Rolle's Theorem/Mean Value Theorem/Newton's Method
First and Second Derivative Tests
Curve Sketching
Optimization
100
These are the two values of the first derivative at a critical number.
What is zero or does not exist.
100
Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then this is guarenteed to exist for at least one number c in (a, b).
What is f'(c) = 0.
100
This is what both the First and Second Derivative Tests are locating.
What is relative extrema.
100
These are the characteristics of a function on an interval when f'(x) = + and f"(x) = -.
What is increasing and concave down.
100
Along with identifying all given quantities and quantities to be determined, this is part of the first step in solving optimization problems.
What is draw a picture.
200
This is what occurs on a function at a point of inflection.
What is the change from concave up to concave down or vice versa.
200
This is the formula used in Newton's Method to approximate the zeros of a function.
What is X = Xn - f(Xn)/f'(Xn).
200
These are the values of a function necessary when finding extrema using both the First and Second Derivative Tests.
What is critical numbers.
200
These are the values of a function that determine the test intervals in the curve sketching charts.
What is critical numbers and points of inflection.
200
The primary equation in an optimization problem is based off of this.
What is quanitity being minimized or maximized.
300
These are the two possible names for the point (x, f(x)) when x = c is a critical number.
What is relative minimum or relative maximum.
300
If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that this is true.
What is f'(c) = [f(b) - f(a)]/[b - a].
300
These are the three possible results of the Second Derivative test and their corresponding types of extrema.
What is a positive result meaning relative minimum, a negative result meaning relative maximum, and zero meaning test fails and must use First Derivative Test to find extrema.
300
This is the relationship between infinite limits at infinite and horizontal asymptotes. (Describe how to know what the horizontal asymptote is by checking the limit at infinite.)
What is the horizontal asymptote is y = L when the limit of the function at infinite is L.
300
This is the purpose of creating a secondary equation in optimization problems.
What is to rewrite primary equation in terms of a single variable.
400
These are the critical numbers of f(x) = x^2(x^2 - 4).
What is 0, 2, -2.
400
Let f(x) = x^4 - 2x^2. These are the values of c in the interval (-2, 2) guarenteed by Rolle's Theorem.
What is 0, 1, and -1.
400
These are the relative extrema of f(x) = (x^2 - 4)^2/3.
What is a relative minimum at (-2, 0) and (2, 0) and a relative maximum at (0, 16^1/3).
400
This characteristic of f(x) is true when f'(x) is increasing and when f"(x) > 0.
What is concave up.
400
This is the point on f(x) = x^2 that is closest to (2, 1/2).
What is (1, 1).
500
These are the points of inflection of f(x) = x^4 - 4x^3
What is 0 and 2.
500
Let f(x) = x^2/3 on [0, 1]. This is the value of c that is guarenteed by the Mean Value Theorem.
What is 8/27.
500
These are the relative extrema for f(x) = -3x^5 + 5x^3.
What is a relative minimum at (-1, -2) and a relative maximum at (1, 2).
500
This is the graph of f(x) = x^2/(x^2 + 3) with the given characteristics.
What is (draw graph...must see individual work).
500
A farmer plans to fence a rectrangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd and does not need fencing along the river. These are the dimensions required to use the least amount of fencing.