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# Calculus and Analysis

### Derivatives and Differential Equations and Leaking Tank Word Problem

Water is pumped_into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of &#8730;(t+1) gallons per minute for 0 &#8804; r &#8804; 120 minutes. At time t = 0, the tank contains 30 gallons water. (a) How many gallons of water leak out of the tank from time r = 0 to r = 3 minut

### Derivatives, Polynomials, Points of Inflection and Equations

See the attached file. 4. Let h(x) be a function defined for all ... such that h(4) = ?3 and the derivative of h(x) is given by .... (a) Find all values of x for which the graph of Ii has a horizontal tangent, arid determine whether 1 has a local maximum, a local minimum, or neither at each of these values. Justify your answer

### Lagrange Multipliers: Example Problem

The plane 4x-3y+8z=5 intersects the cone Z^2=x^2 + y^2 in an ellipse a. Graph the plane, cone and ellipse b. Use Lagrange multipliers to find the highest and lowest points on the ellipse. This problem must be solved using maple 10 (or 9) please show all work and data entries and outputs.

### Double Integral : Region of triangle bounded by y=x, y=2x, x=2.

The problem is: &#8747;&#8747; y/[x^2 + y^2] dA R R is the triangle bounded by y=x, y=2x, x=2

### Limits and Graphs

1.) Determine lim An= (n^2-11n+22)/(3n^2+29n+19) 2.) Sketch graph 2x^2-4x+y^2+4y=10.

### Conditionally Convergent Series

Prove that if Series An (small "a", sub "n") is a conditionally convergent series and r is any real number, then there is a rearrangement of Series An whose sum is r. [Hints: Use the notation of Exercise 39 (I'll show below). Take just enough positive terms An+ so that their sum is greater than r. Then add just enough negati

### Turbulent Boundary Layers, Flow Veolcity, Laminar Flow and Hydraulic Radii

1. DERIVE EQUATION 6-15 2. DERIVE EQUATION 6-48 AND 6-54 3. SOLVE ALL PROBLEMS SHOWN BELOW: 67. Water at 20°C flows through a smooth pipe of diameter 3 cm at 30 m3/h. Assuming developed flow, estimate (a) the wall shear stress (in Pa), (b) the pressure drop (in Pa/rn), and (c) the centerline velocity in the pipe. What is the

### Functions and calculus

(See attached file for full problem description) --- 1) Consider the following function: a) f (x) = 9x2 - x3 b) f (x) = x + 1 x - 2 c) f (x) = x2/3 (x - 5) for each of the above functions complete the following table. Show the work to justify your answers below the table. f(x) is i

### Equation for a plane

Using the given parameters, find the equation for the plane. Please show all work and a diagram if possible. Thank you. The plane through (1, -1. 3) parallel to the plane 3x + y + z = 7

### Locus: Two Trees Located at Grid Points

In a backyard, there are two trees located at grid points A(-2,3) and B(4,-6). a) The family dog is walking through the backyard so that it is at all times twice as far From A as it is from B. Find the equation of the locus of the dog. Draw a graph that shows the two trees, the path of the dog. and the ralationship defining

### Calculus for Business: Nation's Consumption

6) If a nation's consumption function is given by : C(I) = 0.3I + 0.8 &#8730;I + 6 where I is national income, measured in billions of dollars a) Find the nation's marginal propensity to consume b) Find the nation's marginal propensity to save c) Evaluate the marginal propensity to save when I = 64

3) A wholesaler that sells computer monitors finds that selling price "p" is related to demand "q" by the relation p=280 - .02q where p is measured in dollars and q represents number of units sold a. Find the wholesaler's Revenue function as a function of q, using Revenue = (price) (quantity) b. Find the expression for Mar

### Stochastic Differential Equations, Density Functions and Random Variables

We use the notation X ~N(&#956;, &#963;2) to indicate that the density function for the continuous random variable X, fx(x), has the form .... (a) If X ~N(&#956;, &#963;2) show that..... (Hint: you will need to know how to find the density function for X ? &#956; from the density function for X). (b) If ...., and X1 and X2 a

### Vector Normal to a Curve, Electrostatic Potentials and Vector Fields

Please see the attached file for the fully formatted problems.

