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    Graphs and Functions

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    Quadratic Equation Pictured

    A square picture is mounted in a frame 1 cm wide. The area of the picture is 2/3 of the total area. Find the length of a side of the picture.

    Function solutions

    2. Given the polynomial f(x) = 2x 3 -5x2-4x+3, find the solutions if the function is completed as a) f(x) =0 b) f(x+2)=0 d) f(2x) = 0

    Prove A Variation of Fermat's Theorem

    There always exists a real number n such that a^n = b^n + c^n , where a, b and c are any integers. The problem is not Fermat's Last Theorem, but a variation of it with real exponents.

    Functions: Onto and One-to-one, Bijections and Functions

    Please help with the following problems on graphs and functions. Provide step by step calculations. 1. Assuming A,B not equal to no solution, define m1:AxB->A and m2:AxB-> as follows: m1(x,y)=x and m2(x,y)=y. If f: A->B, show that a) f onto=>m2 |f is onto b)f one-to-one=>m2 f is one-to-one 2. Assuming f: A->B and g:

    Functions and Graphs: Trends and Real World Implications

    Plot your data for each disease as points in a rectangular coordinate system. Year...................1985..........1990..........1995......2002 Heart Disease 771,169 720,058 684,462 162,672 Cancer 461,563 505,322 554,643 557,271 AIDS * 8,000 25,188 39,979 14,095 - Use individu

    Limit and Slope of Tangent Line, Midpoint and Trapezoidal Rule

    Use the limit definition to find the slope of the tangent line, f(x)= X^2+2X+1 Use the midpoint rule to approximate the area of the function f(x)=-2x+3 from [0,1] when n=4. Compare this approximation to the actual area by integration to the approximate area using the trapezoid rule. Sketch the and find its average rate

    Plotting a Surface Function

    Consider the function u(x,t) = sin(4 pi x) e^(-pi t). Plot using a graphical tool and explain what you observe. Please see the attached file for the fully formatted problems.

    Statistics

    What is the "causal relationship" between independent and dependent variable?

    Sequence of Functions and Mean Value Theorem

    Let a<b. Let f_n: [a,b] -> R be a sequence of functions such that, for each n in N ( N set of natural numbers),f_n is differentiable on (a,b). Suppose that for all n in N, Sup on [a,b] of | f'_n(x) | < or = to M, where M is in R. ( Sup is supremum = least upper bound) Prove that for all n in N and all x, y in [a,b], one has

    Uniformly Cauchy Sequence of Continuous Functions

    Let f_n : [0,1] -> R be a sequence of continuous functions such that for each n in N (natural numbers), f_n is differentiable on (0,1). Suppose that f_n(0) converges to some number, denoted f(0), and also suppose that the sequence (f'_n) converges uniformly on (0,1) to some function g: (0,1) -> R. Prove that the sequence (f_n) c

    Diffusion Equation : Energy Decreasing as a Function of Time

    3. Suppose that u(x. t) satisfies the diffusion equation ut = kuxx for 0 < x < L and t > 0, and the Robin boundary conditions ux(0, t) ? aou(0, t) = 0 and ux(L, t) + aLu(L, t) = 0 where k, L, a0 and aL are all positive constants. Show that ... is a decreasing function of t. Please see the attached file for the fully for

    Finding the Maximum Height by Graphing

    If a baseball is projected upward from ground level with an initial velocity of 64 feet per second, then its height is a function of time, given by s(t) = -16t squared + 64t. How would I graph this function for 0 ≤ t ≤ 4? And how do I determine the maximum height reached by the ball?

    Sketching a graph of a function

    Sketch the graph of the function y= -x^3+3x^2-4. Be sure to include and label: 1.) x and y intercepts 2.) asymptotes 3.) 1st and 2nd derivatives 4.) increasing and decreasing intervals 5.) intervals of concavity 6.) inflection point(s) 7.) relative extrema (max and min)

    Diminishing Returns (Point of Inflexion)

    Identify the point of diminishing returns for the input-output functions. R=1/50000(600x^2-x^3), 0<x<400 (those are < or = to signs) R=-4/9(x^3-9x^2-27), 0<x<5 (those are < or = to signs).

    Distance from Starting Point

    A car travels along a straight road, heading West for 3 hours, and then travels NE on another road for 2 hours. If the car has maintained a constant speed of 55 mi/hr, how far is it from its starting point?

    Analytic Functions Provided

    Show that if f in analytic in {z: |z| < 1} and if Im f(1/k)=0 for all k=2,3... then Im f(x)=0 for -1<x<1. Please see attached for Hint.