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Calculus and Analysis

Fundamental Theorem of Calculus and Uniform Convergence

Suppose f is Reimann integrable on [a,b] and let F(x)= ∫_a^x▒f(t)dt for all x ∈[a,b]. Prove that F is continuous on [a,b] (hint: f must be bounded) Let F(x) = {█(x^2 sin⁡(1/x) if 0<|x|≤1,@0 if x=0)┤ and let f(x) = F'(x) Prove that F'(x) exists for all x ∈[-1,1] Find f(x) for all x ∈[-1,1], and prove t

Argand diagrams

Explain the geometrical relationship between the points in the Argand diagram represented by a complex number z and a + (z-a)e^i*theta where theta is a fixed real number. The values for a=0.76 and b=1.45. Please refer to the attachment for the proper formatting.

Calculus : Cross Product

I Have a question regarding cross products, if you know one of the vectors to be crossed and the result of the cross product how do you find the other vector. For example Vector A is unknown and Vector B=-2 Z and the result is a vector that 5X+8.66Y

autonomous differential equations

A) Check that z(t) = 1 + sqrt(1 + 2t) is a solution of the autonomous differential equation dz/dt = 1/(z-1) with initial condition z(0) =2 b) Estimate z(4) if z obeys the differential equation dz/dt = 1/(z-1) with initial condition z(0) = 2.

Solving a differential equation: Example

Dy/dx= 4x+ 9x^2/(3x^3+1)^3/2 given point: (0,2) (that's 4x plus 9x squared over (3x cubed + 1) to the 3/2 power) Use the differential equation and the given point to find an equation of the function.

Autonomous differential Equation

The rate at which a bacteria population multiplies is proportional to the instantaneous amount of bacteria present at any time. The mathematical model for this dynamics can be formulated as follows: db/dt = kb where b is a function in terms of the time t, b(t) is the number of bacteria at the time t, and k is a constant.

Determining Marginal Cost

1) The marginal cost (dollars) of printing a poster when x posters have been printed is dc/dx = 1/5√x^4 Find the cost of printing posters 18 through 130 The cost of printing posters 18 through 130 is $ _ (Round to the nearest cent)

Calculus Question

Problem: Find the area of the region in the first quadrant bounded by the line y=6x, the line x=4, the curve y=2/x(squared), and the x-axis. (1) Part A: Solve the above problem using any methods, concepts.

Evaluate the sum

I really need assistance on these three calculus problems.... Evaluate the sum: 7 &#8721; k (6k+5) (Simplify your answer) K=1 Evaluate the sum: 6 &#8721; (-13k) K=1 (Simplify your answer) Find the function y(x) satisfying dy/dx=6x-5 and y(5)=0 is y(x) = ? (Simplify your answer)

Second Partials Test

Find the critical point(s) of f(x,y)= [-4x^2 e^y + 2x^4 + e^4y] and classify each using the Second Partials Test. Does anything seem unusual about your results?

Math

Please show your work: 1. Which of the following are functions? The last two problems, i.e., b & c, are multi part relations consider all parts when determining whether or not these relations are functions. Explain your reasoning for a, b, and c. a. f(x) = x + 5 b. f(x) = 3 if x>2 otherwise f(x) = -2 c. f(x)

Find the Radius, Height, and Volume

Please help with these problems, I am stuck so if you could please post your steps, id appreciate it. 1.Use Newton's method to estimate the one real solution of x(exponent 3)+5x+1=0. Start with x0=0 2. If sinh x= 16/63, find cosh x, tanh x, cothx, sech x and csch x. 3. A right triangle whose hypotenus is square root of

Money Circulation Impact

Suppose that each dollar introduced into the economy recirculates as follows: 85% of the original dollar is spent then 85% of that $.85 is spent, and so on. Find: The economic impact (the total amount spent) if $1,000,000 is spent.

Differential Equation Word Problem: A body falling in a relatively dense fluid, oil for example, is acted on by three forces: the weight W due to gravity (acting downwards), a resistence force R and ...

A body falling in a relatively dense fluid, oil for example, is acted on by three forces: the weight W due to gravity (acting downwards), a resistence force R and a bouyant force B (both actin upwards). The wieght W of the object of mass m is mg. The bouyant force B is equal to the weight of the fluid displaced by the object.

Particular Solution of the Differential Equation

See the attached file. 1. Find the particular solution of the differential equation that satisfies the boundary condition. , y(1) = 2 A) B) C) D) E) 2. Find the particular solution of the differential equation that satisfies the boundary condition.

