a graph that acts as a boundary line.
Holes
when same factor occurs in numerator and denominator