Applications of Derivatives : 10 Derivative Problems, Rate of Change, Pollution and Population Growth

1).Thermal Inversion When there is a thermal inversion layer over a city (as happens often in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 AM. Assume that the pollutant disperses horizontally, forming a circle. If t represents the time (in hour) since the factory began emitting pollutant (t=0 represents 8 AM), assume that the radius of the circle of pollution is r(t)=2t miles. Let A(r) =pi*r^2 represent the area of a circle of radius r .

a. Find and interpret A[r(t)].
b. Find and interpret DtA[r(t)] when t=4.

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2).Arctic Foxes The age-weight relationship of female Artic foxes caught in svalbard, Norway, can be estimated by the function
M(t)=3,120e^-e0.022(t-56) ,

Where t is the age of the fox in days and M(t) is the weight of the fox in grams.

a) Estimate the weight of a female fox that is 200 days old.
b) Use M(t) to estimate the largest size that a female fox can attain.(Hint: Find lim t to infinity M(t).)
c) Estimate the age of a female fox when it has reached 80% of its maximum weight.
d) Estimate the rate of change in weight of an Arctic fox that is 200 days old.(Hint: Recall that Dte^f(t)=f'(t)e^f(t).)
e) Use a graphing calculator to graph M(t) and then describe the growth pattern.
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3. Many biological populations, both plant and animal, experience seasonal growth. For example, an animal population might flourish during the spring and summer and die back in the fall. The population, f(t), at time t, is often modeled by f(t)=f(0)ecsin(t) ,

Where f(0) is the size of the population when t=0. Suppose that f(0)=1,000 and c=2. Find the functional values in parts a-d below:

a. f(0.2)
b. f(1)
c. f' (0)
d. f' (0.2)
e. Graph f(t).
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Rules andApplications of the Derivative
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1. Use the Product Rule to find the derivatives of the following functions:
a. f(X) = (1- X^2)*(1+100X)
b. f(X) = (5X + X^-1)*(3X + X^2)
c. f(X) = (X^.5)*(1-X)
d. f(X) = (X^3 + X^4)*(30

20. The graph below displays growth of a town's population y = P(t) over the next 3 years, where t is time in months.
a. Estimate how fast the population is increasing 5 months, and 20 months from now.
b. Graph y = P'(t).
In graphs of questions 2, 4 and 6, determine which is the f(x) function and which is the derivative?

Y= (1 + 1/x)^1/4
Got the 1st derivative to be Y'= 1/4 (1+1/x)^-3/4 * (-1x)^-2
Is that correct?
Now I need the 2nd derivative, I am completely lost on this. Please work out clearly.

The equation for a wave moving along a straight wire is: (1) y= 0.5 sin (6 x - 4t)
To look at the motion of the crest, let y = ym= 0.5 m, thus obtaining an equation with only two variables, namely x and t.
a. For y= 0.5, solve for x to get (2) x(t) then take a (partial) derivative of x(t) to get the rate of change of

4. A Norman window consists of a rectangle with a semi-circle mounted on top (see the figure). What are the dimensions of the Norman window with the largest area and a fixed perimeter of P meters?
5. A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each pers

S represents weekly sales of a product. What can be said of S' and S'' for each of the following?
(a) the rate of change of sales is increasing
(b) sales are increasing at a slower rate
(c) the rate of change of sales is constant
(d) sales are steady
(e) sales are declining, but at a slower rate
(f) sal