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# Graphs and Functions

### Polar coordinates of the point

Rectagular coordinates of a particular point are x=7, y= negative 24. find the polar coordinates of the point

### Find the vector equation for line of intersection of two planes.

Consider the planes 1x + 4y +3z = 1 and 1x + 3z = 0 (A) Find the unique point P on the y-axis which is on both planes. (0,1/4 ,0 ) (B) Find a unit vector with positive first coordinate that is parallel to both planes. .94869 i + 0 j + -.3162 k (C) Use parts (A) and (B) to find a vector equation for the line

### Parametric Equation of a Line and Point of Intersection

A) Find the parametric equations for the line through the point P = (3, -4, 0) that is perpendicular to the plane 2x + 0y+ 5z = 1 Use "t" as your variable, t = 0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. x = 3+2t y = -4 z = 5t (B) At what p

### Functions : Domain, Intercepts, Symmetry, Asymptotes and Graphing

Given the function R(x) = X^2 + x -12 / X^2 - 4 1. Give the domain 2. Give the X - intercepts 3. Give the Y - intercepts 4. Does it have symmetry with respect to the Y-axis, the origin or neither? 5. Give the vertical asymptotes 6. Give the horizontal asymptotes 7. Graph the function by dividing the axis and te

### Average Cost: Rational Function and Graph

A company manufacturing surfboards has fixed costs of \$300 per day and total costs of \$5,100 per day at a daily output of 20 boards. (A) Assuming that the total cost per day, C(x), is linearly related to the total output per day, x, write an equation for the cost function. (B) The average cost per board for an output of x boar

### What is the difference between functions and relations?

The question ask - in the real world, what might be a situation where it is preferable for the data to form a relation but not a function? Also, if the variables in an equation were reversed, what would happen to the graph of the equation? This was the example I was given, how would the graph of y = x² relate to the graph o

### Estimating Roots by Newton's Method and Rate of Convergence

Please see the attached file for the fully formatted problem. 19. Show that x = tan^-1(x) has a solution alpha. Find an interval [a,b] containing alphasuch ythat for every x E [a,b] the iteration xn+1 = 1 + tan^-1(xn) n>=0 will converge to alpha. Calculate the first few iterates and estimate the rate of convergence.

### Depreciation: linear functions, graphs, slope

Office equipment was purchased for \$20,000 and is assumed to have a scrap value of \$2,000 after 10 years. If its value is depreciated linearly (for tax purposes) from \$20,000 to \$2,000: (A) Find the linear equation that relates value (V) in dollars to time (t) in years. (B) What would be the value of the equipment after 6 yea

### Simple interest: linear functions, graphs and slopes

The simple interest formula says that if \$1,000 is invested at 7.5% (r=0.075), then A=75t+1000, t>=0. (A) What will \$1,000 amount to after 5 years? After 20 years? (B) Sketch a graph of A=75t+1000 for t between 0 and 20. (C) Find the slope of the graph and interpret verbally.

### Use givrn info to draw graph

Use the given information to draw the graph of f(x) on an interval near the indicated point (x=3). If there is not enough information given, then state this fact. f(3) =2 f ' (3) = DNE f ' (x) < 0 for x < 3 f ' (x) > 0 for x > 3 f " (x) < 0 for x < 3 f " (X) < 0 for x>3

### Functions / Graphs : Domain, Range, Distance Between Points

1. Find the domain and range of the function: f(x)=6xsquared+4 2. Find the distance between the two plotted points;(-5,2) (4,-4) 3. Using graph of f(x)=x squared as a guide, graph the function; g(x)=(x-3)squared+4 _____________________________________________________________________ 2

### If r = xi + yj + zk and R denote &#9553;r&#9553;, Verify (1) &#9660;R^n = nR^(n-2) r (2) &#9660;×(R^n r) = 0 Verify (1) &#9660;*(F + G) = &#9660;*F + &#9660;*G (2) &#9660;×(F + G) = &#9660;×F + &#9660;×G

Real Analysis Gradient, Divergence and Curl (I) If r = xi + yj + zk and R denote &#9553;r&#9553; Verify (1) &#9660;R^n = nR^(n-2) r

### Conditional Mass Function

#40. Two dice are rolled. Let X and Y denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of Y given X = i, i = 1, 2, 3, 4, 5, 6. Are X and Y independent? Why? (Question is also included in attachment)

### Polynomials, Equations, and Radicals: Please respond in excel format. See the attached file for complete description.

