Problem 10 - 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. $$f(t)=2200+400 t, g(t)=400 \cdot 2^{1 / 20}$$

Step-by-Step Solution

Answer

Answer: The constant growth rate for the linear function \(f(t)\) is \(400\), and the constant relative growth rate for the exponential function \(g(t)\) is \(\frac{1}{20} \ln(2)\).

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

To find the growth rate of the linear function \(f(t)=2200+400t\), we'll take the first derivative with respect to \(t\): $$\frac{df}{dt} = \frac{d}{dt}(2200+400t)$$ Using the power rule, we find: $$\frac{df}{dt} = 400$$

Step 2: Growth rate of the linear function is constant

Since the first derivative of the linear function \(f(t)\) is just a constant, \(400\), it means that the growth rate of this function is constant.

Step 3: Find the growth rate of \(g(t)\) (exponential function)

To find the growth rate of the exponential function \(g(t)=400 \cdot 2^{1 / 20}\), we'll take the first derivative with respect to \(t\): $$\frac{dg}{dt} = \frac{d}{dt}(400 \cdot 2^{1 / 20 t})$$ We will use the chain rule here: $$\frac{dg}{dt} = 400 \cdot 2^{\frac{1}{20}t} \cdot \frac{1}{20} \cdot \ln(2)$$

Step 4: Find the relative growth rate of the exponential function

To find the relative growth rate, we'll divide the growth rate by the function itself: $$\text{Relative growth rate of} \ g(t) = \frac{\frac{dg}{dt}}{g(t)}$$ Plugging in our values, we get: $$\frac{400 \cdot 2^{\frac{1}{20}t} \cdot \frac{1}{20} \cdot \ln(2)}{400 \cdot 2^{\frac{1}{20}t}}$$

Step 5: Simplify

We can simplify the expression by canceling out the terms: $$\frac{400 \cdot 2^{\frac{1}{20}t} \cdot \frac{1}{20} \cdot \ln(2)}{400 \cdot 2^{\frac{1}{20}t}} = \frac{1}{20} \ln(2)$$

Step 6: Relative growth rate of the exponential function is constant

As we can see, the relative growth rate of the exponential function \(g(t)\) is a constant, which is \(\frac{1}{20} \ln(2)\), which proves that the relative growth rate of the exponential function is constant.

Key Concepts

Linear Functions

Linear functions are fundamental in calculus and mathematics as a whole. In simple terms, a linear function is one where the output changes at a constant rate proportional to the change in the input. The standard form for a linear function is given by: \[ f(t) = mt + b \]where:

• \( m \) is the slope, indicating the rate of change.
• \( b \) is the y-intercept, the value of the function when \( t = 0 \).

For our specific function \( f(t) = 2200 + 400t \), the slope is 400. This means for every unit increase in \( t \), \( f(t) \) increases by 400.

Constant Growth Rate

The growth rate of a linear function is constant because its derivative is a constant value. The derivative, denoted by \( \frac{df}{dt} \), represents the rate of change of the function with respect to \( t \). For our function:\[ \frac{df}{dt} = 400 \]No matter the value of \( t \), the growth rate of \( f(t) \) remains 400, which confirms the constancy of linear function growth rates.

Exponential Functions

Exponential functions are distinctly different from linear functions. They involve growth patterns that increase multiplicatively, rather than additively. The general form of an exponential function is:\[ g(t) = a \cdot b^{ct} \]where:

• \( a \) is the initial value (when \( t = 0 \))
• \( b \) is the base of the exponential, representing growth rate per time unit
• \( c \) is a constant that affects the rate of growth

For the function \( g(t) = 400 \cdot 2^{1/20 t} \), the base \( 2^{1/20} \) indicates a growth pattern that increases by a factor of \( 2^{1/20} \) for each unit increase in \( t \).

Exponential Growth Characteristics

Since the growth is multiplicative, exponential functions can quickly become large and are ideal for modeling phenomena like population growth or radioactive decay. The first derivative \( \frac{dg}{dt} \) gives us the absolute growth rate at any point, but we often use the relative growth rate to understand how quickly the function grows in proportion to itself. In mathematical terms, this is given by:\[ \text{Relative growth rate}= \frac{1}{20} \ln(2) \]This ratio remains constant, illustrating that while the output of an exponential function increases, the proportional growth rate does not change.

Derivatives

Derivatives are a vital concept in calculus and help us understand how functions change. In essence, a derivative represents the rate of change of a function as its input changes slightly. In practical terms, it can be thought of as the slope of the tangent line to the curve of the function at a particular point.

Calculating Derivatives

To find the derivative of a function, different rules apply depending on the function type:

• Power Rule: Useful for polynomial functions. If \( f(t) = t^n \), then \( \frac{df}{dt} = nt^{n-1} \).
• Chain Rule: Applies to composed functions, like exponentials. If \( g(t) = (h(t))^{k} \), then \( \frac{dg}{dt} = k(h(t))^{k-1} \frac{dh}{dt} \).

For the linear function \( f(t) = 2200 + 400t \), the derivative was found straight from the power rule:\[ \frac{df}{dt} = 400 \]For the exponential \( g(t) = 400 \cdot 2^{1/20 t} \), the chain rule helps obtain:\[ \frac{dg}{dt} = 400 \cdot 2^{\frac{1}{20}t} \cdot \frac{1}{20} \cdot \ln(2) \]Derivatives enable us to compute essential properties such as growth rates and can be applied across various disciplines like physics, economics, and biology.