To our athletic compadre

Hey Max, whether you check this during your trip or when you get back I hope you guys did well and came back with your heads held high. I am going to try my hardest to post as nice as you do it but I am not sure if i am quite that blog savvy yet. Thank goodness there was no graphs! I really don't know how to write powers on this so I just put carrot ... sorry about that. Homework: Read- Pgs.128-129, Do questions #1, 3, 9, 11, 23, 25.

QOD

A bacteride was added to a growing population of bacteria. The size of the population in time t, in hours was given by: P(t) = (10^6)+(10^4)x-(10^3)x^2. This is 10 to the power of 6 plus 10 to the power of 4, x plus 10 to the power of 3, x squared. Find:

a)the rate of growth at 3 hours, 5 hours and 8 hours.

b)is the reate of growth increasing or decreasing at t = 7 hours?

a) In order to find the rate of growth we took the derivative of P(t) which is (10^4)-2(10^3)x which come out to: P'(t) = 10000-2000x We then plugged in each number... P'(3) = 10000-2000(3) = 4000 bacteria/hour , P'(5) = 10000-2000(5) = 0 bacteria/hour , P'(8) = 10000-2000(8) = -6000 bacteria/hour

b) In order to find out if the rate of growth was increasing or decreasing we had to (this might be a little confusing but you will understand it hopefully) find the rate of growth or the rate of growth. This would show if it is positive or negative, thus if it is increasing or decreasing. This is the same as d^2y / dx^2. This is finding the derivative of the derivative.

P''(t) = -2000 (beacause the derivative of P'(t) = 10000-2000x = -2000) P''(7) = -2000 (because there is no x so it is constant) bacteria/hour/hour Therefore it is decreasing (because it is negative) by 2000.

Part A tells us: If our starting point if 0, from 0 to 5 hours the population of bacteria is increasing. At t>5 the population of bacteria is decreasing. The maximum population is at t = 5 hours.

Part B tells us: As the population of bacteria decreases the rate of growth increases (in an absolute value sense).
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What is the meaning of f '(a)?

• the slope of a tangent to f at x=a
• the limit of the slopes of the secants to f at x=a
• instantaneous rate of change of f at x=a
  

What is the meaning of f ''(a)?

• the derivative of f '(a)
• rate of change of the slope of f at x=a
• instantaneous rate of change of the instantaneous rate of change of f at x=a

-the value of f ''(a) tells you whether f '(a) is increasing, decreasing or constant

Higher Order Derivatives
1st derivative f '(x) or dy/dx
2nd derivative f ''(x) or (d^2)y/dx^2
3rd derivative f''''(x) or (d^3)y/dx^3
nth derivative f^n(x) or (d^n)y/dx^n

Pg. 133 #29
Cost of Production: a function that gives the cost associated with varying levels of production

Marginal Cost of Production:
"The change in cost when production in increased by 1 unit." - Definition from Mr. A

*Marginal cost of production always varies depending on where you are starting.
For example: If you are starting at zero prodution and want to change your unit to 1000 production, the change in cost is going to be very large because you need to pay for startup costs such as machinery. As opposed to if you are starting at 1000 to 2000 production the change in cost is not going to be as much because you already have payed the startup costs.

The equation for the average rate of change is your classic:

C2-C1 /X2-X1

Where c=cost, x=# of items produced. The numerator is the 2 costs compared (C1 being the starting production cost and C2 being your new production cost). The denomonator is the jump in production (X1 being your starting amount of items being produced and X2 being your new amount of items you wish to produce)

This is an average rate but in Calculus, we want instantaneous. How do you find this...... THE LIMIT!

The equation for the instantaneous rate of change is:

wow, that was super hard to write that and post it. haha. anyways...

This is the marginal cost when x=x1.

So what is marginal cost?

It is setting the level of production to maximize prodution.

Here is an example and some questions to try out:

Total cost in dollars of producing x designer eyeglass frames is given by:
c(x)=5000+10x=0.05x^2

*The cost of the function should be continually increasing.
*You should also see the fixed cost vs. variable cost. A fixed cost may be the cost of a machine to shape plastic, a variable cost may be say cost of screws for the frames because this depends on the amount of frames you are producing.

a)find the average cost of producing 100 frames
b)find the marginal cost at a production level of 100 frames
c)find the actual cost of increasing production from 100-101 frames
-actual cost is found by taking the cost of 100 and 101 and then you will know the cost of this increase.


Good Luck!