(1) Public health records indicate that t weeks after the outbreak of a certain

Question

(1) Public health records indicate that t weeks after
the outbreak of a certain form of influenza, approximately
f (t) =4/(1+7e^(-0.7t)) thousand people had caught the disease.

a) How many people had the disease initially?
b) How many had caught the disease by the end of 3 weeks?
c) If the trend continues, how many people in all will contract
the disease?

(2)f (x) = ln(9x^2+3)

(A) Find all critical values of f .
(B) Find the x-coordinates of all local maxima of f.
(C) Find the x-coordinates of all local minima of f .
(D) Use interval notation to indicate where f (x) is increasing.
(E) Use interval notation to indicate where f (x) is decreasing.
(F) Use interval notation to indicate where f (x) is concave
up.
(G) Use interval notation to indicate where f (x) is concave
down.
(H) Find all inflection points of f .
(I) Find all horizontal asymptotes of f .
(J) Find all vertical asymptotes of f .

Explanation / Answer

1. a) Plug in t=0. since it says INITIALLY. e^0 is still 1 so 4/(1+7)=1/2=500 people had it initially. b) Plug in t=3. You have to use calculator on this one. 1+7e^-2.1=1.86 4/1.86=2.15. About 2150 ppl caught it. c) I think you need to read the book carefully.....Everybook tells u that in a regression equation, the numerator is the maximum population. Or the max value of the graph. In this case its 4. So 4000 ppl caught it in all. 2. a) find first derrivative which is 1/(9x^2+3)times 18x=6x/(3x^2+1)^2 find x when first derrivative is zero, which is 0. so critical point is x=0. b,c) find second derrivative which is (-18x^2+6)/(9x^4+6x^2+1). if you don't know how to find derrivative....then don't even do these problems, go back and review the equations. Plug in two points, one bigger and one smaller than the critical point x=0 to the second derrivative, cuz the theorem says that at critical point, when second derrivative is negative, that's a local max, when second derrivative is positive, that's local min. You only have 1 critical point, so you plug it in, and second derrivative is possitive, so you only have a local min at x=0 and NO local MAX. d, e) look at first derrivative, when first derrivative is positive, its increasing, if its negative its decreasing. so x>0 is increasing, x

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