First Derivative
How does one determine where critical points are, and what do they tell us about the original function?
By finding the zeroes of the first derivative, and they tell you where (possible) minima and maxima of the original function are.
You are given a critical point of a function and told that f''(x) > 0. What do you know about this critical point?
(This is the second derivative test.)
It is an absolute minimum.
What is the derivative of x with respect to x?
1.
List the four steps to solving an optimization problem:
1. Write the equation for the quantity that you wish to maximize or minimize.
2. Use the constraints given by the problem to relate your variables.
3. Rewrite the equation from (1) in terms of one variable using the results of (2).
4. Use candidates test to find extrema.
Which is the largest country by land area?
Russia
A function's derivative is negative, and then zero, and then positive. What does this information tell you? Your answer should have three pieces!
The function was decreasing, reached a local minimum, and then started increasing.
The second derivative of a function goes from positive, to zero, to negative. What do you know about this interval?
The zero is a point of inflection because the graph when from concave up to concave down.
What is the derivative of y with respect to x?
y' or dy/dx
We wish to build a box whose base is a square of dimensions x with height h. Its volume must be 36 cubic feet. The material used to build the bottom of the box costs $8/ft2 and the material for the sides only $6/ft2. Write an equation C(x) which gives the cost of the box as a function of one two variables: x and h.
This is step 1 in the optimization process.
C(x) = 8(2x2)+6(4xh)
Which is the most populous country?
India
Find the critical point(s) of the function
f(x) = 2x2 - 4x + 3
(1, 0)
The slopes of the tangent lines of a function are getting smaller. What does this tell you about the first and second derivatives of said function?
It is known that x2 + y2 = 1. Find dy/dx. (This is the equation of the unit circle.)
y' = -x/y
We wish to build a box whose base is a square of dimensions x with height h. Its volume must be 36 cubic feet. The material used to build the bottom of the box costs $8/ft2 and the material for the sides only $6/ft2. Use the constraint to identify the relationship between x and h.
This is step 2 in the optimization process.
x2h=36, or h = 36/x2
Which is the single largest religion by number of believers?
Sunni Islam
Find the critical point(s) of the function and classify as a local minimum or a local maximum.
f(x) = -1x2 + 6x - 2
(3, 0), local maximum
See insert 1.
II and III.
See insert 3.
C.
We wish to build a box whose base is a square of dimensions x with height h. Its volume must be 36 cubic feet. The material used to build the bottom of the box costs $8/ft2 and the material for the sides only $6/ft2. Write an equation C(x) which gives the cost of the box as a function of one variable: x.
This is step 3 in the optimization process.
C(x) = 8(2x2)+6(4x)(36/x2) = 16x2 + (864/x)
Which is the smallest country?
Vatican City
Find the critical point(s) of the function and classify as a local minimum or a local maximum.
f(x) = (x-2)3+4
(2, 0), neither a local minimum nor a local maximum.
See insert 2.
II only.
See insert 4.
C, 2.
We wish to build a box whose base is a square of dimensions x with height h. Its volume must be 36 cubic feet. The material used to build the bottom of the box costs $8/ft2 and the material for the sides only $6/ft2. Determine the dimensions (x and h) which will minimize the cost of the box.
Within five years, how old was the oldest person who has ever lived?
Jeanne Louise Calment, 2/21/1875 - 04/08/1997 (122 years)