43. (3y+2) (2y^2-y+3) *44. (4y+ 3) (y^2+3y+1) 46. (m^3-4mn^2)(6m^4n^2-3m^6+m^2n^4) *90. (5-6y)(3y^2-y-7) *91. Office Space. The length of a professor's office is x feet, and the width is x+4 feet. Write a polynomial that represents the area. Find the area if x=10ft. 100. If a manufacturer charges p dollars each

### Continuous real value function

34.12 Show that if f is a continuous real-valued function on [a,b] satisfying b &#8747; f(x)g(x) dx = 0 for every continuous function g on [a,b], then f(x) = 0 for a all x in [a,b].

### Determination of Horizontal Asymptotes

Please address the following: How do you determine horizontal asymptotes, if any, given the equation for a rational function?

### Determine vertical & horizontal asymptotes & sketch graph

Determine the vertical and horizontal asymptotes and sketch the graph of the rational function F. Label all intercepts and asymptotes. F(x) = 1__ x + 4

### Examples of Functions : Exponentials, Graphs and Usefulness

1. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph of both the functions and post to the discussion forum. Discuss these functions and their graphs with your classmates. 2. Given the values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, x, and y, form each of the follow

### Graph of polar equation

Inspect the four given graphs, which are plotted in polar coordinates (and shown in an attached .doc file), and choose the graph that corresponds to the polar equation r = 6 - (cos theta).

### Lipschitz functions

A function f:A->R is called Lipschitz if there exists a bound M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x, y belong to A. Geometrically speaking a function f is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of f. a- Show that if f:A->R i

### Graphical Representation of Linear Equation

Solve the following questions on graphical representation of linear equations. Your response should include the graphs for each of these questions. 1. Plot the graph of the equations 3x - 2y = -6 and 4x + 2y = -11 and interpret the result. 2. Plot the graph of the equations 3x - 5y = 14 and -6x + 10y = 9 and interpret

### Divergence Criterion - Corollary

Corollary:(Divergence criterion for function limits).let f be a function defined on A, and let c be a limit piont of A. If there exist two sequences (x_n) and (y_n) in A with x_n not =c and y_n not =c and lim x_n=lim y_n=c but lim f(x_n) not = lim f(y_n), then we conclude that the functional limit lim f(x) as x->c does not exist

### Graphs : Connectedness and Cycles

13. Let G be a connected graph with (please see the attachment). Prove that G contains exactly one cycle.

### Differential Equations - Cross-Eyed Heart

27. y'' + 25y = sin(4t), y(0) = 0, y'(0) = 0. Plot the component curves and the orbit, the latter for the rectangle |y|< 0.25 and |y'|< 1. Any surprises? 28. Hearts and Eyes: Find a solution formula for y'' + 25y = sin&(wt), where w is not equal 5. Plot the solution curve of the IVP with y(0) = y'(0), where w=4. Plot t

### Finding curves

Find the curve that passes through the points (3, 2) and has the property that if the tangent line is drawn at any point P on the curve, then the part of the tangent line that lies in the first quadrant is bisected at P.

X^2 + 1 = 6x

### Computing 'Ambiguity' Equation

1. Calculate the ambiguity function of a signal with an envelope u(t) = Bexp(-t^2)T^2). What should be the value of B that will make the signal of the unit energy? 2. Calculate the ambiguity of a signal with a complex envelope u(t) = Bexp(-t^2/T^2)exp(jpikt^2). Note that this is the same signal as above, except for the additi

### Approximation of Functions

Let f(x)=invertedCOS(x) for EQUATION1 (the principal branch of EQUATION2) Find the polynomial of degree two EQUATION3 which minimizes EQUATION4. *(Please see attachment for all equations)

### Proof f(x)=x

Suppose that F is a continuous function on [0,1] and f(x) is in [0,1] for each x. Prove that f(x)=x for some x.

### Double False Position

Compute by double false position, to three decimal places, the root of: x^3 - 36x + 72 =0 that lies between 2 and 3. * He gave us the following formula to use when using double false position: x(subscript 2)y(sub1)-x(sub1)y(sub2)/y(sub1)- y(sub2) I couldn't figure out how to get subscripts to show up! PLEASE SHOW EACH S