Calculus - Differentiate

I have an exam coming up and I am having problems with these two questions. Please show work and final answer. That way I can rewrite what you do so I understand it. 1. d/dt (x sin x + x^2)/ (x^2 + 1) 2. Find the second derivative if f(x)= sec(4x)

Determining the Limits Involving Infinity

(2) For the function g whose graph is given (see attached), state the following: (a) lim x->∞ g(x) (b) lim x->-∞ g(x) (c) lim x->3 g(x) (d) lim x->0 g(x) (e) lim x->-2+ g(x) (f) The equations of the asymptotes.

Piecewise Function Straight Lines and Parabola

4. Solve: a Solve for x if 2logx = 2 + log25 b Simplify log3 81 - log3 27 + 4log3 v3 c Find the inverse of the function f(x) = e^x + 1 5. This is completing a piecewise function which consists of 2 straight lines and a parabola. Coordinates given are (-6,0) (-3,6)-straight line, (-3,6) (.5, -6) (2,-4) -parabola, and last

Mathematics

1.Find the values of a and b for the polynomial f(x) = 2x^3 + ax^2 - 4x + b, given that f(x) is divisible by x+1 and x-3. Write f(x) in a factorised form. Hint: consider f(-1) and f(3) 2. The population of a country is found to be growing continuously at an annual rate of 1.98% t years after 1 January 1960. The total populat

Statics problems

1. A horizontal circular plate is suspended as shown from three wires that are attached to a support at D and form 30° angles with the vertical. Knowing that the z component of the force exerted by wire BD on the plate is 232.14 N, determine ( a) the tension in wire BD, ( b) the angles ux, uy, and uz that the force exerted at

Find the slope of the function's graph at the given point

1. Find an equation for the line tangent to y= -5-5x^2 at (5,-130) 2. Find the slope of the function's graph at the given point. Then find an equation for the line tangent to graph: f(x) = x^2 + 2, (-2,6) What is the slope of the function's graph at the given point? 3. Using the defenition, calculate the derivative

Calculus and Inequalities in Word Problems

Calculus applications Exercise 6.1 23. Manufacturing A company manufacturing two types of leaf blowers, an electric Turbo model and gas-powered Tornado model. The company's production plan calls for the production of at least 780 blowers per month. a. Write the inequality that describes the production plan, if x represents

Calculus questions

1. What is missing in this surface of revolution as a function of x revolved around the y-axis: ? A) Nothing, the formula is fine as is. B) Nothing, but the limits of integration should be on the interval [c, d]. C) The variable x should be x2. D) The constant 2&#960; is missing. 2. Find the solid of revolu

existence and value of the limit of a function

I need help on these two problems please 1. Does the existence and value of the limit of a function f(x) as x approaches x_o ever depend on what happens at x = x_o? Explain and give examples. 2. What does it mean for a function to be continuous? Give examples to illustrate the fact that a function that is not continuous on

Find the Volume using Calculus

Find the volume of the unbounded solid generated by rotating the unbounded region around the x axis. This is the region between the graph of y= e^-x and the x axis for x> or= 1. [Method: Compute the volume from x=1 to x=b, where b> 1. Then find the limit of this volume as b -->+infinity] What about if y=1/SQRTx.

Functions, Their Differentials and the Corresponding Plots

Find the interval on which the function f is increasing and or is decreasing and label the local maxima and minima if there is a global extrema is should be so identified. 1, F(x)= x^3 + 3x 2. F(x) = (x-2)^2(2x+3)^2 3. F(x)= 1/x 4 F(x)=6-5x-6x^2 5. F(x)=x^3 +4x

Simplifying Calculus Terms

Please help simplify the following: write dy in terms of x and dx y=2SQRTx -3/cubed root x write dy in terms of x and dx y=1/x-SQRTx write dy in terms of x and dx y=x/x^2-4 write dy in terms of x and dx y=1/(x^2-1)^4/3 write dy in terms of x and dx y=x^2 sin x write dy in terms of x and dx y=cos^3 3x.

Supply and Demand Functions in Equilibrium

Consider a market with the following supply and demand functions: QD = a0 - a1PD a0, a1 > 0 QS = b0 + b1PS b0, b1 > 0 (a) Find the equilibrium quantity and price as a function of the parameters (use any method you like). Are there any additional restrictions that you must impose on the parameters for thi

Calculus

Problem 3. POLLUTION CONTROL Commissioners of a certain city determine that when x millions dollars are spent on controlling pollution, the percentage of pollution removed is given by P(x) = (100 Sqrt(x))/0.04x^2 + 12 a. What expenditure results in the largest percentage of pollution removal? Problem 4. Compute the

coordinates in a rotating system

A point is rotating about the circle of radius 1 in the counterclockwise direction. It takes 8.4 minutes to make one revolution. Assuming it starts on the positive x-axis, what are the coordinates of the point in 7.4 minutes?