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Calculus
Please could you differentiate y=X^X (that's X to the power of X)? In sea navigation, given two points A,B with LON and LAT coordinates, find the travel distance from A to B. The distance is actually the great circle passing A and B. The solution is needed in the form of a double integral where the first int covers the LONs and the second the LATs. I guess the difficult part is to form the vector. A high school senior writes: "The horizontal line y=c intersects the curve y=2x-3x^3 in the first quadrant. Area A is bounded by y=c, x=0, and the curve. Area B is bounded above by the curve and below by y=c. Find c such that (area of A)=(area of B)." I can do this using integrals, but I don't want to solve any cubics. There must be an easier way to find c. In my calculus book, it mentions a function that is not differentiable at any point due to the fact that it is not smooth at any point. It does not go any farther, and I was interested in hearing more. What's a conjugate? (I'm in calculus, I've heard the term before, I just had a mental malfunction...) If a "u substitution" is made for different parts of the function, I get different answers. However the back of my book says that only one of the answers is correct. Here is the problem: Integral of secant squared of 3x times tangent of 3x. One of my friends has asked me to solve a problem for her. I arrived at the right answer using my method of doing it. Later, she showed me her method. It seems correct, too. I was just wondering if you can find a mistake for her. Question/answer: antiderivative of secant square 3X times tangent 3X= 1/6 (cos3X) to the negative 2 plus C. This Theorem states that "If f(X) is continuous in [a,b], then there is a 'z' in (a,b) such that the integral from a to b f(x)dx = f(z)(b-a). How does the Mean Value Theorem apply to life? Why do we need this theorem? I have been studying vector calculus, and I was interested in finding more about the principles of the divergence theorem and Stokes' theorem. I am a grade twelve student taking calculus and was wondering if you could help me with this problem: y=2x(6x+5)exp4 - solve to the second derivative. I am a little confused about finding volumes when graphs are revolved around an axis - specifically about finding the volume of a disk and the volume of a washer. Given the following rational function: f(x)= (x-1)/(x^2+3x-1), how do you find the range? Our book gives the answer of: (neg infinity to 0.12] U [0.65 to pos infinity). In calculus, we were looking for the solutions for a third degree polynomial equation. Using my TI-85 to find the solutions, I stumbled upon an interesting observation. Given a cubic of the form ax^3 + bx + c, the absolute value of the sum of any two zeros seems to be equal to the absolute value of the third. However, I have so far been unable to give a general proof. How do I find the slope of: root x = -ln (xy) at (4,2) ? Find all critical points (if any) in the following problem: k(t)=1/the square root of (t-squared +1). Given f(x) = 15 x^(2/3) + 5x, find coordinates of relative max. or min. For what x values is the function concave down? Please show me the steps for solving this problem: integral of (1+sec(x))^2 dx. Why is "e" so important? How significant is "e" compared with "pi?" How did it come about? How is it defined? Why is it taught only at higher level mathematics? Are there other numbers like "e"? I need help with two similar limit of integral problems. Find the value of "a"so that the circles with equations (x-a)^2 + y^2 = 2 and (x+a)^2 + y^2 = 2 intersect at points where their tangents are perpendicular. How do you find the point on the parabola y = x^2 nearest the point (-3,0)? Given y = A sinx + B tanx. Find A and B if the slope of the tangent to the curve at x1 = pi/4 is m1 = 4 + root 2, and at x = 0, m2 = 4. Why do we have to use the natural log for the antiderivative of 1/x? Why doesn't the power rule work on the antiderivative of 1/x? Right now we are learning general exponential and logarithmic functions. In one of my homework problems.... What is the purpose of the derivative? A Norman window is a window in the shape of a rectangle with a semicircle attached to the top. Assuming that the perimeter of the window is 12 feet, find the dimensions that allow the maximum amount of light to enter. A student from Riverdale has interesting integration questions. I need the integral from 0 to 1 of X^4*exp(-x/2). Can you help? How do you integrate the function: (tan 2x)^3 dx, in terms of x? What is the volume of the solid of revolution created by the function, y = cos(x) and y = (x+1)/3, when these are revolved about the x-axis? I can't figure out how to take the limit using L'Hopital's rule on this problem... The problem is to design a more efficient soda can that holds the regular 12 oz. of liquid. The can needs to have the least possible surface area. What are the integrals of sin(ln(4x+5)) and ln(sinx) ? Can e^sinx be integrated ? What is the integral of 5^x dx ? Prove that (1+1/n)^n is bounded from above. Is it possible to integrate [exp(-zcosh(t)] dt