Unit 10 Math Test Flashcards
Extrema happens in two places
- Turning Points
- Endpoints
Absolute (global) maximum f(c)
f(x) ≤ f(c) for all x values on the domain
Absolute (global) minimum f(c)
f(x) ≥ f(c) for all x values on the domain
Relative (local) maximum f(c)
f(x) ≤ f(c) for all x values in an open interval containing c
Relative (local) minimum f(c)
f(x) ≥ f(c) for all x values in an open interval containing c
Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both a maximum and minimum value on the interval
Critical Point
A point in the domain which f’ = 0 or f’ does not exist
Finding absolute extrema
- Determine the critical points
a. take the derivate
b. set the derivate to zero and solve
c. determine any places where the derivative is undefined - Test the critical points and end points. The largest value is the absolute max and the smallest is the absolute min
Relationship between increasing/decreasing functions and derivative
f’ > 0 at each point in (a,b) then f increases on (a,b)
and opposite
Local Max and Local Min
A function f has a local maximum at x = c if f’ changes from positive to negative
A function f has a local minimum at x = c if f’ changes from negative to positive
Are endpoints local extrema? [a, b]
The left endpoint “a”
1. a local maximum if f’ is negative for x>a
2. a local minimum if f’ is positive for x>a
The right endpoint “b”
1. a local maximum if f’ is positive for x<b
2. a local minimum if f’ is negative for x<b
Method to finding all extrema of a function
Number line
Concave up is a smile
Concave down is a frown
Inflection points
Points when curves change concavity
f’’ = 0 or does not exist
Definition of Concavity
f(x) is concave up on an interval if all the tangents to the curve are below the graph
f(x) is concave down on an interval if all the tangents to the curve are above the graph
Concavity: f’ is increasing (THINK SIGN AND NUMBER)
Concave up
Concavity: f’ is decreasing (THINK SIGN AND NUMBER)
Concave down
Velocity is the
First derivative
Change of position over time d(p)/d(t)
Acceleration is the
Second derivative
Change of velocity over time d(v)/d(t)
If velocity and acceleration have the same sign
speeding up
If velocity and acceleration have the opposite sign
slowing down
Five things to remember for word problems
- List needed formulas
- Define variables
- Draw pictures
- Write the equations using variables
- Write the full sentence for the answer
Area of a triangle
A = 1/2 bh
Area of a triangle with trig
A = 1/2 ab sin(θ)
Tangent line approximation
f(x) ≈ f(a) + f’(a)(x-a)
Error
E(x) = f(x) - f(a) - f’(a)(x-a)
Mean Value Theorem
If f is a continuous function on the closed interval [a,b] and differentiable on the open interval (a,b) there exist a number c in (a,b) such that
f’(c) = f(b) - f(a)/b-a
Area of a circle
A = πr^2
Circumference of a circle
2πr
