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# Calculus and Analysis

### Simulation : Skydiver in Free-fall

A skydiver, weighing 70kg, jumps from an aeroplane at an altitude of 700 metres and falls for (T) seconds before pulling the rip cord of his parachute. A landing is said to gentle if the velocity on impact is no more than the impact velocity of an object dropped from a height of 6 metres. The distance that the skydiver falls d

### Setting up a Riemann Sum

I can't figure out exactly how to formulate a riemann sum. For example, when given y=x+2; [0,1], and told to "find the area of the region under the curve y=f(x) over the interval [a,b]. To do this, divide [a,b] into n equal subintervals, caluculate the area of the cooresponding circumscribed polygon, and then let n go to infin

### Evaluating an Integral with 2nd Order Pole : Moivre-Laplace Fomulation

Problem: Evaluate the integral from 0 to INF of: (x^a)/(x^2 +4)^2 dx, -1 < a < 3 We are to use f(z)= (z^a)/(z^2 +4)^2, with z^a = e^(a Log z), Log z= ln|z| + i Arg z, and -pi/2 < Arg z < 3pi/2. I have found the residue at 2i to be: [2^a(1-a)/16]*[cos ((pi*a)/2) + i sin ((pi*a)/2). Please let me know if this is correct and how to

### Calculus : Reimann Sum and Limits and Continuity

34) The function f is continuous on the closed interval [1,5] and has values that are given in the table below. If 2 subintervals of equal length are used, what is the midpoint Reimann sum approximation of integral with 5 on top and 1 on bottom f(x)dx? Please given step by step explaination and answer is 32. x 1 2 3

### Calculus : Antiderivative and Rate of Change

36)If the functions f and g are defines for all real numbers and f is an antiderivative of g, which statements are not true? I If g(x)>0 for all x, then f is increasing. II If g(z)=0 then f(x) has a horizontal tangent at x+a. III If f(x)=0 for all x, then g(x)=0 for all x. IV If g(x)=0 for all x, then f(x)=0 for all x.

### Calculus : Differentiability and Maximizing Area

24) Let f be a differentibale function defined on the closed interval [a,b] and let c be a point in the open interval (a,b) such that. I f'(c)=0 II f'(x)>0 when a<or equal to x<c, and III f'(x)<0 when c<b<or equal to b. Which is true? Then tell why others false. a. F'(c)=0 b. F"(c)=0 c. F(c) is an abs. ma

### Velocity, Continuity, Limits, Differentiability & Integrability

26) The vertical height in feet of a ball thrown upward from a cliff is given by s(t)=-16t^2+64t+200, where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero? 27) If the function f is continuous for all real numbers and the limit as h approaches 0 of f(a+h)-f(a)/h = 7 then which sta

### Critical Points and Maximum and Minimum Values

Please see the attached file for the fully formatted problem. Find the critical points and use your test of choice to give local maximum and minimum values. Give those values. f(x) = x^2 /sqrt(x^2 + 4)

### Differential Equations : Free Fall and Terminal Velocity

An object in free fall in a gravitational field is governed by the ODE m*dv/dt=mg + Fs, where m is the mass of the object, g=9.8 meters/sec is the acceleration of gravity, v(t) is the velocity of the object t seconds after it is released, and Fs denotes external forces acting on the object. In all that follows, assume that v(0)

### Chaos : Conjugacy in Discrete Dynamical Systems

I am taking a course in Dynamics/Chaos and I am trying to prove conjugacy between the logistic and quadratic functions. I have some ideas, but cannot get the proof to work. Attached is a word document with the functions and problem.

### Tangent through a point

A portion of a river has the shape of the equation y=1-x^2/4, where distances are measured in tens of kilometres, and the positive y-axis represents due north. the town of Coopers Crossing is situated on the river at its most northerly point. The town of Black Stump is 10 kilometres due south of Coopers Crossing. the town of And

### Line Equations : Tangents & Normals

I) Find the equation of the tangent to y=x(1-x) at x=1 ii) Find the equation of the normal to y=x(1-x) at x=1 iii) Find the equations of the tangents to y=x(1-x) that pass through (-1, 1/4)

