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 TOPICS This page:   calculus    Search   Dr. Math See also the Internet Library:   sv calculus and   mv calculus COLLEGE Algorithms Analysis Algebra    linear algebra    modern algebra Calculus Definitions Discrete Math Exponents Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean Imaginary/Complex   Numbers Logic/Set Theory Number Theory Physics Probability Statistics Trigonometry Browse College Calculus Stars indicate particularly interesting answers or good places to begin browsing. Minimum Distance Between a Line and an Ellipse [05/30/2000] How can I find the extremum from a point on the line x+y=4 to a point on the ellipse x^2+4y^2=4? Is there a general way to solve such problems in a 3-dimensional space? Minimum Distance to an Ellipse [06/10/1999] What is the minimum distance between a point inside or outside an ellipse and the ellipse? Minimum Distance Using Lagrange Multipliers [11/17/1999] How can I find the minimum distance from (0,0,c) to the cone z^2 = (x^2/a^2)+(y^2/b^2)? Min, Max of 3-Variable Function [8/16/1996] Find all the critical points and determine their nature for the function z = x^3 - 6xy + y^3. Monkey Climbing a Chain [09/22/1999] A 20-lb. monkey is attached to a 50-ft. chain that weighs 0.5 lb. per (linear) foot... Monotone Convergence Theorem [8/16/1995] I assert that the Monotone Convergence Theorem can only help us conclude that the sequence of partial sums converges if we can also show that the sequence of partial sums is bounded above. Is this true? Moving Particle [07/14/1999] 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... Multidimensional Calculus and Vector Geometry [02/09/1999] The depth of iron ore can be approximated by a plane... Multivariable Limits [12/01/1999] How can I find a function f(x,y) such that lim(y->0)lim(x->0) f(x,y) = lim(x->0)lim(y->0) f(x,y)? Nested Sums [6/17/1996] If the sum from j = 1 to infinity of 1/(j^3) equals p, and the sum from k = 1 to infinity of 1/(k^2) equals q... Newton-Raphson Method [06/24/2009] Are there any equations that cannot be solved using the Newton-Raphson method, regardless of the initial estimate? Non-homogeneous Differential Equation Solutions [02/26/2001] Why do we need to include the solution to the homogeneous equation as part of the solution to a non-homogeneous equation? Since the solution to the homogeneous part is by definition zero, can't we ignore it? Normal Distribution Curve [06/12/1997] What is the integral of e^(-x^2)? Odd Bernoulli Numbers Must Be 0 [02/16/2003] Take the second derivative of t/exp(t) -1 and find that it is an even function; show that an even function, when expressed as a Taylor Series, has only even powers of t; draw the conclusion that the odd Bernoulli numbers have to be 0. On the Complex Differentiability of the Hyperbolic Secant [09/24/2010] A student wonders if sech(z) is complex differentiable, and where. Picking up on the student's familiarity with the Cauchy-Riemann equations, Doctor Jordan uses trigonometric identities to examine the function's real and imaginary parts separately, and reveals the conditions under which the function is holomorphic. Optimization: Minimum Area [11/07/1997] 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? Optimizing Advertising [10/15/2003] Doubling the amount spent on advertising increases total income by 20 percent. What is the optimum level of advertising? Ordinary Differential Equation, Second Order [10/26/2002] I've been looking for solutions to the differential equation y'' = x^2 / y^2. Partial Derivatives [03/24/2001] What sense do partial derivatives make in the case where u is given as a function of two variables, say x and y...? Partial Derivatives and Gradient [12/12/1998] What is the relation between partial and directional derivatives and gradients? What do they mean? Particular Solution of Differential Equation [11/03/2007] Set up the appropriate form of a particular solution y_p, but do not determine the coefficients, of the ODE y' + 4y = 3xcos(2x). Phase Difference of Sampled Waves [05/27/2002] Given two sine waves that have been sampled, find the lag/lead of the second with respect to the first. Pi as the Sum of Rational Numbers [02/20/2003] Since pi/4 is the sum of rational numbers (i.e., 1 - 1/3 + 1/5 - 1/7 + ...), doesn't pi have to be rational as well? Polar Coordinates for Velocity and Acceleration of a Particle [03/11/1997] 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. Polar Equation of a Ship's Course [05/30/2000] A ship is following a course along a curve whose polar equation is r = 6 cos(angle) km... Polynomial Function with Matching Maxima [11/17/2008] Let c(t) be a polynomial of degree 2n for which all roots are unique, exist and are located within [0, 1]. I would like to find all the roots such that all maxima of the polynomial are equal. Powers of Matrices, Putzer's Method [03/09/1997] If t is a constant and A is an n x n matrix, how do you evaluate e^ (tA) without using a Taylor Series expansion? Primer on Fourier Transforms [01/27/2003] How does the Fourier transform extract individual sine waves? Proof for the Radius of Curvature [08/17/2003] The equation of a circle of radius R centered at the origin is x^2+y^2 = R^2. Demonstrate that the radius of curvature is equal to R. Proof of Cosine of 36 Degrees [11/26/2001] Prove: cosine {(pi)/5} = {1+5^(1/2)}/4. Proof of Largest Element by Epsilon, Delta — and Some Hidden Choices [09/18/2014] An undergraduate struggles to follow his professor's calculus proof. Doctor Peterson separates the arbitrary choices from the strategic ones. Proof of Roots of Odd Degree Polynomial [05/10/2000] How can I prove that any real polynomial mapping of odd degree has a root? Proof of Stirling's Approximation [03/09/2006] Can you prove that lim ((e^n)(n!)) / ((n^n)(n)^1/2 = (2pi)^1/2 ? Proofs of e [03/21/2002] I found out that the definition of e is: e = lim, as n approaches infinity (1+1/n)^n. Is there a proof for this or for 1+2(e-1/n)? Proof that 2 Equals 1 Using Derivatives [05/10/2000] If d(x^2)/dx = d(x+x+...+x)/dx, then 2x = 1+1+...+1 = x. What is the error in this "proof"? Proving Limits at Infinity Using the Formal Definition [04/11/2004] We are required to find a limit (informally), then prove that our answer is correct by using the formal definition of limits. I'm not sure I'm on the right track. Proving the Derivative of e^x Is e^x [02/20/2006] What is the delta x proof of (dy/dx)e^x = e^x? Ptolemy's Theorem Shows Compound Angle Formula [7/7/1996] How can the Ptolemy's formula be extended to show the Sin(A+B) and the Cos(A+B) formulas? Quaternion Numbers [01/23/1997] How do you divide quaternion numbers? Can quaternion math be extended to transcendental functions? Questions about Fourier Series [11/18/2000] Must the integrals defining the coefficients of a Fourier series exist? If so, what type of integral (Riemann, Lebesgue, etc.) is considered, and how is convergence of the series determined? Page: []

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