analytically? I need to find the minimum value of E for E = be^(-ar) - dr^(-6) where b, a, and d are constants and r is the variable. Find the limit of (x^2 + 2x) / (5x - 5) as x tends to infinity. Jones has a roll of tape for which he knows the thickness of the tape, diameter of the core, and diameter of the entire roll of tape. Is there a formula for determining the length of the tape? The half-life of a radioactive substance is d days. If you begin with a sample weighing b g, how long will it be until c g remains? A colony of N bacteria increases with time according to the formula dN/dT = kN. If initially there are N' bacteria, show that the number at any time is given by N = N'e^kt. Find the limit of [sin 3x / tan (x/3)] as x goes to 0. Using the definition of derivative, find f'(x), in simplest form, for f(x)=x^(1/3). (I know through the "tricks" that the answer is (1/3)*x^(-2/3) but I can't get to that using the definition of derivative.) If f(x)= [x], prove or disprove that f(x) has a limit at x=1. How do I prove this derivation: (f/g)' = fg'-fg'/g^2 ? A student asks questions about convergence and divergence in integrals. We are trying to prove that the "power rule" works for finding the derivative of x^(3/2) (where x>0). What is the domain of f(x)=((2-x)/(x-2))^.5 What is the integral of 1/(x(1-x))^(1/2)? How can I integrate this function using only the derivative of arcsin or arctan? What is the equation of the tangent line to the graph of f(x)=x^(2/5) at the point (-32,4)? I have forgotten the formula for the integral of f(x)*g(x) and f(x)/g(x). Can you help me? For a limit to exist, the right side and left side have to equal each other. Say you have a limit such as "lim as x->4 for function (3x-4). To see if this function exists, you have to check it on both sides. Would you plug in 3 for the left and for the right? The problem of drawing a tangent to a given curve at a given point is closely connected to the problem of finding areas ("areas under a curve"). What is the connection, and when and by whom was it discovered? My question is just like the Norman window question, but I have an isosceles triangle on top. Where do I start? I seem to have 3 unknowns. b, h and x. A steel girder 27 ft long is moved horizontally along a passageway 8 ft wide and into a corridor at right angles to the passageway. How wide must the corridor be for the girder to go around the corner? Neglect the horizontal width of the girder. A closed box with a square base is to contain 252 cubic feet. The bottom costs $5 per square foot, the top costs $2 per square foot, and the sides cost $3 per square foot. Find the dimensions that will minimize the cost. Find the area under the parabola f(x) = x^2 + 1 for x = 0 to x = 2, and y > 0. Also find the length of the curve from x = 0 to x = 2. Integrate ln(r/a)dr - the limits are a to b. Is this possible, or is there some table for it? A telephone company has to run a line from point A on one side of a river to another point B that is on the other side, 5km down from the point opposite A. The river is uniformly 12km wide. The company can run the line along the shoreline to a point C and then under the river to B. The cost of the line along the shore is $1000 per km and the cost under the river is twice as much. Where should point C be to minimize the cost? Find the dimensions of the cylinder of maximum volume that can be inscribed in a cone having a diameter of 40 cm and a height of 30 cm. Show that the maximum area of the cylinder is 4/9 the volume of the cone. A string of length 25cm is to be formed into the arc of a circle so that the area of the segment formed will be a maximum. How do you Integrate: x^n/e^x ? Find all the discontinuities of the following function and label those that are removable... A street lamp whose light is 5 meters from the ground is 6 meters from a vertical brick wall. A person, 2 meters tall... An AP Calculus student asks questions about his chapter test. A circular conical reservoir, vertex down, has depth 20 ft and radius of the top 10 ft. Water is leaking out so that the surface is falling at the rate of 1/2 ft/hr... F(x)=Integral from -1 to 1 of 1/sin(x) Find the area between the curve x = 1-y^2 and the y-axis? Find the shortest distance between the point (4,5) and the circumference of x^2 + y^2 = 9. I have a question about the limit of a sequence ... A rectangle is inscribed in a triangle (with certain dimensions). Find the maximum area of the rectangle. I am trying to find the indefinite integral of the following function: f(x) = x^x. I can't seem to find the anti-derivative of (cos X)^2. Find the total distance travelled by a particle along the path, sketch the path; find the equation of the tangent line at t1, sketch the curve and tangent at t1... Does the function f(x)=3De^(1/x) have a local minimum? The cost of producing x units of a certain product is given by C = 10,000 + 5x + (1/9)x^2. Find the value of x that gives the minimum average cost. Solve 5x^2 + log(x) = 0 for x... I am having difficulty understanding the Taylor Series and Maclauren Series. The number of bacteria in a culture increases from 3000 to 9000 in 8 hours. If the rate of increase is proportional to the number present, estimate the number of bacteria at the end of 24 hours. Looking at the equation: 1 + x*2^(x-1) Use the washer method to find the volumes of the solid generated by revolving the regions bounded by the lines and curves about the y-axis... d/dx sin(sin (2cos(2x/3))) = ? We recently learned about the washer method and the shell method. I was wondering which method to use in different situations. What is the mathematical formula or equation for calculating the speed of a boat as it relates to the water line of the boat.? Water is rising in a tank with a sloping floor. Find how far it has risen after a certain amount of time. