Elements of the Differential and Integral Calculus |
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Common terms and phrases
a²y angle approaches the limit approaches zero Assume asymptote ax² axis b²x² circle concave upwards constant convergent coördinates corresponding cos² critical value curvature curve d³y denoted derivative dx dx dx dy dx dx² dy dy equal EXAMPLES Find the equation formula fraction function f(x given gives max Hence hypocycloid increases without limit increment independent variable indeterminate form infinite integral length limit zero logarithms maxima and minima minimum values parabola parameter parametric equations plane point of inflection positive radius rational rational function Rolle's Theorem sec² Second step Show sin² sin³ slope Solution solving Substituting subtangent surface tangent line Taylor's Theorem Theorem Third step velocity x₁ ди ду дх
Popular passages
Page 50 - The derivative of the quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
Page 37 - x". It is not "dy" divided by "dx" or "d" multiplied by "y" divided by "d" multiplied by "x." In precise mathematical terms a derivative of a function is the limit of the ratio of the increment of the function to the increment of the independent variable when the latter increment varies and approaches zero as a limit.
Page 132 - A Norman window consists of a rectangle surmounted by a semicircle. Given the perimeter, required the height and the breadth of the window when the quantity of light admitted is a maximum.
Page 428 - The differential equation Mdx + Ndy = 0 is said to be homogeneous when M and N are homogeneous functions of x and y, and are of the same degree.
Page 128 - Assuming that the strength of a beam with rectangular cross section varies directly as the breadth and as the square of the depth, what are the dimensions of the strongest beam that can be sawed out of a round log whose diameter is d ? Solution.
Page 173 - Arithmetically it is done, as before shown, by involving the numerator for a new numerator, and the denominator for a new denominator.
Page 49 - C") x L. Proof. The slant height of the frustum of the circumscribed pyramid = L, (Why ?) then S' = i (P + P) x L. (Why ?) [To be completed by the student. HINT. — Use Theorem of Limits.] NOTE. — It can be shown that the limit of the sum of a finite number of variables is equal to the sum of their respective limits. 668. COR. The lateral area of a frustum of a cone of revolution is equal to the circumference of a section equidistant from its bases multiplied by its slant height. For C=2irR, and...
Page 107 - ... function. The derivative of the derivative of a function is called the second derivative of the function ; the derivative of the second derivative is called the third derivative ; and so on.
Page 130 - Prove that a conical tent of a given capacity will require the least amount of canvas when the height is V2 times the radius of the base. Show that when the canvas is laid out flat it will be a circle with a sector of 152° 9
Page 17 - What is the ratio of their radii ? of their apothems ? of their perimeters ? of their areas ? 5. The diameters of two circles are d and d' respectively. What is the ratio of their radii ? of their circumferences ? of their areas ? 6. If the number of sides of a regular inscribed polygon is indefinitely increased, what is the limit of the apothem ? of each side ? of the perimeter ? of the area ? of the angle at the center ? of each angle of the polygon ? 7. How do you find the area of a regular polygon...