Karl's Calculus Forum: January-June 2006

Archived Forum Discussion for January-June 2006

F(x) = x² + 2x
       ------------------
       √3x+1

Without using the Chain Rule or the Difference Quotient

I hope you can help:)

Thanks

Angie







 MN USA - Wednesday, May 24, 2006 at 16:26:13 (EDT)

Sudheendra, Are R, T, a & b constants or do they change with v? What does this equation refer to, as if R, T, a & b change with v there might be some arrangement that is a constant? Peter
Peter

Ireland - Wednesday, May 24, 2006 at 08:20:34 (EDT)

hi i had problem in doing some integration n differentiation for my project. pls help me in this. im typeing the integRal required. I = integration((1/(v^2+2*b*v-b^2))-(R*T*v/((v-b)^2))+(a*v*(2*v+2*b)/((v^2+2*b*v-b^2)^2))) with respect to dv. thanq
Sudheendra

Mangalore, Karnataka india - Wednesday, May 24, 2006 at 06:45:17 (EDT)

Angie, Does your textbook give a definition for a cusp? Have you tried to sketch the graph of the function? You should try a sketch and then try to draw a few tangents to see if there are any places where the tangent is vertical and also to see if there are any places where you could draw two or more lines that appear to be tangents at the same point. Use those x values and plug them into the derivative of f(x) and see what happens. Peter
Peter

Ireland - Tuesday, May 23, 2006 at 14:27:55 (EDT)

Hello, I'm having difficulty determining if a function has a vertical tangent or cusp. Can someone please explain how to find theses? The function I'm working with is f(x) = x^(3/4)÷(x+2). I appreciate any help, as my textbook doesn't explain how to find these. Thanks! Angie
Angie

Canada - Tuesday, May 23, 2006 at 12:13:09 (EDT)

Brad, What do you get for

d/dy(e^x²)

Peter
Peter

Ireland - Tuesday, May 23, 2006 at 06:10:24 (EDT)

i need help passing my calc course maybe if u could give me some reference material it would help
wendell M

USA - Friday, May 12, 2006 at 17:59:00 (EDT)

How do you integrate e^2x/e^x-3
Fiffy

USA - Friday, May 12, 2006 at 09:27:29 (EDT)

Hi; I'm having trouble with this problem: Find the derivative of

y = x²e^x².

Mainly unsure about the rules concerning the derivation of e^x in general... I know as far as this:

x² × (d/dy(e^x²) + e^x² × 2x

However, I'm rather uncertain as to where to go afterwards because I'm not arriving at the given answer, which is

2xe^x²(x² + 1)

Thanks in advance, Brad
Brad H.

Sterling Heights, MI USA - Sunday, May 07, 2006 at 12:55:29 (EDT)

Find the volume of the solid generated by revolving the region bounded by:

x²/a² + y²/b² = 1

- about the x-axis
- about the y-axis

x>=0, y>=0 (I think, if I remember the problem correctly!)

Has to do with integration to solve it (obviously). Thanks - trying to help my kid with homework, altho I was a math major, it was too many years ago!
Joan

USA - Thursday, May 04, 2006 at 11:50:50 (EDT)

I need help with a word problem for my Calculus for Business & Economics class. In planning a small restaurant, it is estimated that a profit of $5 per seat will be made for the first 80 seats. On the other hand, the profit on each seat will decrease by 5 cents for each additional seat above 80 Find the number of seats that will produce the maximum profit and the miminum profit.
Trina Vidal

Frisco, tx USA - Wednesday, May 03, 2006 at 23:43:57 (EDT)

Reply to Antony: See this page for general treatment of acceleration, and scroll down to see your problem (constant acceleration) in particular.
Karl <Click to Send Email to Karl>
USA - Tuesday, April 25, 2006 at 14:55:27 (EDT)

I have a problem with a problem. Maybe someone can help me? Show that for motion in line with constant acceleration a,initial velocity, and initial displacement, the displacement after time t is
antony

murray, ky USA - Monday, April 24, 2006 at 23:15:03 (EDT)

Reply to Elizabeth: For details of the cone-in-the-sphere problem click here.
Karl <Click to Send Email to Karl>
USA - Sunday, April 16, 2006 at 14:34:28 (EDT)

Reply to Yvonne: So you have two pieces, one of length, x, and the other of length L-x. The radius of the circular piece is x/2π. The square has length-of-side, (L-x)/4. So the total area is

