Problem 19 - Applied Calculus

Question

The value of a mutual fund increases at a rate of \(R=\) \(500 e^{0.04 t}\) dollars per year, with \(t\) in years since \(2010 .\) (a) Using \(t=0,2,4,6,8,10\), make a table of values for \(R\). (b) Use the table to estimate the total change in the value of the mutual fund between 2010 and 2020 .

Step-by-Step Solution

Answer

The total change in value is approximately \(6151.09\) dollars.

Step 1: Calculate R for t=0

Substitute \(t=0\) into the formula \(R = 500 e^{0.04t}\). This gives \(R = 500 e^{0}\). Since \(e^0 = 1\), \(R = 500\).

Step 2: Calculate R for t=2

Substitute \(t=2\) into the formula \(R = 500 e^{0.04t}\). This gives \(R = 500 e^{0.08}\). Calculate \(e^{0.08} \approx 1.08329\). Therefore, \(R \approx 500 \times 1.08329 \approx 541.645\).

Step 3: Calculate R for t=4

Use \(t=4\) in the equation \(R = 500 e^{0.04t}\). This becomes \(R = 500 e^{0.16}\). Calculate \(e^{0.16} \approx 1.17351\). Thus, \(R \approx 500 \times 1.17351 \approx 586.755\).

Step 4: Calculate R for t=6

Set \(t=6\) in \(R = 500 e^{0.04t}\). This results in \(R = 500 e^{0.24}\). Calculate \(e^{0.24} \approx 1.27125\). Thus, \(R \approx 500 \times 1.27125 \approx 635.625\).

Step 5: Calculate R for t=8

Insert \(t=8\) into the formula \(R = 500 e^{0.04t}\). This gives \(R = 500 e^{0.32}\). Calculate \(e^{0.32} \approx 1.37713\). So, \(R \approx 500 \times 1.37713 \approx 688.565\).

Step 6: Calculate R for t=10

Plug \(t=10\) into the equation, giving \(R = 500 e^{0.4}\). Calculate \(e^{0.4} \approx 1.49182\). Therefore, \(R \approx 500 \times 1.49182 \approx 745.91\).

Step 7: Create the Table of Values

Based on the previous steps, fill in the table for each value of \(t\): \(t = 0, R = 500\)\(t = 2, R = 541.645\)\(t = 4, R = 586.755\)\(t = 6, R = 635.625\)\(t = 8, R = 688.565\)\(t = 10, R = 745.91\)

Step 8: Estimate Total Change Using the Trapezoidal Rule

To estimate the total change in value from 2010 to 2020, use the formula for the trapezoidal rule: \(\Delta V \approx \frac{\Delta t}{2} [f(t_0) + 2f(t_2) + 2f(t_4) + 2f(t_6) + 2f(t_8) + f(t_{10})]\) with \(\Delta t = 2\).Substitute the values:\(\Delta V \approx \frac{2}{2} [500 + 2(541.645) + 2(586.755) + 2(635.625) + 2(688.565) + 745.91]\)Calculate \(\Delta V \approx 500 + 1083.29 + 1173.51 + 1271.25 + 1377.13 + 745.91\)So, \(\Delta V \approx 6151.09\).

Key Concepts

Exponential Growth

Exponential growth is a mathematical concept describing processes where quantities increase at rates proportional to their current amount. In the context of the original exercise, the mutual fund value increases exponentially over time. This is modeled by the equation \( R = 500 e^{0.04t} \), where \( R \) is the rate of increase and \( t \) is the number of years since 2010. The constant 500 represents the initial rate of growth. The function uses the base \( e \), a mathematical constant approximately equal to 2.71828, which is commonly used in growth and decay problems due to its unique properties. Key features of exponential growth include:

• Faster growth over time
• A consistent relative growth rate
• Appearing in various scientific and financial contexts

This growth model helps in financial contexts like mutual funds, forecasting parameters like expected returns.

Trapezoidal Rule

The trapezoidal rule is a numerical method to estimate the integral, or total change, of a function. In our exercise, it is used to estimate the total change in the value of the mutual fund over a decade. The rule approximates the region under a curve by dividing it into trapezoids rather than exact integrals, which are often complex.The formula used is:\[ \Delta V \approx \frac{\Delta t}{2} [f(t_0) + 2f(t_2) + 2f(t_4) + 2f(t_6) + 2f(t_8) + f(t_{10})] \]where \( \Delta t \) is the interval between each time period. In the exercise, \( \Delta t = 2 \) years. This rule provides a simplified yet adequate estimation of the integral, making it particularly handy in applied sciences and financial calculations where only discrete data are available.

Rate of Change

The rate of change measures how one quantity changes as another quantity changes, fundamentally captured by the derivative in calculus. In this exercise, the rate of change refers to \( R = 500 e^{0.04t} \), the rate at which the fund's value is growing annually. Characteristics include:

• Applied to describe velocity in physics or financial growth as in this example.
• Can be constant or variable, based on other parameters in the equation.
• Helps understand dynamic systems in real-world contexts.

Understanding the rate of change helps investors anticipate how fast their investments might grow, impacting future financial decisions and investments.

Mutual Fund

A mutual fund is a financial vehicle consisting of a pool of money collected from many investors. It invests in securities like stocks, bonds, money market instruments, and other assets. Mutual funds are managed by professional managers who allocate the fund's assets and attempt to produce capital gains or income for the fund's investors. Key features include:

• Diversification, risking the investment across multiple assets, reducing impact if one fails.
• Accessibility and affordability, requiring lower minimum investments than direct buying.
• Professional management, where expert fund managers make investment decisions.

In the exercise, the mutual fund's value is modeled mathematically to determine its growth, illustrating foundational principles in finance and investment strategies.