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You are here: Experts > Science > Mathematics > Advanced Math > urgent math
Expert: Sherry Wallin - 11/3/2009
Question
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, of course I can't answer all these questions so as you asked I'll pick a few to help you out:
Let's do the 30t derivative of cos(2x). Let's examine the first few to see if there is a pattern:
first (cos(2x))' = 2(-1*sin(2x)) = -2sin(2x);
second (cos(2x))'' = -2sin(2x)'= -2(sin(2x))' = -2(*2cos(2x))= -4cos(2x) but you already know that the derivative of cos(2x) is -2sin(2x) so
third (cos(2x))''' = -4*-2sin(2x) = 8 sin(2x)
fourth (cos(2x))'''' = 8 sin(2x)' but you already know that the derivative of sin(2x) is 2cos(2x) so
8(2cos(2x)) = 16 cos 2x
and this pattern will continue to repeat itself. When n = number of derivative is a multiple of 4, i.e., n = 4k then cos(2x) will be positive and when n = 4k + 2 cos(2x) will be negative. The number of the derivative is also used to find the coefficient in front of the cos(2x) and it is 2^n. You now know that 30 = 4*7 +2 so it is of the form 4k + 2 so it has a negative value and it's coefficient is -2^30. It is also true that the nth derivative of cos 2x is -2^n*sin 2x when n = 4k+1 or 2^n*sin(2x) = 4k + 3. Start k = 0 by the way.
The next few derivatives based on what i said above are:
5th -2^5 sin(2x) 5 = 4k + 1 should be sin(2x) and should be negative
6th -2^6 cos(2x) 6 = 4k + 2 should be cos(2x) and should be negative
7th 2^7 sin(2x) 7 = 4k + 3 should be sin(2x) and should be positive
8th 2^8 cos(2x) 8 = 4k should be cos(2x) and should be positive
6.If h(x) is given h(x)=sqrt(7+6f(x)), where f(3) = 7 and f '(3) = 5, find h'(3)
Find the derivative of h(x): h'(x) = f'(x)/[2*(7+6*f(x))^(1/2)]. Plug in the values they gave you for f(3) and f'(3): 5/[2*(7+6*7)^(1/2)] = 5/(2*sqrt(49)) = 5/(2*7) = 5/14 = h'(3).
Good Luck,
Math Prof
---------- FOLLOW-UP ----------
QUESTION: Wow really appreciate it, helps alot. So would you know how to do q#1 or 3?
Answer
#3
μ is called mu and is the greek letter mu. In physics we use it for a coefficient (constant of friction). The rate of change is just the first derivative of the function. μW / μsin(x)+cos(x) and in your case they ask it to be in terms of x, so find the first derivative with respect to x:
a.Find the rate of change of F with respect to x.
μW / μsin(x)+cos(x)
Let's first take the derivative of μsin(x)+cos(x):
[μsin(x)+cos(x)]' = μcos(x) -sin(x)
Note this is in the denominator so you will need to use the quotient or modified product rule:
I will use the quotient rule: the bottom times the derivative of the top minus the top times the derivative of the bottom all over the bottom squared:
{[μsin(x)+cos(x)]*0 - μW[μcos(x) -sin(x)]}/[μsin(x)+cos(x)]^2
= - μW[μcos(x) -sin(x)]/[μsin(x)+cos(x)]^2
for the 2nd part set this derivative equal to zero to find where it is zero and use what you know about the sine and cosine function to determine where it is zero:
You really just want to examine where the numerator is zero: either - μW = 0 or μcos(x) -sin(x)= 0
in the first case that would only happen when there wasn't any friction, ie μ = 0. In the 2nd factor when μcos(x) = sin(x). One value is when μ = 1 and x = pi/4 = 45 degrees, i.e., the rate of change is zero when the angle is 45 degrees.
Math Prof
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