Implicit Differentiation
Related Rates
Higher Order Derivatives
Optimization 1
Optimization 2
100
Determine dy/dx of y^0.5 = x
2 * y^(0.5)
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
100
Write the notation of the second derivative in the following format 1. f(x) 2. y 3. dy/dx
1. f''(x) 2. y'' 3 d2y/dx2
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)
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
Determine dy/dx of y^3 - x^3 = 3
x^2 / y^2
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
200
Find the second derivative of f(x) = 6x^3 + 2x^(1/2)
f''(x) = 36x + -0.5 (x ^ -1.5)
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)
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
Determine dy/dx of 3*x*y^2 + y^3 = 8
- y / (2x + y)
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
300
Given the position equation below, write the velocity and acceleration equations. s(t) = 2t / (t+1)
v(t) = 2 / (t+1)^2 a(t) = -4 / (t+1)^3
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)
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
Given (x + y)^3 = 12x - 4 Determine the equation of the tangent at (1,1)
y = 0
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
400
Draw the s-t, v-t and a-t graphs given the following chain of events 1) I start at s=0 and walk back backwards (in the negative direction) at a constant velocity for 5 seconds 2) I stay at my position and do jumping jacks for 5 seconds 3) Then I turn around and start running forward at an increasing velocity for 10 seconds 4) I continue to walk forward at a constant velocity for 5 seconds 5) I stand still and take a break for 10 seconds
See Teacher Diagram
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
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
Given x + y^2 = 1 Determine the equation of the normal that is perpendicular to the line x + 2y = 0
y = -0.5x + 1
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
500
I'm standing on a balcony of my apartment and reach my hand out to throw a ball up according the equation below h(t) = -4.9t^2 +2t My balcony with the reference point of h=0 If my balcony is 30m above the ground, at what velocity will it hit the ground at?
-24.33 m/s
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
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
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