A Course in Mathematics: Algebraic equations, functions of one variable, analytic geometry, differential calculusGinn, 1907 - Mathematics |
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Common terms and phrases
a₁ abscissa angle approaches zero asymptotes axis of x b₁ bisects c₁ C₂ chord circle x² cissoid coefficients conic conjugate diameters constant coördinate axes curve derivative determined differentiation directrix distance dx dx dx dy dx² dy dx ellipse equal equation ax² factor Find its equation Find the equation Find the locus Find the points fixed point foci focus formulas function given Hence hyperbola increment integral limit line joining locus m₁ negative ordinate origin P₁ parabola y² parallel to OX parametric equations passes perpendicular points of inflection points of intersection polar polynomial positive Prove quadrant quadratic equation quantities r₁ r₂ radius real roots represents respect satisfy Show slope Solving straight line strophoid Substituting tangent theorem triangle velocity vertex whence x₁ y₁
Popular passages
Page 243 - A point moves so that the sum of the squares of its distances from the points (0, 0), (1, 0) is constant.
Page 156 - Show that the locus of a point which moves so that the sum of its distances from two fixed straight lines is constant is a straight line.
Page 69 - ... 72. Prove analytically that the perpendicular bisectors of the sides of a triangle meet in a point. 73. Prove analytically that the perpendiculars from any two vertices of a triangle to the median from the third vertex are equal.
Page 176 - Then the expressions Lim x = a and x == a have the same significance. The expression Lim /(z) = A x==a is read "The limit of /(ж), as x approaches a, is A" 69. Theorems on limits. In operations with limits the following propositions are of importance : 1. The limit of the sum of a finite number of variables is equal to the sum of the limits of the variables. We will prove the theorem for three variables ; the proof is easily extended to any number of variables. Let X, Y, and Z be three variables,...
Page 178 - Any divided difference of the sum (or difference) of two functions is equal to the sum (or difference) of the divided differences of the individual functions.
Page 139 - The ellipse. An ellipse is the locus of a point the sum of the distances of which from two fixed points is constant. The two fixed points are called the foci. Let them be denoted by F and F
Page 177 - The proof is left for the student. 4. The limit of the quotient of two variables is equal to the quotient of the limits of the variables, provided the limit of the divisor is not zero.
Page 157 - A point moves so that the square of its distance from the base of an isosceles triangle is equal to the product of its distances from the other two sides. Show that the locus is a circle. 50. Prove that the two circles z2 + y2 + 2 G,z + 2 Ftf + Cj = 0 and x2 + y...
Page 180 - The derivative of the product of two functions is equal to the first function times the derivative of the second plus the second times the derivative of the first. (4) 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, divided by the square of the denominator.
Page 176 - Lim X+ Lim Y+ Lim Z. 2. The limit of the product of a finite number of variables is equal to the product of the limits of the variables. Consider first two variables X and Y, such that Lim X= A and Limy=A As before, we have X=A + a and Y=B + b.