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Calculus and Analysis

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?

Determining the Maximum Separation

The positions of two particles on the s-axis are s1=sin t and s2=sin(t+ π /3), with s1 and s2 in meters and t in seconds. a) At what time(s) in the interval 0 ≤t ≤ 2π do the particles meet? b) What is the farthest apart that the particles ever get? c) When in the interval 0 ≤t ≤ 2π is the distance between the parti

derivative of the function ..

1. Compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable. (Compute the derivative of the function from the definition only, using limits. More advanced methods are not allowed here. Show your work.) (i) f(x) = x^2 - 1 x = -

Rate of Change for Radio Transmitter

Problem: A radio transmitter is located 3 km away from a straight highway. A truck is traveling away from the transmitter along the highway at a speed of 80 km per hour. How fast is the distance between the truck and the transmitter increasing when they are 5 km apart?

Write down the equation of a sinusoidal function

1. Write down the equation of a sinusoidal function f with the mean value 2, amplitude 4, the period 3 and f(6) = 2 2. Write down the equation of a sinusoidal function f with the mean value: -1, amplitude 5, the period 4 pi and f(0) = -1 Please show all working and explanations.

First Order Differential Equation for Temperature

Please solve the following differential equation: c(dT/dt) + v(T-Teq) = a0sin(wt) In this equation, c is effective heat capacity, T temperature, t time, Teq is equilibrium temperature, v, a0 and w are constants. Please derive an expression for T(t) in terms of the constants and show all the steps in the derivation.

Parabolas and Foci

The parabola C has the equation y^2 = 16x a. Verify that the point P(4t^2, 8t) is a general point on C. b. Write down the coordinates of the focus S of C. c. Show that the normal to C at P has equation y + tx = 8t + 4t^3. The normal to C at P meets the x-axis at the point N. d. Find the area of triangle PSN in terms of t, gi

Computing Consumer Behaviour with a Demand Function

Suppose consumers' demand function a commodity D(q)=50-3q-q2 dollars per unit. A. Find the number of units that will be bought if the market price is $32 per unit. B. Compute the consumers' willingness to spend to get the number of units in part (a).

Finding Maximum Values by Graphing

A Florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. Express the grower's total yield as a function of the number of additional trees planted, draw the grap

Concavity, derivations, and proofs

Please see attachment Determine whether....converge or diverge ...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove... Using binomial coefficients, derive a formula for the nth derivative of the product of two functions. Suppose that f(x) has a continuous first

Determining Concavity, Derivations, and Proofs

Please see attachment ONLY Q1,2,4 Determine whether....converge or diverge ...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove... Suppose that f(x) has a continuous first derivative for all x in R. Prove that f(x) is concave if and only if....

Concavity and derivations

Please see attachment Determine whether....converge or diverge ...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove... Using binomial coefficients, derive a formula for the nth derivative of the product of two functions. Suppose that f(x) has a continuous first

Probability

Find the indicated probability. 1) Two fair dice are rolled. What is the probability that a sum of 6 or 11 is obtained? 1) _______ A) 7/6 B) 17/36 C) 1/66 D) 7/36 Find the simple interest. Assume a 360-day year. Round results to the nearest cent. 2) $ 49,417 at 7.7% for 17 months 2) _______ A) $

Calculus - Volume of a Solid of Revolution

Question 5: Find the volume of the solid generated by revolving the region bounded by the graphs of the equations Please show all work. Please see the attached file for the fully formatted problems.

Average Value on an Interval, Mean Value Theorem

Please see the attached file for the fully formatted problems. Question 7: For the function find: a) its average value, (Average f), on the interval [0,9] b) the value of c in the interval [0,9] guaranteed by the Mean Value Theorem for Integrals. Question9: Evaluate the following limits: a. b.