(L-x)² A = + 16

x² 4

So taking the derivative

dA 2(L-x) = + dx 16

2x = 0 4

Note that I have set this to zero in order to find the optimum. Solve for x.
Karl <Click to Send Email to Karl>
USA - Sunday, April 16, 2006 at 14:31:03 (EDT)

i'm using calculus to find the maximum volume of a cone inside a sphere of radius 5cm. ive got 2 write the (h) height and the (r) radius from the volume of a cone equation in terms of x. and hence express the (v) volume in terms of x. where do i start? xx
elizabeth

west y, UK - Thursday, April 13, 2006 at 17:02:36 (EDT)

This should be straight forward but I cna seem to get a handle on it. Can anyone out there help? A piece of string at lenght L cm is cut into 2 pieces. One piece of length X is made into a circle and the other into a square. A) Find the value of X that makes the sum of the areas a maximum. THIS IS INTUITATIVE - DON'T CUT THE STRING - LET THE ENTIRE LENGTH FORM A CIRCLE. b) Find the value of x that minimizes sum of the the areas. I CAN'T SEEM TO GET STARTED ON SETTING UP THE RELATIONSHIP BETWEEN THE EQUATIONS. CAN YOU HELP?
Yvonne

tucson, az USA - Tuesday, April 11, 2006 at 19:44:50 (EDT)

Reply to Andre: I'm assuming you have already eliminated h by substituting the Pythagorean expression that gives h in terms of r. So now you have the volume equation in terms of r and the constant, R.

If you have

r = R -

θR 2π

then you can substitute the right-hand expression into the volume equation everywhere you see r. You should end up with a volume equation in which the only independent variable is θ. The symbol, R, will still be there, but it, remember, is a constant. Now find dV/dθ by taking the derivative of the volume equation with respect to θ. Whatever experssion you get for dV/dθ, set it equal to zero. Now you have an equation in which the only unknown is θ. Solve for θ and you're done.
Karl <Click to Send Email to Karl>
USA - Saturday, April 08, 2006 at 11:16:12 (EDT)

Hey Karl, When you stated "Use the relationships already shown to first substitute for h using r, then to substitute for r using θ" did you mean to use r, when

r=R-((θ R)/2π)

in the volume formula where "h"? I got all of what you said except for the substitute part. You kinda lost me there with your wording, but I really appreciate the help so far!
Andre
Houston, TX USA - Saturday, April 08, 2006 at 04:44:38 (EDT)

Reply to Andre: The distance from the apex to the hem of the cone will be R, where R is the radius of circle of filter paper that you start with. The radius, r, and the height, h of the cone will be related by:

   R²  =  r² + h²

or equivalently

   h = √R² - r²

To see why this is the case, picture the cone in crossection. You will see that the relationship given above is simply the Pythagorean formula.

If the wedge you cut out of the paper is angle θ radians, then the length of circumference removed is θR. That means that what's left is 2πR - θR. This will be the circumference of the base of the cone (picture in your mind what happens when you form the cone from the paper). Dividing that circumference by 2π gives the radius of the base of the cone, r.

r = R -

θR 2π

The volume of a cone is given by

V =

1 3

π r²h

Use the relationships already shown to first substitute for h using r, then to substitute for r using θ. Now take the derivative, dV/dθ, set it to zero, and solve for θ (remembering that the radius, R of the original filter paper is a constant).
Karl <Click to Send Email to Karl>
USA - Friday, April 07, 2006 at 13:25:11 (EDT)

Reply to Julius: 1) The problem states that

  dV
      =  0.1 ft³/min
  dt

The volume of the sphere is given by V = (4/3)πr³. Taking the derivative of that with respect to t gives

dV dt

= 4 π r²

dr dt

You can find r by using the volume equation replacing V with 1/2 ft³ and solving for r. You know dV/dt from the first equation. Solve for dr/dt.

2) The volume of a cylinder is V = π r²h. The radius, is given as 20 feet. Again dV/dt is given. Taking the derivative of the volume equation, assuming fixed radius:

dV dT

= pi r²

dh dt

Solve for dh/dt.

3) The volume of the cylinder is, again, V = π r²h. Note that V is given, so you can use that to find h in terms of r. The cost function is (two circles of radius, r, at $10 per square meter plus a rectangle that is 2πr by h at $8 mer square meter).

