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    Graphs and Functions

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    Continuity Proofs of Functions

    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

    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 + ?.

    Analytic Functions : Constancy

    7. Let a function f (z) be a analytic in a domain D. Prove that f (z) must be constant throughout D if (a) f (z) is real-valued for all z in D (b) | f (z) | is constant throughout D. (Question also included in attachment)

    Proof for Domains with 2 Elements

    Please see the attached file for the fully formatted problem. 71. 14b. Given the wff .... Show that W is true for any interpretation whose domain has two elements.

    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

    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

    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

    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

    Arrangement of Block Possibilities

    9. A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts the blocks in a line, how many arrangements are possible?

    Tangent Line Curves

    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 Investigated

    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

    Continuous Functions Investigated

    4. Let f(x) = g(x)/h(x), where g and h are continuous functions on the open interval (a,b). Which of the following statements is true for a < x < b? A. f is continuous at all x for which x is not zero. B. f is continuous at all x for which g(x) = 0. C. f is continuous at all x for which g(x) is not equal to zero. D. f is c

    Asymptotes Determined for a Situation

    Suppose that for a particular person enrolled in a typing class, f(x)=55(x+1)/(x+8), for x greater than 0, where f(x) is the number of words per minute the person is able to type after x weeks of lessons. (A) What does f(x) approach as x increases? (B) Sketch a graph of the function f, including any vertical or hor

    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

    Limit of Trigonometric Function

    Find the limit: lim as x approaches 0 of sin 4x/sin 6x Please provide a detailed explanation of what you are doing in each step of the problem. I am not looking for just an answer.....I would like to be able to do similar problems on my own afterwards! Thanks!