Calculus - Limits

** Please see the attached PDF for the complete problem description ** Show all work and a step-by-step solution so that I can clearly see how you arrived at your answer. No calculators or computer programs can be used to solve this problem. Thanks

Global Minima and Maxima

Please see the attached PDF for the problem. Please show all work and step-by-step solution so that I can easily follow your reasoning. Do not use calculators or computer programs to find the solutions. Explain your reasoning and state any theorems used in the calculations.

To establish non-existence of multivariable limit

We examine the above function and consider its limit as (x,y)-> (0,0).We take two different paths in the x-y plane for approaching the point (0,0),and find that f(x,y) approaches two different values .This enables us to conclude that the given limit does not exist.For a detailed discussion see the solution given.

Application of Differential Calculus: Rate of Change

Problem: From your position, an airplane is moving away from you horizontally at 60 km/hr and vertically at 1km/hr. If the position of the airplane is 200km horizontally and 30km vertically away from you, what is the rate of change of distance of the airplane relative to your position?

Parametric Curve and Finding Points

Details of question attached including diagram of graph. The parametric equation of a curve are: x = 2 theta + sine 2 theta y = 4 sine theta Find the value of theta at A and the value of theta at B

Applied Calculus - Revenue Functions and Maximum Revenue

The function given below is a company's price function, where x is the quantity (in thouands) that will be sold at price p dollars and p = 5 - 1n x Find the revenue function R(x). Find the quantity and price that will maximize revenue. Applied Calculus 9th edition by Laurence D Hoffman and Gerald L. Bradley

Rate of Change of Speed and Distance

2) (3 pts) Suzy Sorority agr to give Joe Student a ride lionic for spring break. They agree to meet at the corner of 12 th St. and Willow. Joe is late and Suzy gets tired of waiting, so she starts to drive off down Willow st without him. Joe was already walking down 12th St., and he starts to shout and run towards the corner whe

Differential equations general solutions

See attached (1) Write the general solution for: x2 y" - 2xy+2y=x3 (2) Solve: (3) Solve: x2 y" + xy' - y=0 (4) Calculate L[x sinax] (5) In the vibration of a mass-spring-damper system, the displacement is governed by If f(t) is given by Find the solution if C=0 and x(t=0)=0 and dx

Non-Existent Limit Theorems

Show that the following limits do not exist: use def. of convergence or cauchy or epsilon delta a. lim as x goes to 0 of (1/(x^2)) x>0 b. lim as x goes to 0 of (1/(x^1/2)) same as (1/sqrtx) c. lim as x goes to 0 of (x + sgn(x)) d. lim as x goes to 0 of sin(1/(x^2)) let f: R---> R be defined by setting f(x)= x if x is

Maximum and Minimum Values and Points of Inflection

See the attached file. Question 1 Given the first derivative of the continuous function y = f(x). Find y" and sketch the graph of f. y' = tan x, -pi/2<x<pi/2 I know y"= sec2x - I need the graph and critical, inflection, and max/min points along with a picture of the graph or what need to be entered into m

Mrtgages and Amortization

If a loan of (A) dollars is amortized over n years at an annual interest rate r ( as a decimal) compounded monthly, the montly payments (M) are given by the formula: M = Ai / 1 - (1 + i) to the -12n power a)If a home loan is made for $165,000 at 6% annual interest, compounded monthly, for 30 years, What is the montly pay

Exponential Growth and Decay

Solve each using Q = Q0 e to the rt power A radioactive substance decays exponentially, If 500 grams of the substance were present intially and 400 grams are present 50 years later, how many grams will be present after 200 years? It is estimated that the population of a certain country grows exponentially. If the populatio

Calculus: Differential Equations

** Please see the attached file for the complete problem description ** Solve the differential equation...using the undetermined coefficients method. Solve the differential equation...using variation of parameters method. Solve the initial value problem... Solve by systematic elimination....

Line Integral Over a Path

Please see the attached file for the fully formatted problems. 1. find a formula for the function f(x,y) such that its gradient is (sin y, x*cos y) 2 Use teh answer to part 1 to compute the integral of the gradient over the path from (1,1) to (1,2)