Problem 6 - Essential Calculus Early Transcendentals

Question

Find the most general antiderivative of the function (Check your answer by differentiation $$f(t)=\frac{3 t^{4}-t^{3}+6 t^{2}}{t^{4}}$$

Step-by-Step Solution

Answer

The general antiderivative is \( F(t) = 3t - \ln|t| - \frac{6}{t} + C \).

Step 1: Simplify the Function

First, simplify the given function. The function is \( f(t) = \frac{3t^4 - t^3 + 6t^2}{t^4} \). Simplify this by dividing each term in the numerator by \( t^4 \):\[f(t) = 3 - \frac{1}{t} + \frac{6}{t^2}.\]

Step 2: Integrate Each Term

Find the antiderivative of the simplified function term by term. The antiderivative of \( 3 \) with respect to \( t \) is \( 3t \). The antiderivative of \( -\frac{1}{t} \) is \(-\ln|t|\) and the antiderivative of \( \frac{6}{t^2} \) is \( -\frac{6}{t} \).Thus, the most general antiderivative is:\[F(t) = 3t - \ln|t| - \frac{6}{t} + C,\]where \( C \) is the constant of integration.

Step 3: Differentiate the Antiderivative

To verify that the antiderivative is correct, differentiate it:\[\frac{d}{dt}[3t - \ln|t| - \frac{6}{t} + C] = 3 - \frac{1}{t} + \frac{6}{t^2}.\]This matches the simplified function \( f(t) = 3 - \frac{1}{t} + \frac{6}{t^2} \), confirming that the antiderivative is correct.

Key Concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function given. This opposite process of differentiation helps us find the most general form of a function. In calculus, while finding an antiderivative, we typically add a constant of integration, often denoted by the letter \( C \). This constant comes into play because the process of differentiation can lose constant terms. For example, the derivative of the function \( x + 5 \) and \( x + 10 \) is \( 1 \), which means during integration, we need to account for that lost constant with \( C \).
To find the antiderivative of a function, you can employ different rules and techniques depending on the form of the function. In this exercise, once the function was simplified as \( f(t) = 3 - \frac{1}{t} + \frac{6}{t^2} \), we found each term's antiderivative individually:

• The antiderivative of \( 3 \) is \( 3t \), because the derivative of \( 3t \) brings us back to \( 3 \).
• For \( -\frac{1}{t} \), the antiderivative is \(-\ln|t|\); when differentiated, it results in \(-\frac{1}{t}\).
• The antiderivative of \( \frac{6}{t^2} \) or \( 6t^{-2} \) is \(-\frac{6}{t}\), determined through the power rule for integration.

This demonstrates the process of arriving at the general solution \( F(t) = 3t - \ln|t| - \frac{6}{t} + C \).

Simplification of Functions

Simplification involves breaking down complex expressions into simpler, more manageable components. It is a crucial step in calculus and algebra that makes subsequent calculations more straightforward by reducing the complexity of the function. In the provided exercise, this simplification process involved expanding and breaking down the components of a rational function.
Starting with the original function, \( f(t) = \frac{3t^4 - t^3 + 6t^2}{t^4} \), the process involved dividing each term in the numerator by \( t^4 \). This step made it easier to apply integration rules. By dividing, we obtained:

• \( \frac{3t^4}{t^4} = 3 \)
• \( \frac{-t^3}{t^4} = -\frac{1}{t} \)
• \( \frac{6t^2}{t^4} = \frac{6}{t^2} \)

These transformed the original complicated rational expression into a simplified form \( 3 - \frac{1}{t} + \frac{6}{t^2} \). A well-executed simplification serves not only to ease integration but can often reveal aspects of functions that are not immediately obvious, making them simpler to understand and work with in calculations like integration.

Differentiation

Differentiation is the process of computing a derivative, which represents the rate at which a function is changing at any given point. It's a fundamental concept in calculus handled via various rules and formulas. Differentiation finds significant applications in verifying solutions to integral problems, among many other uses.
In our exercise, after finding the antiderivative \( F(t) = 3t - \ln|t| - \frac{6}{t} + C \), we differentiated it to check our work. Differentiating each term, we used basic derivatives:

• \( \frac{d}{dt}(3t) = 3 \)
• \( \frac{d}{dt}(-\ln|t|) = -\frac{1}{t} \)
• \( \frac{d}{dt}(-\frac{6}{t}) = \frac{6}{t^2} \)
• The derivative of a constant \( C \) is \( 0 \).

Thus, the derivative of the antiderivative is \( 3 - \frac{1}{t} + \frac{6}{t^2} \), which matches the simplified function \( f(t) \). This step reassures us that the antiderivative was computed correctly. Differentiation not only confirms accuracy but also provides deeper insights into the behavior and characteristics of functions.