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Calculus and Analysis

Applications of integration

Let f and g be the functions given by f(x) = ¼ + sin(Pi*x) and g(x) = 4^ -x. Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed by the graphs of f and g. a) Find the area of R. b) Find the areas of S. c) Find t

Finding equations in standard form for conics.

Please explain in as much detail as possible finding an equation in standard form for each conic. 1) Parabola, focus (-4,0), directrix X=2 2) Hyperbola, foci (0, +/_3), vertices (0,+/-1) 3) Ellipse, center (2,2), a focus (0,2), vertex (5,2)

Calculus for Business:

(See attached file for full problem description) --- 10. y = √x + 3 (√x+3 is all squared). Please use the Chain Rule 16. f (w) = w √w + w² (√w is only squared) Please use the Chain Rule 18. y = 4x³ - 8x² Please use the Chain Rule

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 √(t+1) gallons per minute for 0 ≤ r ≤ 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 Differential Equations

4. Let h(x) be a function deflned 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 answers. (b) On what interval


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

A few questions

(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

Equations for planes in space

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:

6) If a nation's consumption function is given by : C(I) = 0.3I + 0.8 √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

Calculus for Business:

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(μ, σ2) to indicate that the density function for the continuous random variable X, fx(x), has the form .... (a) If X ~N(μ, σ2) show that..... (Hint: you will need to know how to find the density function for X ? μ from the density function for X). (b) If ...., and X1 and X2 a

Sturm-liouville problem

1. (a) Find the eigenvalues and eigenfunctions of the boundary-value problem. x2y'' + xy' + λ 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 =

Sketching Contour Maps and Determining Radii for a Hemisphere

The hemisphere is given by f (x,y) = sqrt 64 - x^2 - y^2. Sketch a contour map for this surface using the level curves corresponding to c = 0,1,2,3,4,5,6,7,8 Since this is an example shown in the book, I can see that when c = 8 you draw a circle with a radius of 8 and when c = 0 you have a point at the center of the circle.

Infinite Sequences (20 problems)

Please see the attached file for the fully formatted problems. Please do problems 3 - 8, 10, 12, 14, 16 - 40 even, 42 ,44,45-49, 54 - 60 odd.

Vector-valued function word problem.

At time t = 0 a baseball that is 5 feet above the ground is hit with a bat. The ball leaves the bat with a speed of 80 feet per second at an angle of 30 degrees above horizontal. a) How long will it take for the baseball to hit the ground? b) Use the result in part a to find the horizontal distance traveled by the ball. Ple