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Vocabulary
Optimization
First Derivative Test
Second Derivative Test
100
f(x) is concave up if...
f''(x)>0
100
You have found two equations from the given information in a problem: 2xy=40 and C=20x+10y. You are trying to minimize cost C. Find C in terms of one variable (I did y, but x is ok.)
C(y) = (400/y) + 10y
100
Sketch a number line with a relative minimum at x=2 and a relative maximum at x=6
100
If f''(c)>0, then f has a _____ at x=c.
minimum
200
f has a singular point at x=a is...
f'(a) DNE
200
In optimization, you first find the equation for what you're optimizing in terms of one variable. What do you do next and why? (What is the next step useful for?)
Take the derivative. This is useful because you need to find the critical points.
200
In your own words, what is the difference between a relative maximum and an absolute maximum?
A relative max is the largest in its "neighborhood," and there can be many. There can only be one absolute maximum, and it must be the largest value in the entire domain of the function.
200
When does the second derivative test fail? What do you do if that happens?
It fails if the second derivative is 0 at your critical point. You use the first derivative test instead.
300
f has a critical point at x=a if a is a... (hint: 2-part answer)
stationary point or a singular point
300
Use the following information to build the appropriate number of equations to begin the problem: An airline mandates that for luggage, the sum of the length and the width is at most 36 inches, while the sum of the length, height, and twice the width is at most 72 inches. You want to find out what the dimensions are of the bag with the largest volume.
l+w=36; l+h+2w=72; V=lwh
300
[Julia will draw a number line on the board - example 3 from page 4 of lecture notes 5.1 part 1] Identify the critical point(s) and relative extrema. (x values are ok here)
CP x=3/2; No relative maxima; relative minimum at x=3/2.
300
Find the relative extrema of f(x)=x^3 - 3x^2 - 9x using the second derivative test.
f''(3) = 12 -> min; f''(-1)= -12 -> max
400
Draw an example of a stationary point
400
You are trying to maximize volume. You have already gotten volume in terms of 1 variable, and the derivative is V'(l)=24l-l^2. Find CP(s), judge if they're reasonable, and use the second derivative test to verify that the answer yields a maximum.
CPs: l=0,24; 0 not reasonable; V''(24)=72-144<0
400
Given that g'(x) = 8/(2x-3)^3/5, find the critical point(s) and sketch a number line.
CP at x=3/2; it's a relative min
400
For the first 15 months after the introduction of a new video game, the total sales can be modeled by the curve S(t)=20e^(0.4t) units sold, where t is the time in months since the game was introduced. At what rate is the sales changing after 10 months? At what rate is the growth rate changing after 10 months? Use correct units.
S'(10)=436.79 units sold per month; S''(10)=174.71 units sold per month per month
500
What is an inflection point?
A point where f changes concavity, OR a point where f''(x)=0
500
For tax reasons, I need to create a rectangular vegetable patch with an area of exactly 242 sq. ft. The fencing for the east and west sides cost $4 per foot, and the fencing for the north and south sides cost only $2 per foot. What are the dimensions of the vegetable patch with the least expensive fence? Use the second derivative test to confirm your answer yields a minimum.
22 ft by 11 ft
500
Use the first derivative test to find the relative extrema of the function f(x)=3x^4 + 5x^3 on the domain (-infinity,infinity). Draw the number line and write the coordinates of the extrema. [example 6 from lecture notes 5.1 part 1]
min (-5/4, -2.441), no max
500
Use the second derivative test to find relative extrema of f(x)=3x^2 + 4x - 1
minimum (-2/3 , -7/3)