Once approved, you'll enjoy N CapitalOne $0 FRAUD LIABILITY if ntinum PRSAT STD U.S. POSTAG PAID VALPAK ACCESS OTUD MA SY 217 Families of Implicit Curves LABORATORY PROJECT (b) Illustrate part (a) by graphing the ellipse and the normal line. 69. Show that the ellipse x2/a2 +y/b2 1 and the hyperbola xiA2y2/B2= 1 are orthogonal trajectories if A2 < a2 and a2 b2= A2 B2 (so the ellipse and hyperbola have the same foci). 2 where the slope 75. Find all points on the curve x2y2 xy of the tangent line is -1. 76. Find equations of both the tangent lines to the ellipse 36 that pass through the point (12, 3). 70. Find the value of the number a such that the families of (x c)andy = a(x + k)/3 are orthogonal 77. (a) Suppose f is a one-to-one differentiable function and its inverse function f is also differentiable. Use implicit curves y = trajectories. 71. (a) The van der Waals equation for n moles of a gas is differentiation to show that 1 ( na (V V2 (fx) = nb) nRT f'f (x)) P+ where P is the pressure, V is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a andb are positive constants that are characteristic of a particular gas. If T remains constant, use implicit di fferentiation to find dV/dP. (b) Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of V = 10 L and a pressure of P = 2.5 atm. Use = 3.592 L2-atm/mole2 and b= 0.04267 L/mole. provided that the denominator is not 0. (b) If f(4) 5 and f'(4) , find (f)'(5) 78. (a) Show that f(x) = x + e is one-to-one. (b) What is the value of f(1)? (c) Use the formula from Exercise 77(a) to find (f)'(1). 79. The Bessel function of order 0, y J(x), satisfies the differential equation xy" y' +xy 0 for all values of x and its value at 0 is J(0)= 1 (a) Find J'(0). (b) Use implicit differentiation to find J"(0) a 72. (a) Use implicit differentiation to find y' if 80. The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 4y25. If the point (5, 0) is on the edge of the shadow, how far above the x-axis is the lamp located? x2 xy y2 + 1 = 0 (b) Plot the curve in part (a). What do you see? Prove that what you see is correct. CAS (c) In view of part (b), what can you say about the expression for y' that you found in part (a)? 73. The equation x2-xy + y2 3 represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel. 3 -5 x2+4y2 5 74. (a) Where does the normal line to the ellipse x2-xy y23 at the point (-1, 1) intersect the ellipse a second time? FAMILIES OF IMPLICIT CURVES CAS In this project you will explore the changing shapes of implicitly defined curves as you constants in a family, and determine which features are common to all members of the LABORATORY PROJECT 1. Consider the family of curves c[(y + 1)'(y +9 ) - x] y2 2x2(x+8) section there are. (You might have to zoom in to find all of them.) 5 and c= (a) By graphing the curves with c 0 and c = 2, determine how many points c 10 to your graphs in part (a). Wha (b) Now add the curves with c = notice? What about other values of c?
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
I need help with question 73 in Section 3.5, page 217, of the James Stewart Calculus Eighth Edition textbook.
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