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# Calculus and Analysis

### How do I integrate sine to an odd power of 3 or higher?

The steps for integrating sine to an odd power of 3 or higher are shown using the example Ssin^5(x)dx. The solution is detailed and well presented.

### Differential equation from calculus II

Given the differential equation: (y^4)(e^2x) + y' = 0 NOTE: The differential equation above is attached in a microsoft word document for better legibility. Additionally my work is attached as a jpeg file. The questions: a)Find the general solution. b)Find the particular solution such that y(0) = 1.

### Newton's Law of Cooling relating to differential equations.

At 10:00 AM, an object is removed from a furnace and placed in an environment with a constant temperature of 68 degrees. Its core temperature is 1600 degrees. At 11:00 AM, its core temperature is 1090 degrees. Find its core temperature at 5:00 PM on the same day.

### Increasing functions: Comparing Functions

Explain why the graph of f(x) is rising over an interval a < or equal to x < or equal to b if f '(x) > 0 throughout the interval. What can you say about the graph of f if f '(x) is less than zero on a < or equal to x < or equal to b?

### Sample Question: Word problems

1. The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 320 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decreas

### Geometric applications for calculus

I need an overview of geometric applications for calculus.

### Finding the slope and other various characteristics of a line.

Given the points (3,7) and (-1, 3), find the slope of the line containing these 2 points, find the distance between these 2 points and find the midpoint.

### Calculus

1. A weather balloon is rising vertically at a constant rate of 4 ft/s directly above a straight and level road. When the balloon is 75 ft above the road, a car moving at 55 ft/s passes directly under the balloon. Based on this information find: a. the rate the distance between the balloon and the car is changing 3 sec after t

### Help me to understand the formal setup of the Chain Rule.

The procedure is shown using the easy example y=(5x^4+3x^2-2)^7.

### How do I know when to use the Chain Rule and when not to?

The question is answered by contrasting the procedures for taking the derivatives of f(x)=x^2-3x+7 and f(x)=(x^2-3x+7)^4.

Find two real numbers whose sum is 10 and whose product is maximal?

### A minimization fencing problem.

A rectangular field is going to be enclosed and divided into two separate rectangular areas. (Areas do not have to be equal). Find the minimum fencing that is required if the total area of the field is 1200m2.

### Working with infinite sequences and series.

If series Sum(an) and Sum(bn) with positive terms are convergent, is the series Sum(an*bn) converegent? Note: 1. Sum replaces the symbol for summation 2. an and bn are nth elements of the two series

### Calculating rates of change in a loan situation.

The formula for the loan one can get with a payment of \$P paying monthly for 15 years at an interest rate of r is: L=(12P/r)[1-(1+(r/12))^(-180)] a.) Find dL/dt, the rate of change of the loan with respect to time. (Here, t is the time that is passing, not the t in the original function if you know the loan. Trea

### Exponential growth and decay

A leaking oil tank has a capacity of 500 000 liters of oil. The rate of leakage depends on the pressure of oil remaining in the tank and the pressure depends on the height of oil. When the tank is half-full, it loses 20L/min. How long goes it take to lose 15 000L from half-full?

### Laplace transforms

Please see the attached problem file

### Explaination for derivatives

Explaination for derivatives related to exponential and logarithmic functions ,formulae used to solve them and solutions to some problems. All problems are in the solution file

### What is the equation of three bisecting solid rods centered at the origin?

Hello, What is the equation of three bisecting solid rods centered at the origin? Given 3 solid rods of length 3 and diameter 1. One rod is on the x-axis One rod is on the y-axis and One rod is on the z-axis Each is centered at the origin and is perpendicular to the other rods in each axis. Need equation in rectan

### When to use the Chain Rule

The key is whether or not you are plugging the result of a function into another function. The idea is shown by contrasting the procedures for taking the derivatives of sin(x^2) and x^2*sin(x).

### Maxima: a)how many eggs are produced in one day. b)When are the eggs produced at the fastest rate

Eggs are produced at a rate of R(t)eggs per hour,where t=0 represents 12:00 midnight and R(t)(in thousands of eggs) is :- R(t)= -10cospi/12t+10 a)how many eggs are produced in one day. b)When are the eggs produced at the fastest rate c)A machine can produce eggs at a constant rate. At the end of 1 week the same

### Euclidean space

Compute the distance from a point b = (1, 0, 0, 1)^T to a line which passes through two points (0, 1, 1, 0)^T and (0, 1, 0, 2)^T. Here ^T denotes the operation of transposition, i.e. the points are represented by column-vectors instead of row-vectors.

### Differential Equation

Xy' + (1+x)y = 3 with y(4) = 50

### Calculating first order differential equations.

Y' = (x+2)^2e^y with y(1) = 0

### Calculating rates of separation for related rates.

Northbound ship A leaves the harbour at 10:00 with a speed of 12km/h. Westbound ship B leaves the same harbour at 10:30 with a speed of 16km/h. (a) How fast are the ships separating at 11:30? (b) When is their rate of separation 18.86 km/h

### Chain rule/derivatives HW

What did I do wrong? 1. Find f'(x) when f(x)= 5x(sinx + cosx) My answer: cos(4x^2)- sin(6x^2)/(5x^2) 2. Find f'(x) when f(x)= ((x^3) + 4x + 4))^2 My answer: 6x^2(x^3 + 4x + 4) 3. Find f'(x) when f(x)= (3x + 8)^-3 My answer: -6(3x + 8) 4. Find f'(x) when f(x)= Sq root of (5x + 8) My answer: x/5x + 8 5. Find f'(x) when f(

### Finding the surface area of the intersection of two cylinders.

Find the surface area of the solid that is the intersection of the two solid cylinders: x^2 + z^2 <= k^2 (x squared plus z squared is less than or equal to a constant squared) AND x^2 + y^2 <= k^2 (x squared plus y squared is less than or equal to the same constant squared) What is my f(x,y)? What are my limits of integr

### Having difficulty with algebra in calculus

Find intervals on which the function is: (a) increasing (b) decreasing (c) concave up (d) concave down find any (e) local extreme values and (f) inflection points for the equation y = x to the 4/5 power times(2-x)

### Dissecting a trig function

Let f be the function defined by f(x)=sin squared x - sinx for 0<or=x<or=(3pi)over 2. a. find the x- intercepts of the graph of f b.find the intervals on which f is increasing c. find the absolute maximum and absolute minimum value of f. Justify your answer.

### Tan line and velocity problems

The parabola y = (x^2) + 3 has two tangents which pass through the point (0, -2). One is tangent to the to the parabola at (A, A^2 + 3) and the other at (-A, A^2 + 3). Find (the positive number) ? If a ball is thrown vertically upward from the roof of 64ft foot building with a velocity of 96 ft/sec, its height after t seconds

### Calculating the derivative from mathematical expressions.

Find the derivative of: f(x) = sqaure roo 4 / t^3 f(x) = 5 + 2/x + 4/x^2 f(x) = ((4x^3) - 2)/ x^4