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 'r'. A lump of radioactive subtance is disintegrating at time 't' days after it was first observed to have mass 10 grams and: (dm/dt) = -km (where k is a positive constant). Find the time, in days, for the substance to reduce to 1 gram in mass, given that its half life is 8 days. Evaluate: xdx / (x^2+1)^(1/2), given u^2 = x^2+1. I need to solve the integral from 0 to 1 of ln(1-x) / x symbolically, if possible. Determine the number of arithmetic means between two numbers whose sum is 5.5. If the initial velocity is 834 meter^2/sec at an angle of 60 degrees, and x is 21,000, at what time, t, is the object at x = 21000 meters? (2x^2 + 2y^2 - y)dx + (x^2y + y^3 + x)dy = 0 The arch y=sin x,0 less than or = to x less than or = to pi, is revolved around the line y=c to generate a solid. Find the value of c that minimizes the volume. How can I fit an exponential function to a set of three discrete data points? Problem: y = 4-x2 ; x axis - a) Draw a figure showing the region and a rectangular element of area; b) express the area of the region as the limit of a Riemann sum; c) find the limit in part b by evaluating a definite integral by the second fundamental theorem of the calculus. How many people does it take to maximize your profit? An island is 6 miles offshore... where should you land in order to go from the island to a store on shore in the least possible time? Integrate the following function: x^2 * e^(-2x^2) How do you integrate x^x? I don't know how to do the following problems... What does integral of (cos(x).x(cub).sinh(x)square).dx mean, and what's the answer? What are the formulas for the surface area, total surface area, and volume of a sphere, and volume of a pyramid and cone? (1) The cost of goods and services in an urban area increased 1.5% last month. If the rate continues, what will be the annual rate of increase? (2) Suppose a certain population increases by 30% every 10 years. By what percent does it increase each year? The integral looking somewhat like this: S ((e^u)/u) du has resisted every attack on my part. What is the antiderivative of the function tan(x)? I am now thoroughly confused: we just learned the formulas for volume of a sphere and volume of a pyramid, but, he wouldn't tell me how to do it. Show that the expression for the rate of change can also be given by -cos x cot^2 x. Give a primitive function of (sin x) / (ln (2 + x^2)). Just to check that I can't do this because f'(0) = infinity... The sum of the two lozenges (diagonals of a diamond) are maximum when the angle between the longest lozenge and the side of the diamond is 30 degrees, diamond side length = sqrt 2... If f, g: N -> N+, then either f(x) is O(g(x)) or g(x) is O(f(x)). Why? Show that the graph of every cubic polynomial has a point of symmetry. Find f min and f max where x and y are real numbers and f(x,y) = 2(sin x)(cos y)+3(sin x)(siny)+6(cos x). A boat leaves a harbour, O, position vector (0i+0j) at 9 a.m... at what time will it arrive? What are some examples of functions of a real variable whose derivatives don't have derivatives? log a = -1.3 How can it be that if y=x^2, then y'=2x, but if y = x+x+...+x (x*x) then y'=1+1+...+1 (1*x) = x? For all real x, F(x) = x+3 and G(x) is a polynomial of degree two such that G(F(x)) = x+2. Find G(x). (1) Given three points A,B,C, find the angle between R(AB) and R(AC); the (scalar) area of triangle ABC; a unit vector perpendicular to ABC... Is it possible to show the formula Sin(A-B), Cos(A+B) or Cos(A-B) using Ptolemy's Theorem? How do I explain why logarithms work to non-math-oriented people? Also, what is the history of the development of the concept of logarithm? Why is it called a "logarithm"? If lim(An/n) = L and L>0, how do you show that lim(An) = + infinity? What resources can I use to learn about differential equations? I would like to see the proof of the tangent double angle identity. If y does not equal to 0, prove that |x / y| = |x|/|y|. Point P is moving in the positive direction along the y axis at a constant velocity of 10 m/s. Point Q is always moving with a constant speed of 11 m/s directly towards point P... What is the arc length of sin x from x = [0, 2pi]? Cardan's well known formula solution of cubic equation I need to maximize the volume of a right-circular cylinder that fits inside a sphere of radius 1 m. A loan company is limited to charging maximum 18 percent interest