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The rate of change is a fundamental concept in both mathematics and business finance. In this context, it refers to how quickly the amount of capital invested changes over time. Mathematically, the rate of change of a function is represented by its derivative.

For example, if we denote the amount of capital invested at time \( t \) as \( f(t) \), then the rate at which this capital changes over time is given by the derivative \( f'(t) \). This derivative tells us whether the firm's investment is increasing or decreasing and by how much.

In our scenario, the rate of net investment, or \( f'(t) \), is crucial for the business to achieve optimal investment levels. This rate must be managed carefully to ensure the firm's financial stability and growth.

Understanding the rate of change helps businesses in strategizing and making informed decisions regarding their investments.

Investment capital refers to the funds allocated by a business for the purpose of generating returns. This capital can be used to purchase assets, finance projects, or support operational costs. In our exercise, we denote the investment capital at time \( t \) as \( f(t) \).

Managing investment capital effectively is vital for any business. It involves ensuring that investments are made wisely and that the rate of net investment aligns with the company's financial goals.

The optimum level of investment, represented as \( C \), is an important target set by the firm's management. Hitting this target can help achieve maximum efficiency and profitability.

By understanding the dynamics of investment capital and its management, businesses can better navigate their financial planning and optimize their resources.

A proportional relationship is one where two quantities maintain a constant ratio. In our case, the rate of net investment \( f'(t) \) is proportional to the difference between the optimum investment level \( C \) and the current total capital invested \( f(t) \). This can be expressed mathematically as:

\ ( f'(t) = k (C - f(t)) \

Here, \( k \) is the proportionality constant. This equation indicates that the rate of net investment changes directly with the discrepancy between the optimal capital and what is already invested.

Proportional relationships are useful in many areas of finance and business as they help create predictable models. Knowing how the rate of investment adjusts in proportion to the gap between current and optimal investments allows firms to strategize better and maintain equilibrium in their financial operations.