Derivative Applications Review

Using f'(x)

100

Find the critical points of the function y=x³-9x²-21x-11

x = -1 and x = 7

200

For the function y=3x²-2, is x = 0 a local min or local max? Show how you know using the first derivative test. 

local min (derivative switches from negative to positive)

300

Find the all local extrema of the function f(x) = x³ - 3x - 6 using the first derivative test. Be sure to specify which points are local minimums and which are local maximums. 

x= -1 is a local maximum

x=1 is a local minimum

400

The following is the derivative of a function. Use it to find all local minimums and maximums of the function (state which are minimums and which are maximums). 

f'(x)=6(x+1)(x-2)²

x= -1 is a local minimum, there are no local maximums

500

The following is the derivative of a function. Find the interval(s) on which the function is increasing. 

f'(x) = 2(x-3)(x+1)(x-4)

-1 < x < 3 U x > 4

Using f''(x)

100

Using the second derivative test, find any local minimums of the function f(x) = 2x³ - 6x + 2

local min @ x = 1

local max @ x = -1

200

Find any inflection points of the function f(x) = x³ - 9x² -21x - 11

x = 3

300

Which of the following is NOT an inflection point for the function f(x) = 3x⁵ - 5x⁴  and why?

x= 0 or x = 1

x = 0 is not an inflection point (second derivative doesn't change sign)

400

Given the derivative of a function f'(x) = 6(x+1)(x-2) find any points of inflection of the function. 

x = 1/2

500

On what interval(s) is the function f(x) = 0.25x⁴ +2x³ +4.5x²  concave down?

-3 < x < -1

Absolute Extrema

100

Determine the absolute min and max for the following equation on its given interval f(x) = 5x^2 + 3x -5, -2 ≤ x ≤ 0

Min is (-0.3, -5.45) Max is (-2 ,9)

200

Determine the absolute min and max for the following equation on its given interval f(x) = (1/6)*x^2 + x^(1/3), 0 ≤ x ≤ 2

min is at (0,0) max is at (2, 1.93)

300

Determine the absolute min and max for the following equation on its given interval f(x) = x^4 - (2x+1)^2, -2 ≤ x ≤ 0

min is (0, -1) max is (-2, 7)

400

A farmer has a 1000m of fencing and he wishes to enclose an area on his farm with three equal partitions. What dimensions will maximize the area of the enclosure? What area would that result in?

The dimensions are 125m by 250 m The resulting area is 31,250 m^2

500

Find the length of the shortest ladder that will reach over an 8-ft. high fence to a large wall which is 3 ft. behind the fence https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory/MaxMin.html

17.64 ft Solution https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxminsol3directory/MaxMinSol3.html#SOLUTION 18

Optimization

100

A production facility is capable of producing 60,000 widgets in a day and the total daily cost of producing x widgets in a day is given by: C(x) = 250,000 + 0.08x + (200,000,000 / x) How many widgets per day should they produce in order to minimize production costs?

50,000 widgets Solution http://tutorial.math.lamar.edu/Classes/CalcI/BusinessApps.aspx

200

A window is being built and the bottom is a rectangle and the top a semi-circle. If there is 12 meters of framing materials, what must the dimensions of the window be to let in the most light?

r=1.68m b=3.36m h= 1.68m Solution http://tutorial.math.lamar.edu/Classes/CalcI/MoreOptimization.aspx

300

After hiring a consultant to study the revenue and costs of a shoe production facility, the following relationships are reported where x is thousands of pairs of shoes produced R is revenue in thousands of dollars C is cost in thousands of dollars R(x) = 10x C(x) = x^3+6x^2+15x Given that profit is R-C, what production quantity of shoes would result in maximizing profits? How much profit would that be?

3.528 thousands pair of shoes profit is $13,128 Solution https://www.youtube.com/watch?v=dam16G6cZ8k&list=WL&index=13

400

An offshore oil well, P, is located in the ocean 5km from the nearest point on the shore. A pipeline is to be built to take oil from P to a refinery that is 20km along the straight shoreline from A. If it costs $100,000/km to lay pipe underwater and only $75,000/km to lay pipe on land, what route from the well to the refinery will be the cheapest?

The pipeline meets the shore 5.7km from point A. Cost would be $1.83M Textbook Chapter Review Question 21

500

A store sells portable MP3 players for $100 each and, at this price, sells 120 MP3 players every month. The owner of the store wishes to increase his profit, and he estimates that, for every $2 increase in the price of MP3 players, one less MP3 player will be sold each month. If each MP3 player costs the store $70, at what price should the store sell the MP3 players to maximize profit?

Max profit occurs when units are sold at $204 (68 units sold) $206 (67 units sold) Textbook Chapter Review Question 20

Related Rates

100

Each edge of a cube is expanding at a rate of 5cm/s. At what rate is the surface area changing when each edge is 8cm?

480 cm^2/s

200

Air is being pumped into a spherical balloon at a rate of 10cm^3/s. How fast is the radius of the balloon increasing when it has been filling for 30s?

0.046 cm/s

300

On a dark night, Mr. Lam, who is 1.8m tall, walks past a 10m tall street lamp to get to his car. How fast is his shadow increasing when he is 15m away from the street lamp walking at 1.3 m/s?

0.285 m/s

400

A hopper in the shape of an upside down square pyramid contains grain feed for its livestock on a farm. 0.1m^3 worth of feed is consumed each week. How fast is the level (ie. height) decreasing when the level is at 1.5m. The hopper's height is 6m with a square top of 4m^/2.

-0.6m/week

500

Ship A is 32 miles north of ship B and is sailing due south at 16 mph. Ship B is sailing due east at 12 mph. At what rate is the distance between them changing at the end of 1 hour?

-5.6 mph