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    Calculus and Analysis

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    Functions : Critical Values

    The function f(x)=2x^3 - 33x^2 + 108x - 6 has two critical numbers. The smaller one equals ______ and the larger one equals______.

    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.

    Maximization to Construct a Fence

    A Farmer wants to construct a fence. The area that he is going to enclose is rectangular, but one of the sides is a river (assumed to be straight). If he has 120 m of fence, what is the maximum surface that he can enclose?

    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

    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

    Chain Rule Examples

    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.

    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.

    Infinite Sequences and Series Example Problem

    In the figure (see attachment) there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.

    Infinite Series Convergence

    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?

    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

    Bisecting Rods Centered

    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).

    Rate of production of eggs

    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 - computing distance

    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.

    Related Rates of Ships

    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(

    Find 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