Problem 30 - Calculus

Question

Find the inverse function. $$f(t)=50 e^{0.1 t}$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(t) = 10 \cdot \ln\left(\frac{t}{50}\right) \).

Step 1: Understand the Function

The function given is an exponential function, \( f(t) = 50 e^{0.1 t} \), where the base of the exponent is the natural number \( e \), and the coefficient is 50.

Step 2: Replace \( f(t) \) with \( y \)

To make it easier to find the inverse, replace \( f(t) \) with \( y \). Thus, the equation becomes \( y = 50 e^{0.1 t} \).

Step 3: Solve for \( t \) in Terms of \( y \)

Begin to solve for \( t \) by isolating the exponential expression. Divide both sides by 50 to get: \( \frac{y}{50} = e^{0.1t} \).

Step 4: Apply the Natural Logarithm

Take the natural logarithm (ln) of both sides to remove the base \( e \): \( \ln(\frac{y}{50}) = 0.1t \).

Step 5: Solve for \( t \)

Divide both sides by 0.1 to solve for \( t \): \( t = \frac{1}{0.1} \cdot \ln(\frac{y}{50}) \), which simplifies to \( t = 10 \cdot \ln(\frac{y}{50}) \).

Step 6: Replace \( y \) with \( f^{-1}(t) \)

The inverse function is \( f^{-1}(t) = 10 \cdot \ln(\frac{t}{50}) \).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical expressions where a constant base is raised to a variable exponent. A general form of an exponential function is given by \( f(x) = a \, b^x \), where \( a \) is a constant, \( b \) is the base of the exponential function, and \( x \) is the exponent. In the exercise, the exponential function is \( f(t) = 50 \, e^{0.1 t} \). Here:

• \( e \) is the base, known as the natural exponential base, approximately equal to 2.71828.
• \( 0.1t \) is the exponent which shows how the function increases or decreases over time.
• \( 50 \) is a multiplier or coefficient, scaling the exponential growth.

Exponential functions are vital as they model real-life scenarios such as population growth, radioactive decay, and interest calculations. When dealing with these functions, understanding the role of the base and the exponent is crucial, as they determine the growth rate and direction of the function. For any exponential expression, as \( t \) increases, the function's value will also increase or decrease at an exponential rate depending on the exponent's sign.

Natural Logarithm

The natural logarithm is the logarithm to the base \( e \), denoted as \( \ln(x) \). It is the inverse operation of the exponential function with base \( e \). This means that if \( y = e^x \), then \( x = \ln(y) \). In the context of the original exercise, we utilize the natural logarithm to determine the inverse function.When we encounter an equation like \( y = 50 \, e^{0.1 t} \), taking the natural logarithm of both sides helps us solve for \( t \). This is a crucial step because it allows us to remove the exponential by converting it into a linear expression. Thus, \( \ln(\frac{y}{50}) = 0.1t \). Applying the natural logarithm is particularly useful when dealing with continuous growth or decay models as it simplifies the process of isolating variables embedded within exponential functions. Notably, understanding the natural logarithm as an operator that "undoes" the exponential function is a fundamental concept.

The natural logarithm also has significant applications in various fields such as mathematics, physics, and engineering, particularly when working with uniform growth processes, half-life calculations, and information theory.

Solving Equations

Solving equations involves finding the unknown variable's value that satisfies the equation. In the context of this problem, we start with an exponential equation \( y = 50 \, e^{0.1 t} \). Our goal is to isolate \( t \), the variable of interest. Here’s a simplified breakdown of the solution process:

• Step one involves isolating the exponential part by dividing both sides by 50: \( \frac{y}{50} = e^{0.1t} \).
• Next, apply the natural logarithm to both sides. This step allows you to convert the exponential equation into a more manageable linear form: \( \ln(\frac{y}{50}) = 0.1t \).
• Finally, solve for \( t \) by dividing both sides by \( 0.1 \): \( t = 10 \, \ln(\frac{y}{50}) \).

In general, solving equations with exponential and logarithmic expressions requires careful manipulation and understanding the inverse relationships between these functions. By handling each step methodically, you ensure that the solution is accurate. Properly solving these types of equations broadens your ability to tackle a wide range of mathematical problems, from basic algebra to complex calculus scenarios.