Expert: Paul Klarreich - 8/3/2009

Question
I am in Calculus 131 in MIddlesex County College. 4 Calculus Questions that I do not understand. Can you please show work? You do not need to answer all the problems even if you know just one your help is appreciated :D

1) A storage tank in the shape of a right circular cone has a height of 6 meters and a base of radius 2 meters. The cone is inverted and fluid is pumped in at a rate of .002m^3-minute (2 liters/min). At what rate is the level of the fluid rising when the tank is filled to a height of 3meters? Use V= (pie X r squared X h)/3 and R = (H)/3

Work I Have Tried: I have tried to find the area and divide.

2) Verify the mean-value theorem for
a) f(x) = x^3 - 3x^2 - 13x + 15 on (-3,1)
(b) f(x) = cos 2x - 2 cos^2 x on ((-pie/2),(pie/4))


3) Locate and classify all critical points, points of inflection, intervals where the Function increases/decreases and where the Function is concave up/down for
a) f(x) = x^4 - 4x^2 + 4x^3 + 1
b) f(x) = (1+ squareroot(x) + cuberoot(x))^9
c) f(x) = x/(ln x)^2
d) f(x) = e^x + e^-2x
e) f(x) = e^(x^3 - x)

4) Using differentials, approximate the numerical value of
a) cuberoot(130)
b) ln (3), you know ln (e) = 1
c) log (8), you know log (10) = 1

Answer
Questioner: Anil
Country: United States
Category: Calculus
Private: No
Subject: 4 Hard Calc Questions!
Question: I am in Calculus 131 in MIddlesex County College. 4 Calculus Questions that I do not understand. Can you please show work? You do not need to answer all the problems even if you know just one your help is appreciated

1) A storage tank in the shape of a right circular cone has a height of 6 meters and a base of radius 2 meters. The cone is inverted and fluid is pumped in at a rate of .002m^3-minute (2 liters/min). At what rate is the level of the fluid rising when the tank is filled to a height of 3meters? Use V= (pie X r squared X h)/3 and R = (H)/3

Work I Have Tried: I have tried to find the area and divide.

2) Verify the mean-value theorem for
a) f(x) = x^3 - 3x^2 - 13x + 15 on (-3,1)
(b) f(x) = cos 2x - 2 cos^2 x on ((-pie/2),(pie/4))


3) Locate and classify all critical points, points of inflection, intervals where the Function increases/decreases and where the Function is concave up/down for
a) f(x) = x^4 - 4x^2 + 4x^3 + 1
b) f(x) = (1+ squareroot(x) + cuberoot(x))^9
c) f(x) = x/(ln x)^2
d) f(x) = e^x + e^-2x
e) f(x) = e^(x^3 - x)

4) Using differentials, approximate the numerical value of
a) cuberoot(130)
b) ln (3), you know ln (e) = 1
c) log (8), you know log (10) = 1
...................................................
This is a lot more than four.  I'll limit myself to making some suggestions -- I cannot teach you the whole course -- that is why you have a professor.

1) A storage tank in the shape of a right circular cone has a height of 6 meters and a base of radius 2 meters. The cone is inverted and fluid is pumped in at a rate of .002m^3-minute (2 liters/min). At what rate is the level of the fluid rising when the tank is filled to a height of 3meters? Use V= (pie X r squared X h)/3 and R = (H)/3

-- a standard related rates question:

Variables:  
h,r = height and radius of cone of remaining water
V   = volume ....................................

Relations:

h = 3r, and

V = (1/3) pi r^2 h

Subst:  

V = (1/3) pi r^2 (3r)

Diff, substitute values, etc.

....................................

2) Verify the mean-value theorem for
a) f(x) = x^3 - 3x^2 - 13x + 15 on (-3,1)

Find, f'(x), then find your c such that:
       f(1) - f(-3)
f'(c) = -------------
         12 - (-3)
and make sure  c is in (-3,1)

3) Locate and classify all critical points, points of inflection, intervals where the Function increases/decreases and where the Function is concave up/down for
a) f(x) = x^4 - 4x^2 + 4x^3 + 1
b) f(x) = (1+ squareroot(x) + cuberoot(x))^9
c) f(x) = x/(ln x)^2
d) f(x) = e^x + e^-2x
e) f(x) = e^(x^3 - x)

Each of these is  LOT OF WORK.  Once you realize that, you will be OK.  If you think you can do one in 30 seconds, you are in trouble.

Make sure you know what the vocabulary means.

4) Using differentials, approximate the numerical value of
a) cuberoot(130)

The LINEAR APPROXIMATION is:

y = f(x) = f(x0) + f'(x0)(x - x0),

where:

x is your 130
x0 is a value of x near 130 whose cube root is easy to find, and in this case I would pick 125.

You are on your way.