### Sturm-liouville problem

1. (a) Find the eigenvalues and eigenfunctions of the boundary-value problem. x2y'' + xy' + &#955; y = 0, y(1) = 0, y(5) = 0. (b) Put the differential equation in self -adjoint form. (c) Give an orthogonality relation. 2. Hermite's differential equation y'' -2xy' + 2ny = , n =

### Linear Differential Equations - Finding a Particular Solution

Calculate the unique solution for 7e^x in the following non-homogeneous second-order linear equation, showing all the steps. d²y/dx² + 5dy/dx - 6y = 7e^x.

### Particular Solution for a Non-Homogeneous Second Order Differential Equation

Please demonstrate how to calculate the particular (i.e. not general) solution for: d²y/dx² + 6dy/dx + 9y = e^(-4x)

### Nonhomogenous Second Order Linear Differential Equation

Please find the particular solution for xe^x in this problem and show all details. d²y/dx² - y = xe^x.

### Particular Solution for a Nonhomogenous Linear Differential Equation

I am not sure how to find the particular solution for the sin(t) part of the equation: d²y/dt² - 9y = sin(t) Please show all the details.

### Differential Equations and Green's Functions

Consider the inhomogeneous differential equation -u' + u = r(x) on (a,b) where r(x) is a continuous function on the interval (a,b) Find the solution to this differential equation in the form u = integral from a to b (G(x,s) r(s) ds) and determine G(x,s).

### Particular solution of differential equation

The differential equation is: d²y/dx² - y = xe^x For the first part of the general solution, I got y=Ae^x + Be^-x where a and b are constants. Now I need to find the particular solution. Thanks.

### General solution to nonhomogenous second order differential equation : d²y/dt² - 9y = sint

Please find the General solution to the non-homogenous second order differential equation below. Please show all details in your solution. Thank you d²y/dt² - 9y = sint

### General solution of non-homogenous second order linear differential equations : d²y/dx² + y = cosx

Please find the general solution of the nonhomogenous second order linear differential equation below by following these steps: 1. Find the general solution y= C1y1 + C2y2 of the associated homogenous equation (complementary solution) 2. Find a single solution of yp of above.(particular solution). 3. Express the general so

### General solution of non-homogenous 2nd order linear differential equations : d²y/dx² - 4y = e^(3x)

Please find the general solution of the nonhomogenous second order linear differential equation below by following these steps: 1. Find the general solution y= C1y1 + C2y2 of the associated homogenous equation (complementary solution) 2. Find a single solution of yp of above.(particular solution). 3. Express the general so

### Find the particular solution of the differential equation

Find the particular solution of the general equation below, given the initial conditions. Please show all details in the solution, simplifying if possible. Thank you. y=C(e^(-2x))sinx + C(e^(-2x))cos x y(0)= 1 y'(0)= 0

### Find the particular solution of the differential equation

Please find the particular solution of the differential equation below satisfying the given initial conditions. The equation needs to be put in standard form (s² + bs + c = 0). The roots need to be found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equa

### Find the general solution of the differential equation : 2d²y/dx² + 4dy/dx + 5y = 0

Please find the general solution of the differential equation below. It needs to be converted to standard format (s² + bs + c = 0), the roots found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. Please show all details in the solution. Thank

### Find the General Solution of the Differential Equation : 2d²y/dt² + 5dy/dt + 2y = 0

Please find the general solution of the differential equation below. It needs to be converted to standard format (s² + bs + c = 0), the roots found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. Please show all details in the solution. Thank

### Graphing and Lagrange Multipliers

The plane, 4*x - 3*y + 8*z = 5, intersects the cone, z^2 = x^2 + y^2, in an ellipse. 1. Graph the cone, the plane and the ellipse 2. Using lagrange multipliers find the highest and lowest points on the ellipse. Please show grophs, equation of the ellipse and all work.

### Maximum and Minimum Values and Saddle Points

Use a graph and level curves to estmate the local maximum and minimum values and saddle points of f(x,y) = x^3 - 3*x + y^4 - 2*y^2; then use calculus techniques to find these values precisely.