Question

You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=\frac{12,000 t}{\left(t^{2}+2\right)^{2}} $$

Short answer

The capital accumulation over a five-year period is found by evaluating the expression obtained in Step 3. This represents the amount of investment accumulated over the period.

Step-by-step solution

Identify the expression to be integrated

The expression that needs to be integrated in this scenario is \(\frac{12,000 t}{\left(t^{2}+2\right)^{2}}\), which represents the rate of investment over time. This is a somewhat complex integral because of the quotient involved, so it is necessary to handle it carefully and in a step-by-step manner.

Calculate the integral

To solve the integral \(\int_{0}^{5}\frac{12,000 t}{\left(t^{2}+2\right)^{2}} dt\), it is advisable to use a substitution method. Let's set \(u = t^2+2\), then \(du = 2t dt\). We can rewrite the integral as \(\int_{0}^{5}\frac{6,000 du}{u^{2}}\), which simplifies to \(-\int_{0}^{5}\frac{6,000 }{u} du\). Executing this definite integral gives \(-6,000 (\ln |u|]_{0}^{5})\).

Substitute back

Having obtained the result of the integral in terms of \(u\), it needs to be expressed back in terms of \(t\). Replace \(u\) with \(t^{2}+2\) in \(-6,000 (\ln |u|]_{0}^{5})\) and simplify the expression to obtain the final result.

Key concepts

Rate of Investment Calculus

Understanding the rate of investment through calculus is crucial in financial mathematics, particularly when assessing how an investment grows over time. The rate of investment, symbolized by \( \frac{dI}{dt} \), quantifies the change in investment with respect to time. When we integrate this rate over a certain time period - in this case, five years - we're essentially summing up all the little changes to get the total change in investment, or the capital accumulation.

To shed more light on the process, let's delve into the provided exercise. We consider the formula \( \frac{dI}{dt} = \frac{12,000 t}{(t^2 + 2)^2} \). This expression represents a variable rate of investment over time, as it's not constant but depends on the function of time \(t\). In real-life scenarios, such rates could be influenced by many factors such as market conditions, interest rates, and business performance. The integral of this rate from 0 to 5 years will show us the total amount of investment added or accumulated over these years.

To enhance the understanding of this concept for students, practical examples could help. For instance, drawing a graph of the rate of investment over time and demonstrating how the area under the curve represents capital accumulation can make the concept more tangible.

Capital Accumulation

Capital accumulation is a key concept in economics and finance, reflecting the growth of assets over time. It's the result of reinvesting earnings rather than consuming them, which leads to more assets and, consequently, more future earnings. This can be thought of as the snowball effect in personal savings or compound interest.

In our example, the integral \( \int_{0}^{5} \frac{dI}{dt} dt \) quantifies the capital accumulation over five years by summing up infinitesimal investments accrued over each moment within this time frame. The process of integration, then, gives us a powerful tool to forecast the future value of an investment based on its rate of growth over time.

Incorporating this concept into our exercise, the solution will provide students with the exact amount of capital accumulation from the given rate of investment. The exercise can be made more relatable by comparing it to everyday savings account scenarios, where compound interest plays a similar role to capital accumulation in the growth of initial deposits over time.

Substitution Method Integration

The substitution method, also known as u-substitution, is a technique used in integration to simplify complex integrals. It is particularly useful when dealing with integrals of functions that are products or quotients involving composite functions. The substitution method often transforms an integral into a simpler form that is easier to evaluate.

In our exercise, we approached the definite integral \( \int_{0}^{5}\frac{12,000 t}{(t^2 + 2)^2} dt \) by letting \( u = t^2 + 2 \) and \( du = 2t dt \). This substitution helped us transform the integral into a simpler form, \( \int_{0}^{5}\frac{6,000}{u^2} du \), which then simplifies into \( -6,000 (\ln |u|]_{0}^{5}) \). The steps taken to execute this method are essential for students to understand not just the procedure of integration but also the strategic application of mathematical techniques to solve problems.

Improving the Substitution Method Understanding

To further improve a student's grasp on the substitution method, practice problems with varying levels of complexity should be provided. Visual aids, such as mapping the transformation from \(t\) to \(u\), and step-by-step walk-throughs highlighting the logic behind each substitution move, can solidify understanding of the concept. By mastering the u-substitution method, students will enhance their ability to tackle a wide array of integral problems with confidence.