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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)

### Suppose a function f has an inverse.

These problems deal with the concept of inverse of functions. See attached file for full description.

### 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

### The function f is defined by the following function table.

The function f is defined by the following function table. Graph the function. x f(x) -2 4 -1 0 0 4 1 4 2 -2

### Cost Functions and Marginal Cost

For the cost function below, determine the marginal cost when x = 10. 200 + 15x - 0.5x^2.

### Finding the Values of T

2t 42 - ------= - ---------------- t+9 t^2 +5t -36

### 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!

### Increasing and Decreasing Functions

(See attached file for full problem description)

### Show that the product of two step functions is a step function

Show that the product of two step functions is a step function. Notes from section of book attached.

### 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=&#8730;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-

### Square Root of an Identity

Is √x=x an identity (true for all values of x)? See the attached file.

### Draw the response of the system for P, PI and PID controller

Consider the following system in the attached file. Where Gc(s)= 10 for P controller = 10(1+1/s) for PI controller = 10(1+s+1/s) for PID controller Draw the response of the system for P, PI and PID controller using Simulink.

### 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 &#945; 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.

### Determinant: Finding Values for When det(A)=0

Let... If... are collinear, for what points ... is det(A)=0? Please see the attached file for the fully formatted problem.

### Is (&#8730;x)^2=x an identity (true for all values of x)?

Is (&#8730;x)^2=x an identity (true for all values of x)? Please explain.

### 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)}.

### Pointwise Operations of Functions : Function Addition

(See attached file for full problem description) Please help with the following questions: 4, 12, 16, 24, 36, 38, 42, 48, 50, 60, 66

### 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. ---