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About Azeem Hussain
Expertise
I can answer virtually any kind of question dealing with of Math 536 and below, my forte being in parabolic functions and analytical geometry. I'm currently learning Linear Algebra, and cannot answer questions that deal with subject matter more advanced than that.

Experience
I am neither a professor nor a teacher of this subject. I am merely a student who is gifted at mathematics and enjoys being of service to his community. I frequently tutor people in math and the results are usually great.

Publications
Reflections, Riverside School Board (2005, 2006)

Education/Credentials
Diploma of Secondary Studies from Chambly Academy High School, and IBO-MYP certificate as well. My lowest mark on a high school math final was 97%, peaking at 99% in 2006 and 2007 (second-highest Math 436 mark in the province). Being a Quebecer, I am fluent in English and French and can respond to questions easily in both languages.

Awards and Honors
Pascal Math Competition, School Champion(2007)

You are here: Experts > Science > Math for Kids > Geometry > Calculus I - Various

Geometry - Calculus I - Various

Expert: Azeem Hussain - 11/3/2009

Question
Hey there I have a test tomorrow nigh for Calc and I just can't seem to get the answers for these questions, i've been trying hard, but always come out wrong. Please reply by tomorrow any time if you can it will be GREATLY appreciated. PLEASE IF YOU CAN"T GET ALL SHOW ME HOW TO DO ANY THAT YOU CAN DO. IF YOU THINK ITS TOO MUCH PLEASE DO WHAT YOU CAN.

1.A baseball diamond is a square with side 90 ft. A batte1r hits the ball and runs toward first base with a speed of 22 ft/s.

a)At what rate is his distance from second base decreasing when he is halfway to first base?

b)At what rate is his distance from third base increasing at the same moment?

2.Whats the 30th derivative of cos(2x)?

3. An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle x with the plane, then the magnitude of the force is given by the following equation, where μ is a constant called the coefficient of friction.

μW / μsin(x)+cos(x)

a.Find the rate of change of F with respect to x.

b.When is this rate of change equal to 0?

c.If W = 60 lbs and μ = 0.8, draw the graph of F as a function of x and use it to locate the value of x for which dF / dx = 0. (Round the answer to two decimal places.)

4.Each side of a square is increasing at a rate of 8 cm/s. At what rate is the area of the square increasing when the area of the square is 49 cm^2? (in cm^2/s)

5.Scientist can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 74% as much 14C radioactivity as does plant material on Earth today. Estimate the age of the parchment in years.

6.If h(x) is given below, where f(3) = 7 and f '(3) = 5, find h'(3)

h(x)=sqrt(7+6f(x))

7.Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 300 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 40 kPa/min. At what rate is the volume decreasing at this instant? cm^3/min

THANKS SO MUCH! George

Answer
Hi George,

2.
To find the 30th derivative, begin by finding the first few.
f(x)=cos(2x)
f'(x)=-2sin(2x)
f''(x)=-4cos(2x)
f'''(x)=8sin(2x)
f''''(x)=16cos(2x)

There is a pattern. Odd derivatives have sin(2x), even derivatives have cos(2x). They are a multiplied by a factor of 2^n, n being the number of the derivative. There is a pattern -sin, -cos, sin, cos pattern which is happening. The 28th derivative would be a cos, the 29th a -sin, and the 30th a -cos. The thirtieth derivative is ultimately:
-(2^30)cos(2x)

4.
Let f(x) be the length of the side.
Let f'(x) be the rate at which the length of the side changes.
Let A(x) be the area.
Let A'x) be the rate of change of the area.

It's a square, so A(x)=[f(x)]². When A(x)=49, f(x)=7.
Differentiate both sides.
A'(x)=2f(x)f'(x)

Now, let x be the time time when the area is 49. The side, f(x) is equal to 7, and f'(x)=8, as given in the problem. You're being asked to find A'(x). Substitute.
A'(x)=2(7)(8)
A'(x)=112

5.
This doesn't require calculus (unless if you mean by linear approximation, but that's not what's written). The general form for a half-life equation is:
R=I(0.5)^(t/h)
where R is the remaining, I is the initial, t is time (variable), and h is half-life (in the same units as time). Plug in your values and solve for t.

6.
Find dh/dx.
h(x)=√(7+6f(x))
h'(x)=6f'(x)/2√(7+6f(x))

Take x=3. Substitute what was given the problem.
h'(3)=6f'(3)/2(√7+6f(3))
h'(3)=3(5)/√(7+42)
h'(3)=15/7

7.
General strategy for related rates is identifying your two quantities/dimensions, and then their derivatives as rates. Then find an equation linking the two variables. Substitute what is given in the problem in order to get a numerical value. Now, take your equation (in x) and differentiate both sides. Solve for V', or whatever you're being asked. This shouldn't be too hard to follow:
P(x)V(x)=C
(300)(150)=C
C=45 000

P(x)V(x)=C
V(x)=C[P(x)]^-1
V'(x)=C[-[P(x)]^-2]P'(x)
V'(x)=(45 000)[-150^-2][40]
V'(x)=-80

Thanks for asking,
Azeem

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