Problem 9 - Calculus Early Transcendentals

Question

Two functions \(f\) and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. $$f(t)=100+10.5 t, g(t)=100 e^{t / 10}$$

Step-by-Step Solution

Answer

Answer: The growth rate of the linear function f(t) is constant at 10.5, and the relative growth rate of the exponential function g(t) is constant at 1/10 or 0.1.

Step 1: Derive the growth rate of the linear function f(t)

To find the growth rate of the linear function, we need to find \(f'(t)\). Differentiating the function \(f(t) = 100 + 10.5t\) with respect to \(t\), we get: $$f'(t) = \frac{d}{dt} (100+10.5t) = 10.5$$ Since the derivative of f(t) is a constant value, this means that the growth rate of the linear function is constant.

Step 2: Derive the growth rate of the exponential function g(t)

To find the growth rate of the exponential function, we need to find \(g'(t)\). Differentiating the function \(g(t) = 100e^{t/10}\) with respect to \(t\), we get: $$g'(t) = \frac{d}{dt} (100e^{t/10}) = 10e^{t/10}$$

Step 3: Calculate the relative growth rate of the exponential function g(t)

To find the relative growth rate of the exponential function, we need to find the ratio of the growth rate (i.e., \(g'(t)\)) to the current value of the function (i.e., \(g(t)\)). So we have: $$\frac{g'(t)}{g(t)} = \frac{10e^{t/10}}{100e^{t/10}} = \frac{1}{10}$$ Since the relative growth rate of the exponential function is a constant value, this means that the relative growth rate of the exponential function is constant. In conclusion, we have shown that the growth rate of the linear function \(f(t)\) is constant, and the relative growth rate of the exponential function \(g(t)\) is constant.

Key Concepts

Linear Functions

Linear functions are among the simplest types of functions, which have a constant rate of change. The general form of a linear function is \[ f(t) = mt + c \]where \(m\) is the slope and \(c\) is the y-intercept. A slope indicates how steep the line is. In the function \(f(t) = 100 + 10.5t\), the slope is 10.5. This tells us that for each unit increase in \(t\), the function increases by 10.5 units.

• **Constant Growth Rate:** The key characteristic of a linear function is its constant growth rate. This means that the function changes at a uniform rate over time.
• **Graph Representation:** On a graph, a linear function is represented as a straight line.

Derivatives can help identify this constant growth. The derivative of a linear function, such as \(f'(t) = 10.5\), is a constant value, reinforcing the idea that the function's growth rate does not change as \(t\) changes.

Exponential Functions

Exponential functions grow at a rate proportional to their value. The general form is \[ g(t) = a e^{bt} \]where \(a\) is the initial amount and \(b\) is the growth constant. When \(b\) is positive, the function grows, and when negative, it decays. In the function \(g(t) = 100e^{t/10}\), the constant \(1/10\) governs the rate of growth.

• **Rapid Increase:** These functions can increase rapidly because the rate of change multiplies as the function value gets larger.
• **Exponential Curve:** On a graph, this function is a smooth curve, which bends upwards with increasing \(t\), showing accelerated growth.

The derivative \(g'(t) = 10e^{t/10}\) shows that the function's rate of change also increases exponentially with \(t\). This property sets them apart from linear functions, where the rate of change remains constant.

Derivatives

Derivatives are a mathematical tool used to find how a function changes with respect to one of its variables. In simpler terms, it can be understood as the 'instantaneous rate of change' of a function.

• **Linear Function Derivative:** For linear functions, such as \(f(t) = 100 + 10.5t\), the derivative \(f'(t) = 10.5\) is constant. This indicates a fixed growth rate.
• **Exponential Function Derivative:** For exponential functions like \(g(t) = 100e^{t/10}\), the derivative \(g'(t) = 10e^{t/10}\) represents the growing rate of change, showing how quickly the function value is accelerating.

Working with derivatives simplifies understanding the behavior of different types of functions over time. They are crucial in fields like physics, economics, and any application needing analysis of change.