on a loan. The amount of money available for loans is proportional to the interest rate the company will pay its investors... Let x+y=1 and x^3+y^3=19. What is the value of What's the maximum area I can fence with a thousand yards? A water tank has a shape of a right circular cone... how fast is the water level falling? Otavia's note: High school: calculus Suppose you grab the end of a chain that weights 3 lb/ft and lift it straight up off the floor at a constant speed of 2 ft/s: determine the force as a function of height; how much work do you do in lifting the top of the chain 4 feet? How do I calculate the area and circumference of a given ellipse? How much money would I need to win in the lottery to have an income of $75,000.00 year until I die? That should be in about 30 more years. Is there an alternative to Newton's method of approximation to solve The Bay of Fundy in Canada is reputed to have the largest tides in the world, with the difference between low and high water level being as much as 15 meters... Solve for x... Given the rational function y=r(x)=(4x^6-x^4)/(x^5+5) describe its end behavior... How do I find the equations for all parabolas containing the origin whose vertex is at (2,1)? What is the integral of f(x)=(tanx)^-1/2? What's the volume of water in a cylindrical tank 72" long and 36" in diameter, filled only to 4.25"? In the equation P=V*R/(1-(1+R)^(-n)), how do you solve for "R"? Is it possible to numerically integrate What is the integral of 3(x^2)-5x+9 from 0 to 7? How can I solve the following problem: limit as x-> infinity of x^(1/ x)? Find the rate at which angle BXA is changing in radians per second... If f(x)=1/sin(x), what is the integral of f(x)? Find all the critical points and determine their nature for the function Since we can never really get to zero by reducing something by halves, does that mean that we are floating on air? Regarding Raabe's result for the convergence of a numerical series of non-negative terms - I am looking for is a convergent series having m = 1. Where can I find a site with information about the AP calculus test? How do I integrate x^(-ln x).e^(-x.lnx + x).dx ? Is there a symbolic (not a numerical method) solution to the definite integral of sin(x)/x from 0 to some value? Find the curve whose slope at (x,y) is 3x^2 and which passes through (1,-1). Is the absolute function continuous? How can I apply the chain rule to find du/dx for dy/dx=(nu^n-1)*(du/dx)? How do I find the limit (n-> -infin.) of (sqrt(2n^2+1))/(2n-1)? Why does the integral from -1 to 2 of x^2/3 = x^3/9 |from -1 to 2? A particle moves on a straight line with a velocity v(t)=sin(w)cos^2w; f(0)=0; w is constant; find f(t). How do I use the formula for arc length solve for the length of the curve? What is the integral of [sqrt(1-x^2)]dx from [-1,1]? What's a plain English meaning of the derivative? How do I find dy/dx for ((xy)^(1/2)) + ((x + 2y)^(1/2)) = 4? I have a question on the "chain rule" when finding the derivatives of polynomials. What is the limit of (x^2 -4) / (2x) as x approaches infinity? I know that velocity components are dr/dt and rdo/dt and that components of acceleration are d^2r/dt^2 - r(do/dt)^2, etc. I would appreciate any help at all in understanding how these are generated and/or how to use them. Are the Stokes-Greens-Gauss) theorems related? What is their significance? Minimizing functions and costs. Is there an algorithm for working out the cube root of numbers without a calculator? Find the volume of the solid formed if the curve y = cos -1 x is rotated about the x-axis between x=0 and x=1. A rectangular sheet of cardboard measures 16cm by 6cm. Equal squares are cut out of each corner and the sides are turned up to form an open rectangular box. What is the maximum volume of the box? Given that (x^p)*(y^q) = (x+y)^(p+q), prove that: dy/dx = y/x. Given a region defined by five relations simultaneously, find its area. What dimensions should a rectangular piece of paper have to maximize the volume of the box made by cutting the corners out and folding? How do you integrate (1/x)dx? What is the volume of the shape formed by rotating the parabola y=x^2 around the line y=x? (From x = 0 to 1). How do I evaluate Integral[x tanx dx]? Find the extrema of f(x)=(ln(1+exp(-a-bx-cx^2)))/((x+d)^2)... How do you integrate f(x) = x^2/[(x^4) + 1]? How do you find the positive roots of x^9+3x^8-5x^3+4x+6=0? Show that x^n- 1=0 has exactly 2 roots if n is even, and only 1 real root if n is odd Find the polar equation of x^3+y^3=3axy with x+y+a=0 as an asymptote. Find the slope of the tangent line at the point (3,4) on the circle x^2+y^2 = 25. How do I find the rotations necessary to create a raytrace of a star sapphire so that the star always faces the camera? To make a funnel, we take a circular piece of metal, cut out a sector, and connect the two radial edges together to make an open cone. What should the angle of the sector be to maximize the volume of the cone? When using interval notation to describe when a function is increasing and decreasing, how do I know whether to use brackets or parentheses? How do you solve Int [sin (ln x)] dx? How