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 TOPICS This page:   calculus    Search   Dr. Math See also the Internet Library:   calculus HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Calculus Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Chain rule.     Maximizing the volume of a box.     Maximizing the volume of a cylinder.     Volume of a tank.     What is a derivative? Rate of Change Constant? [03/20/2002] The rate of change of e^x is e^x. Does this mean that the rate of change is constant? Why are sinx,cosx,... and sinhx,coshx,.... similar? Rate of Change of a Function [6/10/1996] Show that the expression for the rate of change can also be given by -cos x cot^2 x. Rate of Change of Angle [8/12/1996] Find the rate at which angle BXA is changing in radians per second... Rate of Change of Radius of a Cone [03/07/1999] 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. Ratioing Errors [01/07/2002] How do you calculate the error bars for ratios? Rational Function [04/03/1997] Given the rational function y=r(x)=(4x^6-x^4)/(x^5+5) describe its end behavior... Raytrace of a Star Sapphire [12/11/1996] How do I find the rotations necessary to create a raytrace of a star sapphire so that the star always faces the camera? Rectification of a Quartic [12/02/1997] 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? Recurrence Relation Resolution [11/25/2013] A student struggles to determine the limit points and explicit formula for a recurrence relation complicated by powers and other operations. Exploiting derivatives and ratios, Doctor Vogler shows the way. Related Rates [08/05/1997] 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. Related Rates [03/11/1998] The rate of change of the edges of a cube, given the rate of change of its volume. Related Rates [03/01/1999] 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... Related Rates [10/16/2003] A related rate problem involving a 2 cm long hair lying on a spherical balloon as the balloon is inflated. Related Rates and a Decreasing Radius [11/18/1998] 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. Related Rates - Clock Hands [04/05/1999] How fast is the distance between the tips of the hands on a watch changing at one o'clock? Related Rates Problems [01/15/1999] A plane flying horizontally at an altitude of 1 mi and a speed of 500 miles/hour... The Relation of the Speed of a Boat to the Water Line [2/27/1996] What is the mathematical formula or equation for calculating the speed of a boat as it relates to the water line of the boat.? The Relationship of the Tangent to a Curve and the Area under a Curve [10/25/1995] 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? Relative and Absolute Extrema of a Function [01/07/2004] What is the difference between the absolute extrema and the relative extrema in calculus? Relativity [05/22/2002] How do you demonstrate algebraically that [the following expressions] are proportional to each other even when they describe 'events' that are not on the expanding wavefront of light? The Riemann Sum of Sin(x) [02/07/2001] How do you evaluate the integral of sin(x) from 0 to pi by Riemann sum? Riemann Sums and Antiderivatives [07/11/1999] Are there any other ways of illustrating the link between Riemann sums and antiderivatives? Riemann Sums and the Integral [01/18/1999] Why does the limit of Riemann sums turn out to be the integral? How is that derived? What is behind it? A Rumor Spreads [04/01/1999] 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... Sea Navigation: Great Circles [11/4/1994] 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. Second Derivative of a Function With Absolute Values [02/27/1998] Let f(x) = |x|^3. Find f"(0). Second Order Differential Equation [11/04/2001] By letting p = dy/dx, solve y(d2y/dx2) = 2 (dy/dx)2. Sequences and Series [4/27/1996] Determine the number of arithmetic means between two numbers whose sum is 5.5. Setting Interest Rate to Maximize Profit [7/25/1996] 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... Shakespearean Differential Equations [10/27/1998] Can you help me solve these differential equations: dG/dt=aJ-bG and dJ/ dt=cG? Shortest Distance between a Point and a Circle [1/22/1996] Find the shortest distance between the point (4,5) and the circumference of x^2 + y^2 = 9. Simplifying and Solving a Natural Log Problem [11/15/1995] Integrate ln(r/a)dr - the limits are a to b. Is this possible, or is there some table for it? Simpson's Rule [04/12/1997] 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. Simpson's Rule - Alternating Pattern [09/20/1997] 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? Simpson's Rule for Cubics [10/23/2000] Why is Simpson's rule exact for cubic polynomials? Simpson's Rule/Trapezoid Rule [02/01/1998] How is Simpson's rule derived and why it is a better approximation of the integral than the midpoint (rule?) or trapezoidal method? Sketching a Function [07/06/1998] Can you help me piece a function together so that the following hold? It is increasing and concave up on (-infinity, 1) ... Sketching a Graph Given Information about Its Derivatives [07/30/2005] We've been learning how to analyze a function by using the first and second derivatives to test if the graph is increasing/decreasing and concave up/down. But now we have to sketch the graph given some information about the derivatives and some specific points on the graph. Slope of 3-Dimensional Equations [11/12/1997] Is it possible to find the slope of three-dimensional equations? Slope of root x. . . [12/14/1994] How do I find the slope of: root x = -ln (xy) at (4,2) ? Page: []

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