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    Graphs and Functions

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    Algebra : Graphing, Distance between Points and Equations of Lines

    Please see the attached file for the fully formatted problems. Can you please help me with the following circled problems? Page 187 1. a) 12, b) 14, c) 16, d) 18 (Check for all four of our symmetries SY, SX, SO, SI; consult in WEEK7 NOTES, in COURSE CONTENT. Practice graphing these using the downloaded graphing utility Gr

    Duality and Saddle Points

    Please see the attached file for the fully formatted problem. I am working on a way to find the minimum of a function J(Y) with the constraint set C = {X E R^N such that gt(x) =<0 Vi E [1,n]} Let L(Y, mu) = J(Y) + SIGMA m --> i = 1 muigi(Y) be the lagrangean of the problem. I am having trouble proving the following

    Lines through Non-Colinear Points

    Given three points, there is one line that can be drawn through them if the points are colinear. If the three points are noncolinear,there are three lines that can be drawn through pairs of points. For three points, three is the greatest number of lines that can be drawn through pairs of points. Determine the greatest number of

    Conic graph

    Identify and sketch the graph of the conic described by the quadratic equation x^2 + 4xy + y^2 - 12 = 0. Do this by writing this equation in matrix form; then change the equation to a sum of squares of the form x'^T Dx' where D is a diagonal matrix.

    Critical Point : Non-Degenerate

    Please see the attached file for full problem description. Show that f(x) = x1x2 + x2x3 + x3x1 has a non - degenerate critical point at x = 0 and describe the shape of f as concretely as possible.

    Division of functions

    Divide 6x^3 - 29x^2 + 36x - 4 by x-2 in the traditional way and find the quotient. Note that the remainder will come out to be zero

    Formulas

    If g(x)=x^2+1, find the formulas for g^3(x) and (gogog)(x).

    Cantor's Diagonal Process

    I am trying to use Cantor's diagonal process to prove that there are uncountably many functions from N into the set {e, pi}.

    Quadratic Function : Word Problem

    Here is the word problem: The cables supporting a straight-line suspension bridge are nearly parabolic in shape. Suppose that a suspension bridge is being designed with concrete supports 160 ft apart and with verticle cables 30 ft above road level at the midpoint of the bridge and 80 ft above the road level at a point 50 fee

    Graphing a Piecewise Function

    Sketch the graph of each function.  Be sure to label three points on the graph. If f(x) =x^3 if x <0 and 3cubed + 2 if x &#8805; 0 Find: (a) f(-1) (b) f(0)   (c) f(1)

    Finding the Equation of a Reflecting Line

    Determine if the following orthogonal matrix represents a rotation or a reflection of the plane with respect to the standard basis. Find the equation of the reflecting line. - - |3/5 4/5 | |4/5 -3/5 | - -.

    Graph Theory.

    Show that it is impossible for an odd number of people in a group to each know exactly 2k+1 other people in the group for any integer k.

    Vector Problems : Plane Angle and Line of Intersection

    Please see the attached file for the fully formatted problems. The plane &#928; 1 has equation x + 2y - z = 5 and the plane &#928; 2 has the equation 3x + y + 2z = 10. (a) Find the angle between the planes. (b) Find the equation of the line of intersection of the planes.

    Behaviour Cubic Functions

    Investigate the cubic functions of f(x) = ax^3 + bx^2 + cx + d which will pass through the points of A = (1,4) B = (2,2) C = (4, 1.5) Now explore the effect of 'd' on the behaviour of the cubic functions. Identify a value of 'd' that gives a cubic function which closely matches the quartic function that passes through these

    Vectors : Force and Line Equations

    (1) A force F of magnitude 6 in the direction i - 2j + 2k acts at the point P = (1,-1, 2). a. Find the vector moment M of F about the origin. b. Find the components of M in the direction of the (positive) x - axis, y -axis and z -axis. c. Find the component of M about an axis in the direction

    Functions: Mapping

    For the functions f defined below, determine which are 1:1, onto or both. 1) f: R onto R, f(x) = |x| 2) f: R onto R, f(x) = x^2 + 3 3) f: R onto R, f(x) = x^3 + 3 4) f: R onto R, f(x) = x(x^2-4) 5) f: R onto R, f(x) = |x| + x 6) f: N onto N, f(x) = x + 1 7) f: N onto NxN, f(x) = (x,x) 8) f: NxN onto N, f(

    Two Segment Graph : Equation of Tangent and Calculation of Points

    Please note: On the attached graph the scale is that each line represents one unit. Please show all work, thanks!! The graph of F consists of a semicircle and two line segments as shown (please see the attachment). Let g be the function given by: g(x)= def.integral from 0 to x f(t)dt. a Find g(3). b Find all value