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# Calculus and Analysis

### Uniform convergence of sequences of functions

1. Show that the sequence x^2 (e^-nx) converges uniformly on [0, infinity). 2. Show that if a is greater than zero then the sequence (n^2 x^2 (e^-nx)) converges uniformly on the interval [a, infinity) but does not converge uniformly on the interval [0, infinity). For problem 2 text gives a hint that if n is sufficiently larg

### total mass of the lamina

1) A lamina has the shape region  in the xy-plane bounded by the graphs of y = (25-x^2)^(1/2) , y = 0, x = 3, x = 5 If the density (in kg/m^3) at each point P in  is inversely proportional to the square of the distance from P to the y-axis, with (5,0,0) = 1 kg/m^3. Find the mass of the lamina [Assume a constant thick

### Finding the equation of a line parallel to the given one.

A) Find a vector equation for the line that passes through (1,-2,7) and is parallel to the line x= 3+2t, y= -t, z= -1-t B) Write a different vector equation for the line you found in part A...

### Example Differential Equation

Find the solution to the differential equation given below. y'=t*exp(3*t) - 2*y I can not arrive at the exact solution using separation of variables and integration. The exact solution makes use of convergence, y(t)=e^t .

### Find the area using calculus

6. Find the area of a circle of radius r. 7. One strip of pink roses will be planted at the tip of the rose garden shown in the figure. Find the area of the strip of pink roses. Please refer to the attachment for more questions and mentioned figure. Please give suitable explanation along with answers that will help me u

### Instantaneous Rate of Change of Demand with Respect to Price

Suppose that the demand for a product depends on the price (p) according to: D(P) = 40,000/p^3 - 1/4, p>0 where p is in dollars. Find and explain the meaning of the instantaneous rate of change of demand with respect to price when p = 50. Please provide a step by step help.

### If the cost of C (in dollars) of removing p percent of the impurities from the waste water

If the cost of C (in dollars) of removing p percent of the impurities from the waste water in a manufacturing process is given by: C(p) = 10,000p/(100-p) Find the rate of change of C with respect to p.

### Solving differential equations.

Solve the following differential equations. 1a. 11x - 6y sqrt(x^(2)+1)*(dy/dx) = 0 w/ y(0) = 2 1b. Find f(x) if y = f(x) satisfies (dy/dx) = 160x^15 and the y-intercept if the curve y = f(x) is 5. 1c. ((x^2)-(y^(2)-8))*(dy/dx) = (1/2y) w/ y(1) = sqrt(9) 1d. 3e^(7x) * (dy/dx) = -49(x/y^(2)) w/ y(0) =

### First-Order Differential Equations

Two Snowplows - Differential Equations (First-Order Differential Equations) One day it began to snow exactly at noon at a heavy and steady rate. A snowplow left its garage at 1:00pm, and another one followed in its tracks at 2:00pm. a)At what time did the second snowplow crash into the first? To answer this question, assum

### Calculas

Please show all woCarlos is blowing air into a soap bubble at the rate of 7 cm3/sec. Assuming that the bubble is spherical, how fast is its radius changing at the instant of time when the radius is 11 cm? cm/sec. How fast is the surface area of the bubble changing at that instant of time? cm2/sec. The Millers are plann

### divergence and curl of the given vector field

Calculate the divergence and curl of the given vector field F. F(x,y,z)=3xi-2yj-4zk F(x,y,z)=(x^2 e^(-z) )i+(y^3 ln⁡〖x)〗 j+(z cosh y)k Evaluate ∫_C▒〖P(x,y)dx+Q(x,y)dy〗 P(x,y)=xy, Q(x,y)=x+y; C is part of the graph of y = x² from (-1,1) to (2,4) Show that the given line integral is i

### Differential equation, Area of bounded region

1. Solve the following differential equation: -2yy' +3x^2 SQRT(4-y^2) =5x^2 SQRT(4-y^2) , -2 < y < +2 2. Let f(x) = ax^2 , a>0 , and g(x) = x^3 Find the value of a which yields an area of PI (i.e. 3.14159) for region bounded by figure, y-axis and line x=1.

### Differential Equations Function for Region Bounded

1. Solve the following differential equation: -2yy' +3x² √(4-y²) =5x² √(4-y²) , -2< y<+2 2. A calculus instructor has determined that the arc of an individual diving into a swimming pool is defined by the function, f(x) = sin (.4x). Determine how far the diver has traversed in his dive as he passes thr

### piece wise continuity

Consider the functions f(x)= {2, x != 5 and g(x) = {3x+1, x != 5 {1. x = 5 {-16 x = 5 in each part, is the given function continuous at x = 5? Justify you answer by showing a) g(x) b) f(x) g(x) c) g(f(x)) !=not looking to

### Newton's Method Roots

Hi, Using Newton's Method, please find the root of y = x^3 - 7x - 2 that exists between 2 and 3. Please show all work. Thank you.

### Differential Equations using Separable Variables

An equation of the form dy/dx=f(ax+by) can be transformed into an equation with seperable variables by making substitution z=ax+by or z=ax+by+c. use this technique to solve. y' = sqrt (4x + 2y - 1)

### Ratio of the Radius on the Bottom to the Height of the Can

A soft drink bottler wants to design a can which will hold a definite volume V o. She also wants to use the minimum amount of aluminum possible in order to hold down costs. Assuming the can will be a cylinder, find the ratio of the radius on the bottom to the height of the can which meets the bottler's constraints.

### Finding Maximum and Minimums of Functions

Consider the function f(x) = x^3 * e^-4x for x is greater than or equal to 0. (a) Find the maximum value of f(x) on the interval [0,infinity). (b) Find the minimum if f(x) on the interval [0,infinity). (c) Find, if there are any, the points of inflection on the interval [0,infinity). (d) For what values of x on the interva

### Find the Cost of Catering

The marginal cost for a catering service to cater x people can be modeled by: Dc/dx = (5x)/(sqrtx^2 + 1000) When x = 225, the cost is \$1136.06. Find the cost of catering to 500 people and 1000 people.

### Solve: Power Series Expansion

A sound wave is given by the function f(t) = 0.5e^-2t cos 4t Write down the first four terms of the power series expansions of e^-2t and cos 4t. Determine the cubic (up to and including the third power of t) approximation of f(t) and calculate and show the accurate and approximate values of f(0.02) stating the differen

### Tax Formula: How much in taxes is paid by individuals with incomes of \$10,000, \$30,000, and \$50,000? What are the average tax rates for these income levels? At what income level does tax liability equal total income?

Here's the problem I'm having trouble with: Taxes in Oz are calculated according to the formula T=.01I^2 where T=thousand of dollars of tax liability and I=income measured in thousands of dollars. How much in taxes is paid by individuals with incomes of \$10,000, \$30,000, and \$50,000? What are the average tax rates for thes

### second order differential equation with convolution term

Solve 4y'' + 4y' +17y=g(t) y(0)=0 , y'(0)=0 your answer will involve a convolution term with g(t). If there is any confusion in the question just assume yourself what must be right .

### Demand Function: Finding the Revenue

There are two different problems of calculus (see the attachment).

### independent, inconsistent or dependent

Please assist with these practice questions. Question 1 Solve the equation. The solution is x= (round to four decimal places as needed) Question 2. Solve the equation. The solution is x= (round to four decimal places as needed) Question 3 Solve the system by addition. Determine whether the system is independe

### Writing a Differential Equation and Finding a Particular Solution

The amount of money M in an account is increasing at the rate of 9% per year. (a) Write a differential equation that describes the amount at any time t. (b) Find the particular solution for M, given that the account begins with \$2,000 in it. (c) When will there be \$24,000 in the account?

### Estimate the distance.

A piece of tissue paper is picked up in a gusty wind. The table below shows the velocity of the paper at 2 second intervals. Estimate the distance the paper travelled using left-endpoints. Show all possible work! Time Velocity (sec) (ft/sec) 0 0 2 8 4 12 6 6 8 26

### Calculating Income Based on Salary and Commission

Please help with the given problems including all the steps used to achieve the solution. Isaac earns a base salary of \$1250 per month and a graduated commission of 0.4% on the first \$100,000 of sales, and 0.5% on sales over \$100,000. Last month, Isaac's gross salary was \$2025. What were his sales for the month?

### Laplace Transform help

Example 1: Solve using Laplace Transform Answer: First, apply the Laplace Transform Knowing that , and we get After easy algebraic manipulations we get , which implies Next, we need to use the inverse Laplace. We have (see the table) For the second term we need to perform the partial dec

### Differientiate the function and tangent line

Please answer the attached problems and circle the answers. thanks 1. Differientiate g(x) = (7x - 5)/(5x + 1) + x^3 2. Find dy/dx, y = x^0.75 3. Differientiate the function y = sqrt(x^2 + 6) 4. Find an equation for the line that is tangent to the curve y = 4x^3 - 4x at the point (1, 0) 5. For the function, find t