do you find the area of an ellipse? Find the area bounded on the right by x+y = 2, on the left by y = x^2, and below by the x-axis. How do you find the domain and range of the function f(x) = 2x^2-3x+1? (Both with and without calculus.) Is there a formula into which you can enter a starting point, a starting slope, an ending point, and an ending slope that gives the equation of the curved line between these points? Given the starting latitude, longitude, distance, and course of a plane, what formula gives its destination latitude and longitude? What is a derivative? What is the geometrical interpretation of the gradient? Find the slopes of two lines tangent to the parabola y = x^2 that pass through the point (2,1). How do you express the equation y = xcosx in terms of y? Why is the following true: lim x --> oo (1-1/x)^x = 1/e? A wire is cut into two pieces. One piece is shaped into a square, the other into an equilateral triangle. How should the wire be cut to maximize the area enclosed by the the two pieces? What is the difference between the area under the curve from a to b and the definite integral from a to b? What is calculus and how does it work? Find the shortest crease when folding a piece of paper 6 units wide by 25 units long. Why does lim n --> oo (n+1)^.5/(n-1)^.5 = 1? The cubic polynomial h(x) = x^3 - 3bx^2 + 3cx + d has a local maximum and a local minimum... Find the domain, intercepts, asymptotes, critical points, points of inflection, and graph of the function: y = 3x^4-4x^3-12x^2. What is the integral of sec^3(x)? How do you solve the equation x^x = 100? How do you break 1/x^2*(x+2) into partial fractions? Integrate sqrt(x^3-1)/x dx. A conical drinking cup is made from a circular piece of paper of radius r by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup. Your professor has been murdered. Using exponents and differential equations, prove that it did not happen while you were in his office. Why is the area under the parabola y = x^2 between the y-axis and the line x = b equal to b^3/3? I'm soon to be a Calculus student in high school. Could you please teach me some Calculus over the internet? The radius of a right circular cylinder is decreasing at the rate of 4 feet per minute, while the height is increasing at the rate of 2 feet per minute. Find the rate of change in the volume when the radius is 2 feet and the height is 6 feet. Is there an exponential pattern for how many squares there are on a checkboard? I was wondering how to calculate the surface area of a sphere in n dimensions. A tank contains 150 litres of brine solution. The concentration is 0.7 kg of salt per litre... Find the time when the salt content will be 90 kg. Can you tell me about Picard's iteration method of solving differential equations? Write the equation of the specified circle in general form... I have a question regarding how to set up integrals for the volume of revolution. At what first quadrant point on the parabola y = 4 - x^2 does the tangent together with the coordinate axes determine a triangle of minimum area? Estimate all points of intersection of the graphs f(x) = x + sin x and g(x) = x^3 by using a calculator or computer. How do you know you have found them all? How do you integrate the following function: sqrt(1-x^2) from 0-1? How do you find the area of a parabola? (I just finished Algebra 2.) I'm trying to apply Simpson's rule to a computer program, but I don't understand it well enough - can you tell me how it works? For the following functions f(x) decide if the funcion is invertible as a function from R to R... How do I find the integral of a number raised to the -1st power? From f:n ---> 3 + (-1/2)^n where n belongs to the set of natural numbers, describe a strip that contains all but a finite number of points of the graph of f. What exactly is nonlinear math, and what is it used for? How is the surface area of a sphere calculated, and why? Graph the following... Besides at x = 3, where is f continuous? I cannot do a problem where I need to convert into the form 0/0 and then use L'Hopital's Rule... I need to find an algorithm to determine any root of a number. I was told I could determine the estimated value by using Newton's Method... Where are imaginary numbers used today in real life, as in the work force or other areas that use math? Is this spherical trigonometry or can it be solved using simpler geometric equations? I have been given the solution in the form of Frensel's Sin, but it explains nothing about how it was integrated. I am not looking for an equation, I am looking for a reason! How do you fold a piece of paper (rect. with width a and unlimited length) so one corner just reaches the righthand side for minimum area? How would you prove that the integral from n=0 to infinite of sin(x^ 2)dx or cos(x^2)dx converges? Is it possible to find the slope of three-dimensional equations? Could you give me the chain rule in easy terms, not a formula? I am trying to find the proof for the sum of the alternating harmonic series. I did find out that it is ln(2), but please tell me why? Please give me a definition and several examples of an implicit