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Chapter 4: Problem 13

Future Value The future value of an investment after \(t\) years is given by $$ F(t)=120\left(1.126^{t}\right) \text { thousand dollars } $$ a. Calculate the future value and the rate of change of the future value after 10 years. b. Write the linearization of \(F\) after 10 years. c. Use the linearization to estimate the future value after 10.5 years.

Short Answer

Expert verified
The estimated future value after 10.5 years is approximately 413.505 thousand dollars.

Step by step solution

01

Calculate Future Value after 10 years

To find the future value after 10 years, substitute \( t = 10 \) into the function \( F(t) \). \[ F(10) = 120 \times (1.126)^{10} \] Use a calculator to solve:\[ F(10) \approx 120 \times 3.253 \approx 390.36 \text{ thousand dollars} \]
02

Calculate the Rate of Change using Derivative

First, find the derivative of \( F(t) \) with respect to \( t \) to determine the rate of change. \[ F'(t) = \frac{d}{dt}[120(1.126)^t] = 120 \times \ln(1.126) \times (1.126)^t \] Now substitute \( t = 10 \) to find \( F'(10) \): \[ F'(10) = 120 \times \ln(1.126) \times (1.126)^{10} \] Calculate the value:\[ F'(10) \approx 120 \times 0.1184 \times 3.253 \approx 46.29 \text{ thousand dollars per year} \]
03

Write the Linearization of F at 10 years

The linearization of \( F(t) \) at \( t = 10 \) can be expressed using the formula:\[ L(t) = F(10) + F'(10)(t - 10) \]Substitute the previously calculated values:\[ L(t) = 390.36 + 46.29(t - 10) \]
04

Estimate the Future Value after 10.5 years

Use the linearization to estimate \( F(10.5) \):\[ L(10.5) = 390.36 + 46.29(10.5 - 10) \] \[ L(10.5) = 390.36 + 46.29 \times 0.5 \approx 390.36 + 23.145 = 413.505 \text{ thousand dollars} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate of Change
When examining the future value of an investment, understanding its rate of change is crucial. The rate of change tells us how quickly the value is increasing or decreasing over time.
This can be illustrated using calculus, specifically the derivative. In this context, the derivative of a function gives us the instantaneous rate of change at any given point.
  • The derivative, denoted here as \( F'(t) \), represents the speed at which the future value is changing at time \( t \).
  • In our example, after calculating, we find \( F'(10) \) to be approximately 46.29.
    This means that at 10 years, the future value is increasing by 46.29 thousand dollars per year.
This information is key for investors to understand how their investment is performing over time and to make informed decisions about future investments.
Linearization
Linearization is a handy tool when we want to approximate the value of a function near a specific point using a straight line.
This is particularly useful when dealing with complex functions.
  • To linearize a function like \( F(t) \) at a certain point, for example \( t = 10 \), we use the function value and its derivative.
  • The formula for linearization is \( L(t) = F(10) + F'(10)(t - 10) \), where \( L(t) \) represents the linear approximation of the function.
In this example, linearization allows us to estimate future values easily, without complex calculations. This is especially helpful for quickly estimating the value at times such as 10.5 years.
Derivative
In mathematics, the derivative of a function represents the rate at which a function is changing at any given point in time.
The derivative is a central concept in calculus, often used to understand how quantities change.
  • For our function \( F(t) = 120(1.126)^{t} \), the derivative \( F'(t) \) is calculated to track how the future value grows.
  • It involves finding \( \frac{d}{dt}[120(1.126)^t] \), which gives insight into changes in value as time progresses.
The derivative is not only crucial for understanding how investment values change over time but also helps in constructing linearizations and approximations for better predictions.
Exponential Growth
Exponential growth describes a process where the quantity increases at a rate proportional to its current value.
This means as time goes on, the quantity grows faster and faster.
  • Our function, \( F(t) = 120(1.126)^t \), is a classic example of exponential growth, where the base \( 1.126 \) indicates growth by approximately 12.6% per year.
  • This type of growth is often more realistic for financial scenarios like investments, as they tend to grow by a fixed percentage over time.
Such growth models are important for investors to predict potential returns. Understanding exponential growth allows for better financial planning and evaluation of long-term investment strategies.

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