  C  =  $10 2π r²  +  $8 2π rh

Substitute the expression for h in terms of r into the above. Then take the derivative, dC/dr, set it to zero, and solve for r. The back-substitute to find h.
Karl <Click to Send Email to Karl>
USA - Friday, April 07, 2006 at 13:04:23 (EDT)

Question on how to start this problem. I don't know if it is ask to make use of a circle or a cone. Someone wants to make s Melita style coffee filter from a round piece of coffee filter paper. They want to cut out angle x and glue the two edges of the wedge to do so. 1)Find a formula for the volume of the resulting cone as a function of the angle x. 2)How should they choose the angle x if they want the volume of the resulting cone to be as big as possible. Ok, I know they want to use a circle, but I'm confused after that part.
Andre

Houston, TX USA - Friday, April 07, 2006 at 10:34:32 (EDT)

Hey, i have a few word problems that I'm having trouble with. 1) A little boy buys a spherical balloon of total volume 1 cubic foot. he starts blowing to fill the ballon at a rate of .1 cubic feet per minute. how fast is the radius of the balloon increasing when he has the balloon halfway blown up? 2) A cylindrical swimming pool is being filled from a fire hose at a rate of 5 cubic feet per second. if the pool is 40 feet across, how fast is the water level increasing when the pool is half full? 3) A cylindrical can is to hold 20 (pie) m^3. the material for the top and bottom cost $10/m^2 and material for the side is $8/m^2. Find the radius and height of the most economical can.
Julius

houston, tx USA - Thursday, April 06, 2006 at 17:27:11 (EDT)

Reply to Wendy: Suppose p/q = log{base 2}(7). Taking 2 to the power on both sides:

  2^p/q  =  7

where p and q are both positive integers. Raise both sides to the q power:

  2^p  =  7^q

Consider that 2^p has a prime factorization of nothing but 2's and 7^q has a prime factorization of nothing but 7's. How can those both be equal to the same integer?
Karl <Click to Send Email to Karl>
USA - Sunday, April 02, 2006 at 16:43:11 (EDT)

Prove (by contradiction) that log(base 2)7 is irrational. help please!!! Wendy
Wendy

AUS - Friday, March 31, 2006 at 23:32:21 (EST)

Reply to Blake: Picture it in cross section. That would be a rectangle inscribed in a circle. If the upper right corner of the rectangle is at (x,y) and the circle's radius is r, then

   y = √r² - x²

If the y-axis is also the axis of the cylinder, then x is the radius of the cylinder and 2y is its height. Presumably the "best fit" would be to maximize the volume of the cylinder. The volume of a cylinder is given by V = π R²h, where R is the radius of the cylinder and h is its height. Replacing R and h with the expressions given for them above:

   V  =  2π x² √r² - x²

Now take the derivative, dV/dx, of the above. Set it to zero and solve for x. You have to apply the product rule.

   0  = 4π x √r² - x²  -  2π x³ / √r² - x²

Multiply through by √r² - x² and divide by 2π to get

   0  = 2x(r² - x²) - x³

Divide this by x and solve the resulting quadratic. Back-substitute to find y.
Karl <Click to Send Email to Karl>
USA - Tuesday, March 28, 2006 at 13:19:38 (EST)

I need to find out how to inscribe a cylinder that is inside a sphere. The sphere has square root of 3 as its radius and i need to find out how to find the base and height of the cylinder that best fits the sphere. Asked my teacher but he says "I do not have enough time for you" please help.
Blake

Ont Canada - Sunday, March 26, 2006 at 22:30:57 (EST)

I need to differentiate: f(t)=(1 + ln t )/ (1 - ln t )
Keith

FT Hall, Id USA - Monday, February 27, 2006 at 12:48:47 (EST)

Cancel the first problem on there. I had a "brain-flatuation". Because the cross section is a square (not a rectangle) I should have

2int(f(x)*f(x)) between 0 and 2, instead of (x*f(x))

Still need assistance with the second one however.
Tony

Phoenix, AZ USA - Wednesday, February 22, 2006 at 13:53:28 (EST)

I'm trying to get an explanation to a couple of the sample problems from the AP Calculus Practice Book at the college board website. They are both on page 30 in the following pdf link.

http://apcentral.collegeboard.com/repository/05836apcoursdesccalc0_4313.pdf

First is #22. The base of a solid S is a semicircular region enclosed by the graph of y=sqrt(4 - x²) and the x-axis. If the corss sections of S perpendiculr to the x-axis are squares, then the volume os S is?

I set this up as 2*int(x*f(x)dx) between 0 and 2. I keep getting half of what they get.