### Differentiability and Limits

Please see the attached file for the fully formatted problems. Let f: I &#8594;&#8476; where I is an open interval containing the point c, and let k &#8712; &#8476;. Prove the following 1. f is differentiable at c with f &#8242;(x) = k iff lim h&#8594;0 [f(c+h) - f(c)]/h=k 2. If f is differentiable at c with f &#8242;(c) =

### Multivariable calculus

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you Sketch the solid bounded by the graphs of the given equation and find its volume by triple integration: z = x^2, y + z = 4, y = 0, z = 0

### Multivariable calculus

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you Sketch the solid bounded by the graphs of the given equation and find its volume by triple integration: z = y, y = x^2, y = 4, z = 0

### Multivariable Calculus: Mass and Centroid of a Plane Lamina

Find the mass and centroid of the plane lamina with the indicated shape and density: The region bounded by the parabolas y = x^2 and x = y^2, with &#61540;(x, y) = xy &#61540;: is the density symbol

### Multivariable calculus

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you. ( &#61682; ^n_r means that n is on the top of the &#61682; and r is on the bottom) Evaluate the iterated integral : &#61682; ^1_0 &#61682; ^(x^2)_0 xy dy dx &#61682;: is the integral symbol

### Multivariable calculus

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you. ( &#61682; ^n_r means that n is on the top of the &#61682; and r is on the bottom) Evaluate the iterated integral: &#61682; ^2_0 &#61682; ^2x_0 (1 + y) dy dx &#61682;: is the integral symbol

### General calculus

20) If the function f is continuous for all real numbers and lim as h approaches 0 of f(a+h) - f(a)/ h = 7 then which statement is true? a) f(a) = 7 b) f is differentiable at x=a. c) f is differentiable for all real numbers. d) f is increasing for x>0. e) f is increasing for all real differentiable ans is B. Explain

### Differential Equation

Most drugs are eliminated from the body according to a strict exponential decay law. Here are two problems that illustrate the process. 1. The drug Valium has a half-life in the blood of 36 hours. Assume that a 50-milligram dose of Valium is taken at time t=0. Let m(t) be the amount of drug in the blood in milligrams t hours

### Curve in polar coordinates and find the area of the inner loop.

Sketch the curve in polar coordinates given by r = 2-4 sin theta Find the area of the inner loop. Find the area of the inner loop for the general case: i.e. r = b - a sin feta (0 less than b less than a )

### Find the area bounded by one loop of a curve.

Find the area bounded by one loop of the curve given by x=sint, y=sin2t You should provide suitable notes to justify you solutions.

### LAPLACE TRANSFORMS

Please see attachment. Require problems solving, also explanations etc for better understanding.

### Calculus 3 and 4

Show all work. Please DON'T submit answers back to me as an attachment. Thank you. Determine whether the function is homogenous. If it is, state the degree: f(x, y)=5x^2 + 2xy

### Find ∫ e^(-x^2)dx by using Laplace Transform.

Laplace Transform Inverse Laplace Transform To find the value of ∫ e^(-x^2)dx by using Laplace Transform, where the range of integration is

### Relation between the roots and coefficients of an equation

Theory of Equation Relation between Roots and Coefficients Harmonical Progression Arithmetical Progression Problem

### Oscillating Inflow Concentration

Oscillating Inflow Concentration A tank initially contains 10 lb of salt dissolved in 200 gallons of water. Assume that a salt solution flows into the tank at a rate of 3 gal/min and the well-stirred mixture flows out at the same rate. Assume that the inflow concentration oscillates I n time, however, and is given by ci(t

### Volume of a Rotating Solid

Let f and g be the functions given by f(x)=e^x and g(x)=ln x. b) Find the volume of the solid generated when the enclosed region of f and g between x = ½ and x = 1, is revolved about the line y = 4. c) Let h be the function given by h(x)=f(x) - g(x). Find the absolute minimum value of h(x) on the closed interval ½ &#88

### Functions : Tangent, Increasing or Decreasing and Area under a Curve

2. Let f be a function defined on the closed interval -3&#8804;x&#8804;4 with f(0) = 3. The graph of f', the derivative of f, consists of one line segment and a semicircle. a) On what intervals, if any, is f increasing? Justify your answer. b) Find the x-coordinate of each point of inflection of the graph of f on t