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Graphs and Functions

Straight Line and slope

A) Explain what is the y-intercept and which letter represents it in the equation y = mx +b. b) If a and b are positive numbers, what is all the information we can deduce about the relationship between the lines: L1: y = - ax + b and L2: y = (1/a)x + b?

Polar Function : Graphing an Ellipse

Graph an ellipse as a polar function with a focus at the pole and parameterized by the eccentricity e and the distance d between the focus and a vertical directrix. ----------------------------------------------------- Please show me how step-by-step on how you would graph this. Thanks

Polar function

Graph an ellipse as a polar function with center at the pole and parameterized by the lengths of the semi-major and semi-minor axes. Can someone please show me step-by-step on how to do this?

Functions : Radius of Convergence and Approximations

Please see the attached file for the fully formatted problems. Let f be the function defined by f(x) = sigma (starting at n=1 ending at infinity) xnnn /(3n n!) for all values of x for which the series converges. a) Find the radius of convergence of this series b) Use the first three terms of this series to approximat

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


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}.

Findin 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.

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