Next is Question 24.

                pi*e^x
If f'(x) = sin( ---- ) and f(0)=1, then f(2) = 
                  2

Having difficulty getting started and any pointers as to the right direction would be appreciated. Also, I could not find any solutions to these online. If anyone has any links, they would be appreciated as well.
Tony

Phoenix, AZ USA - Wednesday, February 22, 2006 at 13:37:34 (EST)

Keith, You have to use the quotient rule which is the ((bottom function multiplied by the derivative of the top fcn) minus (the top fcn multiplied by derivative of the bottom fcn)) (all divided by the bottom fun squared). Are you familiar with the prime notation? (1 + sin x)' means the first derivative of (1 + sin x) which is (cos x). The best way to tackle this is to call the top functin U for upper and the bottom fcn V then the quotentint rule is for y = U/V is y' = ((VU') - (UV'))/(V^2). Hopefully this will get you started, if you need further assistance please post your work and I will comment on it. Peter
Peter

Ireland - Wednesday, February 15, 2006 at 18:02:32 (EST)

My question pertains to text found on this page: http://www.karlscalculus.org/calc1.html#s1_4 At the end of the second paragraph in section 1.4, we find the text: "... on all subsequent days the amount will be that close ...". From my point of view, this phrase requires that there be two consecutive days on which the amount of milk delivered is the same. But this cannot be true if the amount delivered on day n is 1+1/n gallons. For example, let's say that on a particular day I want the amount delivered to be 1.01 gallons. So, on day 100 I get 1.01 gallons. Now, the phrase above says that: "... on [a] subsequent [day] the amount will be that close ...". I think that any subsequent day will be associated with an amount that is LESS than 1.01 gallons. Am I mistaken? ~ Mark
Mark Morse

Seattle, WA USA - Wednesday, February 15, 2006 at 17:37:15 (EST)

I am trying to differentiate this: Y = (1 + sin x)/(x + cos x) I was trying to use the quotient rule but am not sure if i performed the right steps.... and Y = (x)(sin x)(cos x) I didn't know where to start
Keith

ft hall, id USA - Wednesday, February 15, 2006 at 15:31:53 (EST)

Keith, Please put brackets around the equation to indicate the order. For example is it y = (1+sinx)/(x+cosx) or y = ((1+sinx)/x)+cosx or y = 1+(sinx/x) +cosx and so on. Peter
Peter
Ireland - Wednesday, February 15, 2006 at 13:07:09 (EST)

trying to differentiate y = 1+sinx/x+cosx
keith

fort hall, id USA - Tuesday, February 14, 2006 at 18:11:48 (EST)

trying to differentiate y = 1+sinx/x+cosx
keith

fort hall, id USA - Tuesday, February 14, 2006 at 17:44:57 (EST)

Karl, Have you got any information regarding the Wronskian and Abels formula in the solution of 2nd order differential equations and also the proof of the use of the Wronskian. I am at sea with this at the moment and hopefully I will be able to come back with more specific information. Regards, Peter Gibney
Peter Gibney

Ireland - Friday, February 10, 2006 at 15:11:20 (EST)

I have a question that if you can help that would be wonderful. It's a chemistry problem which deals with 2nd order reaction rates. I have to integrate:

  
             dx
       --------------
         [A-x][B-x]

which somehow I'm supposed to come up with:

    1         [B - x][A]
--------- ln -------------
[B] - [A]     [A - x][B]

I tried doing partial fractions, but I had in the ln just the

    [A - x]
ln -------------
    [B - x]

which doesn't correspond to what I'm supposed to get and there isn't even a [A] or [B] in the ln either! If you can help me solve this "mystery" that would be great :) Thank you so much!
Matthew

Davis, CA USA - Sunday, January 29, 2006 at 04:09:46 (EST)

Suppose that S(q) is the price per unit (in dollars) of widgets which will induce producers to supply q thousand widgets to the market, and suppose that D(q) is the price per unit at which consumers will buy q thousand widgets A. Which is larger , S(100) or S (150), and why(be wordy)? B. Which is larger, D(100) or D(150), and why ( be wordy)? C. If D(100)=10 and S(150)=10, what will you predict about the future selling price widgets (currently at 10$)? Justify your prediction
Malcolm Vella

Philadephia, PA USA - Thursday, January 26, 2006 at 18:36:29 (EST)

Two questions about function continuity: 1) if a function is continious everywhere, its limit at minus infinity and plus infinity are both 0, does it necessarily have a maximum? and if the function is non-negative everywhere, what then? 2) how do you prove that if a function is continuious between point A and infinity and its limit at infinity is a constant L then that function is uniform continuious between A and infinity? Appreciate your help, Ron.
Ron