function. I am trying to find a parametric equation for a helix (in 3 space) lying on the cylinder... A picture is two meters high and is hanging so the bottom of the picture is one meter above eye level. How far from the wall should you stand so the angle of vision occupied by the picture is a maximum? How do I find the infinite product of 3^1/3 * 9^1/9 * 27^1/27 * ..... (3^n)^1/3^n ? Why does my textbook define the limit of (1/x) as x approaches zero as infinity? A light shines from the top of a pole 50 ft. high. A ball is dropped from the same height at a point 30 ft. away from the light... Is it possible to construct two cans with different volumes but the same surface area? If y = xsin(3x) prove that y''+9y = 6cos(3x), where y'' is its second derivative. I've been told that some 4th order equations can be rectified. Are there any Cassinian ovals whose perimeter can be calculated without looking up tables? In Calculus today we were trying to evalulate the limit of (1-cos x)/ (x^2) as X->0. Determine the limits... I need to calculate the true length of a cubic curve. I am having difficulties in the washer and disc methods of finding volumes, as in the problem y=x^2, x=3 about the y axis. What is the integrated area if the anti-derivative is undefined? I have a problem with Lagrange Multipliers - can you help? Please help me differentiate (y^2) * cos (1/y) = 2x + 2y. Is there a way to solve the equation e^(i*pi)+1 = 0 without using or using very little calculus? Assume a perfect circle filled with grass and a cow tied with a rope to the fence around it... What is a cycloid and what does it do? Why does the natural log of x equal the integral of 1/t dt from 1 to x? Why is INT[(1/t)dt] from 1 to x the natural log of x, or why was it defined this way? I'm trying to figure out if there is a way to determine the points at which a derivative intersects a function. I figured this worked out to -1/2: lim (x - (x^2 + x)^(1/2)) (x going to infinity). I need to find the derivative of sin(x) when x is in degrees, not radians. I've tried inspection and partial integration and don't seem to get the right results - I wonder if this is a function whose integral cannot be solved algebraically? I am trying to find the volume of a cap of a sphere with radius of 5. The cap has a height of 3 - it is as if the top of the sphere, 3 meters from the top, was severed from the rest of the sphere. How is Simpson's rule derived and why it is a better approximation of the integral than the midpoint (rule?) or trapezoidal method? Why isn't the result what I expect when I use the Chain Rule and the Product Rule to differentiate? I'm trying to find a GOOD definition for "differentiation." Let f(x) = |x|^3. Find f"(0). Evaluating a definite integral and determining an enclosed area. Finding the area of a sector of an ellipse, given the semiminor and major axes and the angles of the 2 vectors bounding the sector. How do you find the range of a function like g(x) = (x+1)/(x^2-1)? Find the equation of the tangent to the ellipse that forms, with the coordinate axes, the triangle of smallest possible area. The rate of change of the edges of a cube, given the rate of change of its volume. I'm so surprised at how often the number e comes up. Where did it come from? Who first derived it? Why is it so common in the field of biology? Why must a function that is differentiable in an interval be continuous in that interval? Would you please explain why the area under a curve is exact and not an approximation? What is a continued fraction and what makes it different from the types of fractions or ratios I'm used to? Integrate [tan 2x dx] by u-substitution. Proofs that the volume of a cone or pyramid is (1/3)b*h. Assuming logistic growth, find how many people know the rumor after two weeks. Can you show me the proof of the formula for the distance between a point and a plane? Using derivatives to find the conditions that ensure that an equation of degree 3 has 3 distinct real-valued roots. How do you integrate: Int[sin^4(x)*cos^6(x)]dx? Can you derive the formula for the surface area of a sphere? We need to prove three trigonometric functions in various ways. How do you solve the integral of e^(x^2)dx? Why does e = 1 + 1/2! + 1/3! + 1/4! + ... and lim (1 + 1/n) ^ n, as n --> infinity? What are dominant terms, and how do you obtain their values? Find the critical number of y = 2 cos x + sin 2x where x is greater than or equal to 0 and less than or equal to 2 Pi. Is zero divided by zero: a) zero, b) undefined, or c) one? I need to show that Sigma(rx^r) = (x-(n+1)x^(n+1)+nx^(n+2))/(1-x)^2. Can you help me derive and prove the formula for the volume of a sphere? I'm having difficulty understanding how my math professor proved the formula for the surface area of this curve... When the second derivative of a point is a negative, is the point said to be a maximum? What values of "a" and "b" will make this piecewise function continuous and differentiable? For what values of k and p will the function be continuous and differentiable? On what interval will it be increasing? Find all points of inflection. The formula for a circle's circumference is the derivative of the formula for its area. What is the significance of this? A population P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k... Using limits to prove that 1^infinity, infinity^0, and 0^0 are indeterminate forms. I think I need to use l'Hopital's Rule and the natural logarithm function to find lim (x goes to 0) [(cos (2x))^(3/(x^2))]. Substituting twice and using trig to integrate sqrt(1 + sin(x)). Integrating INT(e^(-|x|)) dx by treating x > 0 and x < 0 as separate cases. Can you help me find all of the maxima and minima of the function: 5e^(-x/2)sin(2x)? I'm trying to solve the logistic model of population growth for P(t). I need to find out about third derivatives for a project. Can you give me some information, including uses, of derivatives? Can you give me some tips to help me integrate functions? How would you prove the following three limits? ... Does n! = the integral from 0 to infinity of (x^n)(e^-x)dx hold true for all real numbers? If so, can we find the derivative of n!? How can you calculate specific points of a great circle on a sphere? Can you help me find the parametric equation? I need help with these max/min applications of derivatives. If 40 passengers hire a train... Can you show me a derivation of Simpson's Rule? Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle? What is the concise definition of a derivative? How do we use derivatives? I'm having trouble with the following integration. Is substitution needed here? Find two numbers such that their sum is 20, and the sum of their squares is as small as possible. In the following problem, would you use the chain rule or the quotient rule first to differentiate? Problem: (2x+1)^2/(2x+4). Can you help me piece a function together so that the following hold? It is increasing and concave up on (-infinity, 1) ... What is an ordinary differential equation? How do you integrate f'(x) = (x-1)^4? Can you help us find the limit of x (ln x)^n as x goes to infinity, for all n? Do we use L'Hopital's Rule? Find the arc length of a curve in symbolic form and decimal approximation. A ladder 4m long is leaning against the vertical wall of a house... how fast is the top of the ladder sliding down that wall...? A curve is given parametrically by the equations: x = (3-2k)^2 and y = (2+k)^2. Find dy/dx, the x-intercept, the cartesian equation... Can you write x^2 = x+x+...+x (x times) if x is not a positive integer? How would you test to see if the integral of x^x can be expressed in a finite number of elementary functions? What is the limit of (sqrt(2-t) - sqrt(2))/t as t->0? Can you help me find the equation of the tangent of y^2 = 2x using Descartes' method...? Can you help me with some delta-epsilon proofs? How do you choose delta? Can you help me find the smallest distance between two yachts using vectors? What is the difference between finding the derivatives of a function (dy/dx), and finding its differentials (dy,dx)? Is there a general formula to find the derivative of f(x)^g(x)? Can you explain the intuition behind the formal definition of a limit? derive the equation of the tangent line of a function at a given point? How do you use the Squeeze Theorem to find the limit of f(x) = x * sin(1/x) as x approaches 0? Could you please explain the meaning and purpose of the monotone convergence theorem? Why does the sine of 1/x have no limit as x approaches 0? Can you help me integrate functions using the arclength formula, e.g. x^(2/3)+y^(2/3)=1 and y=e^-x? Can you help me solve these differential equations: dG/dt=aJ-bG and dJ/ dt=cG? Can you help me on find the volume for an elliptical cone by using a triple integration? Can you show me an easy example of how to find the first derivative of a function using the definition? How do you diffrentiate an equation that cannot be expressed in the form y = f(x)? Find the volume of a solid formed by revolving the region bounded by y=x^2+1, y=0, x=0, and x=1, about the y-axis. I only have where the tangent intersects the y axis... A nitroglycerin lump is decreasing at a rate of 2 cu.cm/sec. Use related rates to find how fast the lump's radius decreases at 9pi/4 cu.cm. When using the Lagrange Multiplier method, how do you determine which of the two equations is the constraint? Four ladybugs travel from the corners of a square towards the center. Find the equations that describe the path. Can you explain how finding square roots by hand relates to Newton's method for approximating the zero of a function? I need help differentiating expressions such as: y = 2 csc^3(sqrt(x)) and y = x/2 - (sin (2x))/4. Where does the chain rule fit in? Why is 1^infinity indeterminate? How can you show this using limits? Why is it not equal to 1, as intuition leads us to believe? What is the relation between a sphere's surface area and its volume? How does their ratio change? Hints for finding specified x-coordinates... A person on a boat 9 km from the shore must go 12 km down the shoreline in the shortest time possible. Find a function for where the person should hit land... Can you show me a proof of L'Hopital's Rule and say how it relates to the different