Analysis and Calculus

Here's a problem I recently came across in a very old calculus book. Unfortunately, it was an even-number, and I can't quite figure out for sure how to solve it. A bag of sand originally weighing 144 lbs is lifted at a constant rate of 3ft/min. The sand leaks out uniformly at such a rate that half the sand is lost when the bag has been lifted 18 ft. FInd the work done lifting the bag this distance. The thing about it is this would be easy to solve, except that we don't know the weight of the bag *alone* and can not just assume that it's negligable. I know that the work is force times distances, and that you use infinite sums to find the work over that particul…

A bit of homework here that I have no way of checking so I figured someone here can quickly refute or verify what I found. Imagine an equalateral triangle 1 unit on a side. Inscribed in the triangle in one big circle. Then on each side of the cirle (moving towards each vertex) is another circle inscribed in the area enclosed by the two sides of the trianlgle plus the perimeter of the circle. Then moving towards each vertex again are three smaller cirlces. Then three more and so on for an infinite number of circles. The question is what is the area of these cirlces? I got 13pi/36...I can show work better if this is incorrect but if it happens to be write I'd rather…

Please could you tell me how to integrate tan^3[x] without integration by parts. i know that tan^2x = sec^2 x - 1 thanks in advance.

9 dots are arranged in the form of a square Without altering the arrangement , You have to draw 4 straight lines may be equal or unequal in length so as to touch all the points atleast once. Your 2nd line would start where the first line ends and so on The diagram is like this . . . . . . . . . Though i tried many times i was not able to find the answer. Hope you all help me

Given the ratio of a function to its derivative, is it possible to find the function?

After studying the functions on my scientific calculator, I came across sin and cos. I've heard people talking about them on NASA TV, but never learned what it was. Could somebody explain them to me?

I'm a seventeen year old junior in high school who's taking the required Algebra 2 as his third math. I need one more math, but I get a choice as to what I get to take next year. I want to take pre-calculus and then take either statistics or college algebra as an elective. I want to impress my teacher with some knowledge of the class beforehand, but I don't have the money for a "pre-calculus for dummies" book. Can anyone link me to a website where I can get a little knowledge of Calculus before I take it this August? Thanks.

I was asked this question and it has stumped me for a week so i figured i'd come ask you all to see if anyone knows how to solve it. let p(z) = z^n + iz^(n-1) - 10 if wi are the roots of this polynomial where i= 1,2,3,...,n Compute: Sum(wj) from j=1 to n Product(wj) from j=1 to n where sum is capital Sigma (summation) and product is capital pi (product) and the j is a subscript and Z is a complex number

I've been trying to work out this problem but I can't do it! It says: Find (d squared y)/(dx squared) as a function of x if sin y + cos y =x I try: cosy(dy/dx) - siny(dy/dx)=1 (dy/dx)(cosy-siny)=1 dy/dx=1/(cosy-siny) well after that I've tried many things but none of them appear to work... so... could anybody help me please!! thank you very much!

I understand the Ricci tensor is the trace of the Riemann tensor, but I don't fully understand the Riemann tensor; it's the tangent tensor of the curved area? What is the ricci tensor, then, exactly in layman's terms? The reason I ask is because I am trying to understand the lefthand side of Einstein's field equation; I understand the stress energy tensor (as explained in the Feynman Lectures in Physics). What exactly does the Einstein tensor ([math]G_{ab} = R_{ab} - \frac{1}{2}g_{ab}R[/math] where [math]R = g^{ab}R_{ab}[/math]) signify in layman's terms?

Well we work with spheres in derivative and integral problems It's interesting to note that derivative of volume (wrt radius) [math] V = \frac{4}{3}\pi r^3 [/math] [math] V' = 4\pi r^2 [/math] Is the formula for surface area. And circles are the same: [math] A = \pi r^2 [/math] [math] A' = 2\pi r [/math] Area goes to circumference. Now I know this isn't just a coincidence. If someone could explain why this is that'd be great. And also, what happens if you found the integral of the Volume of the sphere (or the Area of a circle) ? (they're [math] \int \frac{4}{3}\pi r^3 dr = \frac{\pi r^4}{3} [/math] and [math] \int \pi r^2 dr = \frac{\pi r^3}{3} [/math…

ok, i was REALLY bored. i decided to try to derive the newtonian analogue of GR with absolute space and time-no space-time. i started like this: you have a piece of matter. this piece is the center of an imaginary sphere(not in the i sense, but in the sense of not existing). the gravity would cause a volume reduction. this could be seen by having the edge of the sphere be composed of matter. [math]V=\frac{4}{3}{\pi}r^3[/math] [math]\frac{dV}{dt}=4{\pi}r^2{\frac{dr}{dt}}[/math] [math]a{\frac{dV}{dt}}=4{\pi}GM\frac{dr}{dt}[/math] now, i know acceleration has something to do with curvature, but i'm not sure how to express it. the acceleration causing …

I am conducting a project where I grew different dilutions (1:10-1:10^6) of 6 different strains of cells in 4 different drugs to induce cell death and grew them on petri dishes. I am not sure if there are different types of siginificance tests but what type of significance test would I need to use in this project? What other types of statistical analysis would be needed? Help is apreciated, thankyou.

A baseball diamond is a square 90 feet on a side. A player runs from first base to second base at a rate of 15 feet per second. At what rate is the player's distance from third base decreasing when the player is halfway between first and second base? We've already set part of this problem up. If we let x be the distance between the player and second base, and y be the distance between the player and third base, then = - 15 feet per second, and will tell us what we want to know. Use the picture to find a relationship that will help you answer the question. so I got the answer - 6.71 after some 10-15 minutes of working around....... lmao and it says wrong …

What is the slope of the tangent line to the curve xy^2+5tan(xy)-2y+x=4 at the point (4, 0)? ok, I used implicit differentiation to find dy/dx then plug in those 2 points to find the slope of the tangent line but I don't know how to find the derivative of 5tan(xy).

Hey, I am having trouble with almost all of the things on this forum because I don't know some advanced math types, such as trig or calculus. I am trying to learn them now on my own, but am having little luck. I have gone through books, tore apart the search engines, but found nothing that works right. Is there any good math websites that cover everything from the ground up and are free?

integrate sin(4x)^2 I know sin(x)^2=1-cos(x)^2 cos(x)^2=sin(x)^2+cos(2x) ok I am lost, problem is sin(4x) so but the identities are in sin(x)

ODE/Sturm-Liouville problem. consider the Sturm-Liouville problem: y" + [ lambda p (t) - q(t)] y= 0 in (0,1), alpha y(0) + beta y'(0) = 0 gamma y(1) + delta y'(1) = 0, Where alpha, beta, gamma, delta are real constants, and p: [0,1] -> R and q:[0,1] -> R are continuous functions with p(t) > 0. (a) Suppose alpha*beta doesn't equal 0. Show that if f_n(t) and g_n(t) are eigenfunctions associated with a given eigenvalue lambda_n of the Sturm-Liouville problem, then f_n(t) = c g_n(t), t belongs to [0,1], for some constant c in R. (b) Can one remove the restriction alpha*beta doesn't equal to 0 in part (a)? [ Explain and justify yo…

I am doing some analysis of results from an impact crater caused by steel balls. I am measuring the diameter of the impact crater and need to link it to the volume of the crater using the volume of resolution. Can anybody help I have a formula that I derived it is correct? Volume of moved by impact = (2/3) π r3 – (1/12) π(4r2-D2)3/2 where r is the radius of the steel ball and D is the diameter of the crater. This i think is linking the diameter of impact crater to the volume moved... I also have two other calculus problems: I need to find out the surface area of the crater in terms of the diameter of the impact crater can anybody help??? Then I need the r…

Hey could someone please help me with this question? the length of 3 sides of a trapezoid is 3 cm, Find the length of the 4th side if the area is maximized.

I am thinking about purchasing Michael Spivak's Calculus book with the text solutions from his website as I've heard great things about it. From what I have read elsewhere, it seems it's a step beyond most Calculus textbooks, and has many difficult problems which help one to get a better understanding of Calculus. I just finished a semester of Calculus, and enjoy it quite a bit. However, the book I'm using (Stewart's), while not terrible, doesn't get very in depth with any topic. Has anyone here had a chance to use this book? Thoughts?

Prove or disprove: If the boundary of set omega in R^d has an outer measure zero, then omega is Lebesgue measurable. I was trying to come up with a counter example, but I couldn't. Then I tried to prove it, yet I was not able to do so. Please help me

When I let x=A+4/3t, why can't I get the answer?

This is one optimization problem that I just cant figure out. I'll post what I have... A hiker at point A on a straight road wants to reach, in the shortest time, a point B located 6 miles from the road and 10 miles from point A. The hiker's speed on the paved road is 4 mph and only 2 mph off the road. How far should he continue on the road before heading in a straight line for the point B? I am pretty sure I would have to use the pythagorean theorem because if you draw the problem you get a triange with two sides given. Plus I denoted [math] dr/dt [/math] to be the speed on the road which is 4 mph, and [math] do/dt [/math] to be the speed off road which is 2…

here is the problem: A particle moves along y=x^3 in the 1st quadrant in such a way that its x-coordinate increase at a steady 12 m/s. How fast is the angle of inclination (theta) at the line joining the particle to the origin when x=2? here is what I currently have dx/dt=12 dy/dt=2x^2 * dx/dt x=2 y=8 I can't find a relation to solve for derivative of theta.