Tel Aviv, Israel - Thursday, January 26, 2006 at 16:00:01 (EST)

I'm wondering if anyone can help me with the following question. I have already completed it and handed it in but I am now preparing for my exam and feel that being able to do a question like this would help. If someone could give me a step by step tutorial it would be greatly appreciated. A construction company has been offered a build-operate contract for $7.8 million to construct and operate a trucking for five years to transport ore from a mine site to a smelter. The smelter is located on a major highways and the mine is 3 km into the bush off the road. The gravel road will not ber perpendicular to the highway. Construction (capital) costs are estimated as follows: Upgrade to the highway (i.e. repaving) will be $ 200 000/ km New Gravel road from mine to highway will be $500 000/km Operating conditions are as follows: There will be 100 return trips each day for 300 days a year for each of the five years. Operating costs on the gravel road will be $65/h and the average speed will be 40 Km/h Operating costs on the highways will be $50/h and the average speed will be 70 km/h. Use calculus to determine if the company will accept the contract and the distances of the paves and gravel road sections producing optimum conditions (max. profit). What is the max profit? Do not consider the time value of money.
Mark

Minesing, Ont Canada - Wednesday, January 25, 2006 at 21:07:29 (EST)

Could someone tell me what a limit is? I just dont quite get it. Like what is the basic definition or concept of limit
Kobe
USA - Tuesday, January 24, 2006 at 12:04:18 (EST)

Hi, Regarding my request for assistance with:


             1  dx
integrate -----------
          (x² + a²)^3/2

I have now figured out where I was going wrong and I cave sucessfullly carried out the integration. Best regards, Peter Many thanks, Peter
Peter

Ireland - Sunday, January 22, 2006 at 17:40:09 (EST)

Hi, Regarding my request for assistance with:


             1  dx
integrate -----------
          (x² + a²)^3/2

I have now figured out where I was going wrong and I cave sucessfullly carried out the integration. Best regards, Peter Many thanks, Peter
Peter

Ireland - Friday, January 20, 2006 at 16:26:06 (EST)

Hi, I would be obliged if I could get some assistance with an integral that has me really stumped:


             1  dx
integrate -----------
          (x² + a²)^3/2

Many thanks, Peter
Peter

Ireland - Friday, January 20, 2006 at 14:37:01 (EST)

Reply to Michelle: Let r be the radius of the cylinder and let h be its height. Then your constraint equation is the volume equation:

V = 24π inches³ = πr²h

From this you can easily get h in terms of r:

h =

V πr²

Your cost function is proportional to (and we'll just say it's equal to) the area of the sides (2πrh) plus three times the area of the bottom (3πr²).

C = π (2rh + 3r²)

Now replace h with its equivalent from the constraint equation:

C = π

2rV + πr²

3r²

=

2V + r

3πr²

Now you find dC/dr of this, set it to zero, and solve for r.

dC = 0 = - dr

2V + r²

6πr

Substituting 24π inches³ for V (as that's what volume is given in the problem), applying the commutative law, and multiplying through by r²:

0 = 6πr³ - 48π inches³

which you can solve easily for r. Back-substitute the solution for r into the constraint equation to get h.
Karl <Click to Send Email to Karl>
USA - Tuesday, January 10, 2006 at 12:43:47 (EST)

Hi i need help with this question: A circular cylinder container, open at the top and of capacity 24(pie) cubic inches, is to be manufactured. If the cost of the material used for the bottom of the conatainer is 3 times that used for the curved part, and if there is no waste of material, find the dimensions that will minimize the cost. thanks in advance
Michelle

Toronto, Canada - Monday, January 09, 2006 at 00:33:28 (EST)

Sorry - here we go:

Couldn't get that mad email link to work so here is a temp mail of mine: bromfieldcourtpathway@hotmail.com

Cheers!
Mason
London, UK - Tuesday, January 03, 2006 at 19:13:03 (EST)

Hi, I am working on my maths to gear up for an Economics masters I am starting this Autumun. One of the questions I am working on has got me stumped though. The text says that:

             Q²
            ----
             10

should differentiate to equal:

             Q 
            ----
             5

I cannot seem to arrive at this answer - can anyone assist????

Best Regards, Mason
Mason
London, UK - Tuesday, January 03, 2006 at 18:57:33 (EST)

You can email me by clicking this button: