Critical Points & Extrema
What is a critical point?
A point where f′(x)=0
or f'(x) is undefined
if f(x) > 0 the function is
Increasing
Which derivative determines concavity?
The Second Derivative
A rectangle has length 8 and with 5, find its area
40cm
What does the Mean Value Theorem guarantee?
At least one point where the instantaneous rate equals the average rate.
What theorem guarantees an absolute maximum and minimum on a closed interval?
Answer: The Extreme Value Theorem or EVT
if f(x) < 0 the function is ?
Decreasing
if F"(x) > 0, the graph is?
Concave Up
A square has side length x. Write an equation for its area
A = x^2
What conditions are required for the Mean Value Theorem?
Continuous on a closed interval and differentiable on an open interval.
What must you check to find absolute extrema on a closed interval?
Answer: Critical points and endpoints.
What derivative test determines intervals of increase and decrease?
The first Derivative test
If f"(x) < 0 the Graph is?
Concave Down
Rectangle has a perimeter 20. Write the are function in terms of x
y=10-x
A=x(10-x)
What do related rates problems involve?
Two or more changing quantities.
If f'(x) changes from positive to negative, what occurs?
Its a Local Maximum
if f(x) = 0, does it guarantee a max or a min?
No
What is an inflection point?
A point where concavity changes.
Find the critical point of A= x (10-x)
A'= 10-2x
10-2x=0
x=5
When solving related rates problems, what should be done before differentiating?
Write an equation relating the variables.
if f(x) changes from negative to positive, what occurs?
A Local Minimum
A function changes from increasing to decreasing at x=3. What occurs at x=3?
A Local Maximum
What must happen to f"(x) for an inflection point to exist?
It must change sign different from its original one.
Mrs. Jackson's classroom is 200ft and wants to enclose a rectangular area. What dimensions maximize the area?
50 ft by 50 ft
Why is implicit differentiation often used in related rates?
Because multiple variables change with respect to time.