Chapter 6MA1102 Business Mathematics

6.1 Integration by Parts

Integration by Parts

If u(x) and v(x) are both differentiable functions of x, then

Integration by Parts Formula

Since , then

Examples.1. .

2. .

3. Find the area of the region bounded by the curve y = ln x, the x-axis, and lines x = 1 and x = e.

4. .

6.2 Introduction to Differential Equations

Differential Equation

Sometimes the mathematical formulation of a problem involves an equation in which a quantity and the rate of change of that quantityare related by an equation.

Since rates of change are expressed in terms of derivatives or differentials, such an equation is called a differential equation.

Example.

The simplest type of differential equation has the form

General and Particular Solutions

A complete characterization of all possible solutions of the equation is called a general solution.

A differential equation coupled with a side condition is referred to as an initial value problem, and a solution that satisfies both the differential equation and the side condition is called a particular solution of the initial value problem.

Examples

1. .

2. .

Rate of Depreciation

The negative of the rate of change of resale value of the machine is called the rate of depreciation.

Another Example

An oil well that has just been opened is expected to yield 300 barrels of crude oil per month and, at that rate, is expected to run dry in 3 years. It is estimated that tmonths from now, the price of crude oil will be 𝑝 𝑡 = 28 + 0.3 𝑡 dollars per barrel.

If the oil is sold as soon as it is extracted from the ground, what is the total revenue generated by the well during its operation?

Let R(t) be the revenue generated during the first t months after the well is opened, so that R(0) = 0.

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Separable Equation

Example.

Find the general solution of the differential equation 𝑑𝑦

𝑑𝑥=

2𝑥

𝑦2.

A Price Adjustment Model

Let S(p) denote the number of units of a particular commodity supplied to the market at a price of p dollars per unit, and let D(p) denote the corresponding number of units demanded by the market at the same price.

In static circumstances, market equilibrium occurs at the price where demand equals supply. However, certain economic models consider a more dynamic kind of economy in which price, supply, and demand are assumed to vary with time.

The Evans price adjustment model assumes that the rate of change of price with respect to time t is proportional to the shortage D - S, so that

𝑑𝑃

𝑑𝑡= 𝑘(𝐷 − 𝑆)

where k is a positive constant.

An Example

Suppose the price p(t) of a particular commodity varies in such a way that its rate of change with respect to time is proportional to the shortage D - S, where D(p) and S(p) are the linear demand and supply functions D = 8 - 2pand S = 2 + p.a. If the price is $5 when t = 0 and $3 when t = 2, find p(t).b. Determine what happens to p(t) in the “long run” (as t → )......

6.3 Improper Integral

Improper Integral

The definition of the definite integral 𝑎𝑏𝑓 𝑥 𝑑𝑥 requires the

interval of integration 𝑎 ≤ 𝑥 ≤ 𝑏 to be bounded, but in certain applications, it is useful to consider integrals over unbounded intervals such as 𝑥 ≥ 𝑎.

We denote the improper integral of 𝑓 over the unbounded interval 𝑥 ≥ 𝑎 by

𝑎∞𝑓 𝑥 𝑑𝑥.

If 𝑓 𝑥 ≥ 0 for 𝑥 ≥ 𝑎, this integral can be interpreted as the area of the region under the curve 𝑦 = 𝑓 𝑥 to the right of 𝑥 = 𝑎.

Although this region has infinite extent, its area may be finite or infinite, depending on how quickly 𝑓 approaches zero as 𝑥increases indefinitely.

Counting Improper Integral

Examples.Evaluate the improper integrals.

1. 1∞ 1

𝑥2𝑑𝑥 .

2. 1∞ 1

𝑥𝑑𝑥 .

Present Value of Perpetual Income Flow

A donor wishes to endow a scholarship at a local college with a gift that provides a continuous income stream at the rate of 25,000 + 1,200tdollars per year in perpetuity.

Assuming the prevailing annual interest rate stays fixed at 5% compounded continuously, what donation is required to finance the endowment?

Other Example

The fraction of patients who will still be receiving treatment at a certain health clinic t months after their initial visit is f(t) = e-t/20. If the clinic accepts new patients at the rate of 10 per month, approximately how many patients will be receiving treatment at the clinic in the long run?

Continuous Probability

Improper integrals also appear in the study of probability and statistics.

For example, the life span of a lightbulb selected at random from a manufacturer’s stock is a quantity that cannot be predicted with certainty. The process of selecting a bulb is called a random experiment, and the life span X of the bulb is a continuous random variable.

Other examples of continuous random variables include: • the time a randomly selected motorist spends waiting at a traffic light, • the weight of a randomly selected person, or • the time it takes a randomly selected person to learn a particular task.

Probability

The probability of an event that can result from a random experiment is a number between 0 and 1 that specifies the likelihood of the event.

Example.

In the lightbulb example, one possible event is that a bulb selected randomly from the manufacturer’s stock has a life span between 20 and 35 hours. If X is the random variable denoting the life span of a randomly selected bulb, then this event can be described by the inequality 20 X 35 and its probability denoted by P(20 X 35).

The probability that the bulb will burn for at least 50 hours is denoted by P(X 50) or P(50 X ).

Probability Density Function

An Example

Example.

A possible probability density function for the life span of a lightbulb.

Uniform Density Function

Expected Value

If a random experiment is performed repeatedly and the results are recorded, then the arithmetic average of the recorded results will approach the expected value, so in this sense, E(X) can be thought of as the “average” of the random variable X.

Example:

• the average highway mileage for a particular car model,

• the average waiting time to clear security at a certain airport, and

• the average life span of a particular appliance.

An Example

A certain traffic light remains red for 40 seconds at a time. You arrive (at random) at the light and find it red.

Use an appropriate uniform density function to find the probability that you will have to wait at least 15 seconds for the light to turn green.

Find the expected value of the uniformly distributed random variable.

(This says that the average waiting time at the red light is 20 seconds.)