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Graphs and Functions

Singular Point : Pole and Residue

2. Show that the singular point of each of the following functions is a pole. Determine the order m of that pole and the corresponding residue B. {please see attachment for functions} Please specify the terms that you use if necessary and clearly explain each step of your solution.

Functions : Maximizing Profit, Diminishing Returns and Maximizing Volume

1. The demand for a product in dollars is given by p(x) = 53/(x)^1/2 Fixed cost are $608 and the cost to produce each item is $0.53. Find the production level of x that maximizes profit within the range of 0<(or equal to)x< (or equal to)7530. 2. An efficiency study of the afternoon shift (12:00-4p.m.) at a factory shows th

Parametric Equations : Canonical Form, Values of t

Consider the line with parametric equations, x = 2t + 3 and y = -4t + 1 a) Find the equation of the line in non-parametric form. b) Find the values of the parameter t which correspond to the points A (3, 1) and B (7, -7) on the line. c) Write down the range of values of t which, together with the given parametr

Intercepts, Vertex, Line of Symmetry and Image Set

This question concerns the parabola which is the graph of the function: f(x) = [1/4(x-2)^2] -1 a) Explain how the graph of the parabola can be obtained from the graph of y =x[squared] by using appropriate translation and scalings. b) Using your answer to part (a), or otherwise, write down the coordinates of the vertex of th

Length of Curve / Length of Arc of Curve

2. Find the length of the arc of the curve y=f(x) on the intervals given: {see attachment} 3 and 4. Find the length of the curve defined by: {see attachment}, between the points: {see attachment} 5. Find the surface area generated when the graph of each function on the interval is revolved about the x-axis. (Give answer to

Polynomial functions, inverses, half-life, investments

Please find the attached. 1) Fit a polynomial Function f(x) to the graph. The scale on the x-axis is 1 and the scale on the y-axis is 5. The point (1,12) is on the graph, Assume that if the graph appears to cross the x-axis at a mark, it really does. (1,12) 3 2 1 4 2) Noise level in decib

Fixed Point : Mean Value Theorem

A number (a) is called a fixed point of a function (f) if f(a)=a. Prove that, if f'(x) does NOT equal 1 for all real numbers (x), then f has at most one fixed point.

Inequalities and Line Equations (5 Problems)

1) A line L1 has a slope of -7/10. Determine whether the line through (5,3) and (-2,-7) is parallel or perpendicular to L1 2) Graph: x+y=4 3) What is the slope of the line 6x+2y=48 4) Graph: Y &#8804; x-1 5) Graph: 3x &#8804; 4y

Graphing Inequities

I don't need to be shown how to graph the below, But I do need help knowing what to graph. 1. Graph: Y &#8804; 3x-6 2. Graph: -4x &#8805; 5y 3. Graph: -8x &#8804; 2y

Graphing Questions

I have a lot of the attached problems to do. I know how to put points on the graph (for example (4,3)) but I am not sure how to get the information I need to graph on the attached. So I don't need to see these problems actually graphed out, just how to get to the stage before graphing. One: Graph: X &#8805; 1 Two:

Parallel Lines

Find the pair of parallel lines: 1: -y=-x+2 2: -2y-2x=2 3:-2x+2y=2 Not sure how to do the above problem.

Continuity Proofs

1. Prove that any function f: Natural Nos. --> R is continuous (N --> R). 2. Prove that if a function f: I --> R is continuous and I is an interval then the image f(I) is an interval.

Equations of Lines, Slopes, Intercepts and Word Problems (15 Problems in Total)

Please see the attached file for the fully formatted problems. 1. Find a linear function perpendicular to the function y= -5x + 12 at the point (2,5) in standard form, point slope form, and slope-intercept form. The orginal line is y = -5x + 12 (slope is -5), so the perpindicular line will be y = 1/5x + ? 5 = (1/5)2 + ?.

Find the Vector Equation for the 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 of

Vector Equation for Line

Find a vector equation for the line through the point P = (-4, -1, 1) and parallel to the vector v = (1, 4, 3). Assume r(0) = -4i -1j +1k and that v is the velocity vector of the line.

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


What is the y coordinate of the point on the curve y = 2x^2 - 3x at which the slope of the tangent line is the same as that of the secant line between x = 1 and x = 2?

Differentiability of Functions

Let f(x) be differentiable for a < x < b. Which of the following statements must be true? A. f is increasing on (a,b) B. f is continuous on (a,b) C. f is bounded on [a,b] D. f is continuous on [a,b] E. f is decreasing on [a,b]

Differentiability of Functions

5. Let f be twice differentiable on (a,b). If g is an antiderivative of f" on (a,b), then then g ' (x) must equal : A. f(x) B. f(x) C. f"(x) D. f(x) + C, for some C not necessarily 0 E. f"(x) + C, for some C not necessarily 0