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    Computing Values of Functions

    Inverse functions: True or false questions

    True or false. Determine whether the statement is true or false. If false, explain why and give an example that shows its false. 1) If the inverse function of f exists, then the y-intercept of f is an x-intercept of f^-1 (f inverse). 2) There exists no function f such that f = f^-1 (f inverse)

    Increasing Functions

    Which of the functions below is increasing for all x-values? f(x) = x - 7 f(x) = x2 - 7 f(x) = x3 - 7 f(x) = x4 - 7

    Substituting variable

    Please show how to solve y'' - 3y^2=0, substituting v=y' so y'' = v dv/dy Initial conditions are y(0) =2 and y'(0)=4 I got it as far as dy/dx = (y^3 +c)^1/2 but that might be wrong!

    Open Neighborhood of A

    (See attached file for full problem description) If A is a nonempty set in a metric space X and if r>0 show that is an open neighborhood of A.

    Closed set in a metric space

    (See attached file for full problem description) If A is a closed set in a metric space (X,d) and , show that d(x,A)>0.

    Metric Spaces

    (See attached file for full problem description) 1. Show that the functions d defined below satisfy the properties of a metric. a. Let X be any nonempty set and let d be defined by The d is the call the discrete metric. b. If X is the set of all m-tuples of real numbers and, if for and , then (X,d) is a metric spac

    Use Matlab to calculate correlation between heartbeat record

    Load ECG data from exercise 1 into MATLAB. Select a segment of recording encompassing at least 10 cycles of the heartbeat. Use Matlab to coherently average 10 cycles to produce a plot of one full cardiac cycle with an improved signal-to-noise ratio. Include plots of the average cycle with labeled axis. Use Matlab to calculat

    Manufacturing Costs and Cost Functions

    Manufacturing Cost The weekly production cost C (in dollars) of manufacturing x hand calculators is given by the formula C = 6000 + 8x - x 2 / 1000. What is the cost of producing 1000 hand calculators?

    Is the square root of x2=x an identity (true for all values of x)?

    Is the square root of x2=x an identity (true for all values of x)? For the equation x-squarex=0 perform the following a. solve for all values of x that satisfies the equation. b. graph the function y=x and y=√x on the same graph (by plotting points if necessary). Show the points of intersection of these two graph

    PID Controller : Ziegler-Nichols Tuning Method

    Consider the following system shown in Fig.2 (attached file) in which a PID controller is used to control the system. The PID controller has the transfer function Gc(s) = Kp(1 + 1/Tis + Tds) Design a PID controller for this system using Ziegler-Nichols tuning method for determination of Kp, Ti and Td. Then obtain a unit-

    Bessel Functions and Sturm-Liouville Problem

    (See attached file for full problem description) --- Use the following table to solve 3 and 4. J0(x) J1(x) Y0(x) Y1(x) 2.4048 0.0000 0.8936 2.1971 5.5201 3.8317 3.9577 5.4297 8.6537 7.0156 7.0861 8.5960 11.7915 10.1735 10.2223 11.7492 14.9309 13.3237 13.3611 14.8974 3. Find the first four α i

    Uniqueness of a Reflector

    Show that if Qx=y, where Q= I- ruu^T (r can be denoted as gamma) , then u must be a multiple of x-y. In other words, prove the uniqueness of reflector Q.

    Consider the Initial Value Problem

    Consider the initial value problem on [1,2]: x^2*y'' + xy' - K^2*y = 0, y(1) =1, y'(1)=0 Find the solution y(x,K). Is it a continuous function of K? Can it be differentiated with respect to K? K is a constant. See the attached file.

    Relations and Functions: Domain and Range

    Determine each of the following based on the relation {(-5, -3), (-2, 1), (2, 2), (-5, 8)}. 1. Is the relation {(-5, -3), (-2, 1), (2, 2), (-5, 8)} a function? 2. Identify the domain of the relation {(-5, -3), (-2, 1), (2, 2), (-5, 8)}. 3. Identify the range of the relation {(-5, -3), (-2, 1), (2, 2), (-5, 8)}.

    Unit-Step Response using MATLAB

    Using MATLAB, obtain the unit-step response of the following system: C(s)/R(s) = 10/s^2 +2s + 10 where R(s) and C(s) are Laplace transforms of the input r(t) and output c(t), respectively. Hint: Use the following command; step(num,den)

    Define the function A

    We are using the book Methods of Real Analysis by Richard R. Goldberg (See attached file for full problem description) --- 12.6-3 Let be a complete orthogonal family in . Define the function A from into .( This means: In order to manufacture our metric space we must therefore regard any two function whose valu

    Bessel Function, proofs

    USING THE BESSEL FUNCTION OF ORDER ZERO: Verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.

    Harmonic and Analytic Function

    Give an example (and explain why it works) of an analytic function u on a harmonic function v such that the composite function u o v is defined but NOT harmonic. Please see the attached file for the fully formatted problem.

    Use the values for approximation to M.

    (See attached file for full problem description with proper equations and exponents) --- 1. Suppose that N(h) is an approximation to M for every h > 0 and that M = N(h)+K1h2+K2h4+K3h6+..... For some value K1, K2, K3... Use the values N(h), N(h/3), and N(h/9) to produce an O(h6) approximation to M. ---