Problem 11 - Calculus: An Intuitive and Physical Approach

Question

Show that if a phenomenon obeys the law \(y=e^{k x}\), then for successive values of \(x\) that are in arithmetic progression the corresponding values of \(y\) are in geometric progression. Suggestion: Suppose that the successive values of \(x\) are \(0, h\), \(2 h, \ldots\), and calculate the corresponding values of \(y\).

Step-by-Step Solution

Answer

For x values in arithmetic progression, the corresponding y values, calculated as e^(kx), have a constant ratio e^(kh) for any successive pair, thus forming a geometric progression.

Step 1: Identify Successive Values of x

Assume the successive values of x are in arithmetic progression with a common difference h. Therefore, the series of x values are 0, h, 2h, ...

Step 2: Calculate the Corresponding y Values

Using the given law y = e^(kx), calculate the y values for the x values 0, h, 2h, ... which are e^(k*0), e^(k*h), e^(k*2h), ... respectively.

Step 3: Show the Ratio of Successive y Values

To establish a geometric progression, we find the ratio of successive y values. The ratio of y corresponding to h and 0 is e^(kh)/e^(k*0) = e^(kh). The ratio of y corresponding to 2h and h is e^(k*2h)/e^(kh) = e^(kh). Showing these ratios are constant proves the y values form a geometric progression.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers wherein each term after the first is obtained by adding a constant difference to the previous term. This difference is typically denoted as 'd' or 'h' in some textbooks. For example, consider the sequence 2, 5, 8, 11, ... Here, we can observe that the difference between consecutive terms is always 3. Therefore, the sequence is an arithmetic progression.

In the context of the exercise, we consider successive values of 'x' that are in arithmetic progression, such as 0, h, 2h, 3h, etc. Each term adds the common difference 'h' to the previous term. This fact forms a key part of our problem, which links how an arithmetic progression in the domain relates to a geometric progression in the range.

Geometric Progression

A geometric progression (GP), unlike an arithmetic progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the 'ratio'. This ratio is frequently denoted by 'r'. For instance, in the sequence 3, 6, 12, 24, ..., each term is twice the preceding term, giving us a common ratio of 2.

In solving our calculus problem, our goal is to show that if 'x' follows an arithmetic pattern, the corresponding 'y' values, calculated using the exponential function, will actually form a geometric progression. Understanding what constitutes a geometric sequence helps us to recognize patterns and demonstrates the connections between the exponential behavior of the function and multiplicative patterns in sequences.

Exponential Functions

Exponential functions are characterized by the form \(y = a^{bx}\), where 'a' is the base (a positive number) and 'bx' is the exponent. These functions grow (or decay) at a rate proportional to their current value, making them extremely significant in representing processes with constant proportional growth rates, such as population growth or radioactive decay.

The exercise provided uses the natural exponential function \(y = e^{kx}\) where 'e' is the base, the natural logarithm's base (approximately 2.718), and 'k' being a constant that determines the growth rate. The exponential nature of this function plays a crucial role in linking arithmetic progression with geometric progression, as seen in the calculations of the corresponding 'y' values for progressively increasing 'x' values.

Calculus Problem Solving

Calculus offers a framework for understanding changes and allows us to tackle a wide variety of mathematical problems. Problem solving in calculus often involves identifying patterns, making use of functions, and manipulating these functions to demonstrate certain properties.

In our exercise to connect exponential functions to progressions, calculus doesn't only help in solving the problem but also in describing natural phenomena and predicting outcomes. By transforming an arithmetic progression in the 'x' domain to a geometric progression in the 'y' range through the exponential function, we highlight the fascinating connections that calculus exposes. It demonstrates not just quantitative skills, but also an understanding of the intrinsic nature of the relationships within sequences and how functions translate one form of progression to another.