versions of the Rule? How does barycentric calculus compare with trilinear or cartesian calculus? What is the difference between a general solution and a particular solution of a differential equation? Given a sequence, I can guess what the limit it. How do I actually prove my guess is the limit? A plane flying horizontally at an altitude of 1 mi and a speed of 500 miles/hour... Prove that e^(-1/3) is the limit for y =(arctan(x)/x)^(1/(x^2)) when x -> 0. The depth of iron ore can be approximated by a plane... A solution containing 3 lb/gal of dye flows into a 100 gal tank at 5 gal/min ... The surface area of a cube is changing at a rate of 8 in./s^2. How fast is the volume changing when the surface area is 60 square inches? Prove the following identity: a . (grad(a . v) - curl (v x a)) = div v... Solve: (integral sign) sin 2x/ sq rt (9-cos^4 x) dx . Convert to a differential equation: A snowblower throws 30 cu. ft. of snow per minute... Integrate x*tan^2 x with respect to x. Find the center of mass of a thin plate of constant density covering the region bounded by the parabolas y = 2x^2-4x and y = 2x-x^2. Can you help me integrate (cos[x])^4? Two questions on three-dimensional figures. Different types of integrals: Riemann, Riemann-Stieltjes, and Lebesque. A circle is inscribed in a square tangent to each side of the square. The circumference of the circle is increasing at a constant rate of 6 inches per second... Integrate (2 - x^2)^4x dx. Find the vertex of a catenary curve. Prove that (nC0)^2 + (nC1)^2 + (nC2)^2 + ... + (nCn)^2 = (2nCn), where nCi = n!/((n-i)!*i!). Find dy/dN if y= A*(1 - e(-cN))* (1 - e(-cM)). I am writing a program to find the moment area of any shape... Using the measured rate of change in volume of water in a cone, calculate the rate at which the circular area of water reduces when the radius is r centimetres. Calculate the surface area of a football with a circumference of 22 inches and an arc length of 14 inches. A man walks toward a light... at what rate does the tip of his shadow move and at what rate does its length change? My first substitution was u^2 = tan x ... Oil from an offshore rig is to be pumped to a location on the edge of the shore... determine how the pipe should be laid to minimize cost. What are some real-life applications of factoring polynomials? How does one find the integral of [(sec x)^3 dx]? Taking the derivative of a function of a function. Find the general integral of x*ln(x)-1. A rumor spreads through a community at the rate dy/dt = 2y(1-y), where y is the proportion of the population that has heard the rumor at time t... What is the formula for calculating the Kth order derivative of the function: f(x) = exp(A(x))? A metal pole L ft long is pushed on the floor from one corridor into another corridor at a right angle... How fast is the distance between the tips of the hands on a watch changing at one o'clock? How does one make a cylinder that holds the greatest volume if the entire surface area is 600 cm squared? I would like to know how to find the volume of a torus using integrals. I would like to know how to find the volume of a torus using integrals. Which number satisfies the conclusion of the Mean Value Theorem for f(x) = sin (x/2)? Differentiate with respect to x: f(x) = x^8 + 3x^5 + (3-5x)^4. Find the distance from Earth to the sun when t = 90 days... The rate of consumption of Cola in the U.S. is S(t)= Ce^kt where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980... Which of the following inequalities are correct? How does one solve the differential equation (t^4)*x'' + x = 0 for x? What does the term curl mean in math, and how do we define it in terms of fluid mechanics? How do you find the maximum area of a right triangle? Do you have a proof of the equation e^(i*Pi) + 1 = 0? Assume a particle moves on the x-axis according to the formula x = t^ 3-6t^2+9t+5. Find: the velocity when t = 3... Are there any other ways of illustrating the link between Riemann sums and antiderivatives? How can I integrate sin(x)sin(2x).dx ? Can you give some real life problems involving Newton's cooling method, logistic growth, and exponential growth and decay? How can I determine the position of a point P on the line y = 2 such that the line segments OP and PQ and arc OQ of the curve x = y^2 enclose minimum total area? Is there a standard formula I can use to know where the center of mass of a semicircle is? How can I find out when a plane, whose position approaching an airport is described parametrically by P_t = (1000,500,900)+ t[-100,-50,-90], will be closest to the traffic control center, located at (24,11,13)? If I have a rectangular card of size AxB, how can I find the size of the square cutout that maximizes the volume of the box produced when the edges are folded up? How can I find the derivative of y = e^x, using the definition of the derivative? How can you calculate the amount of work needed to lift a bucket of water to the top of a well if there is a hole in the bucket causing water to leak out at a rate proportional to the height of the water in the bucket? |
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