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?

Velocity and acceleration

On another planet, Jim throws a ball directly up into the air. The velocity of this ball is a linear function of t. He finds that the velocity of the ball at 1 second is 44 meters/second. At 4 seconds, it is -49 meters/second. a.) Find v(t), the velocity of the ball at time t seconds. b.) Find a(t), the acceleration of t ...continues

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 ...continues

Minimization problems dealing with lengths.

A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to have 1.5 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used.

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

First and second derivatives

Find the first and second derivatives of the function y=((1-x)/x^2)^3

Infinite sequences and series

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.

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.

Maximization

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

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.