Problem 16 - Essential Calculus Early Transcendentals

Question

Find the antiderivative \(F\) of \(f\) that satisfies the given condition. Check your answer by comparing the graphs of \(f\) and \(F .\) $$f(x)=4-3\left(1+x^{2}\right)^{-1}, \quad F(1)=0$$

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = 4x - 3\tan^{-1}(x) + \frac{3\pi}{4} - 4 \).

Step 1: Identify the Antiderivative Components

To find the antiderivative, we need to identify the antiderivatives of the components of the function. The function is given as \( f(x) = 4 - 3(1+x^2)^{-1} \). We are dealing with two separate parts: \( 4 \) and \( -3(1+x^2)^{-1} \). The antiderivative of a constant \( a \) is \( ax + C \).

Step 2: Antiderivative of a Constant

For the constant part \( 4 \), the antiderivative is straightforward: \( \int 4 \, dx = 4x + C_1 \).

Step 3: Antiderivative of the Rational Function

The function \( -3(1+x^2)^{-1} \) can be tackled using knowledge of standard integrals. The antiderivative of \( (1+x^2)^{-1} \) is known to be the inverse tangent function, \( \tan^{-1}(x) \). Hence, the antiderivative \( \int -3(1+x^2)^{-1} \, dx = -3 \tan^{-1}(x) + C_2 \).

Step 4: Combine the Antiderivative Components

Combine the antiderivatives calculated in Steps 2 and 3: \( F(x) = 4x - 3 \tan^{-1}(x) + C \), where \( C = C_1 + C_2 \) is the constant of integration.

Step 5: Use Initial Condition to Find C

We are given that \( F(1) = 0 \). Substitute this into the equation: \( 0 = 4(1) - 3 \tan^{-1}(1) + C \). Since \( \tan^{-1}(1) = \frac{\pi}{4} \), the equation becomes \( 0 = 4 - \frac{3\pi}{4} + C \). Solve for \( C \): \( C = \frac{3\pi}{4} - 4 \).

Step 6: Write the Final Antiderivative Function

Substitute \( C \) back into the antiderivative equation: \( F(x) = 4x - 3 \tan^{-1}(x) + \frac{3\pi}{4} - 4 \). This gives us the particular solution satisfying \( F(1) = 0 \).

Step 7: Verify by Graphing

To verify, graph both \( f(x) \) and \( F(x) \). The graph of \( F(x) \) should look like a potential energy function with \( f(x) \) as its derivative. Check that \( F(x) \) crosses the x-axis at \( x = 1 \), confirming \( F(1) = 0 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the values of trigonometric functions are known. Specifically, the inverse tangent function, denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), allows us to determine the angle whose tangent is \( x \). These functions are essential components in calculus, particularly in integration.
When finding the antiderivative or integral of functions like \( (1+x^2)^{-1} \), the inverse trigonometric functions play a crucial role.

• The antiderivative \( \int (1+x^2)^{-1} \, dx \) equals \( \tan^{-1}(x) + C \), where \( C \) is the constant of integration.
• This property is widely used to solve integrals involving rational functions of certain types, especially when the denominator matches the form \( 1 + x^2 \).

Such inverse trigonometric functions enable us, for example, to integrate more complex expressions and handle initial condition problems efficiently.

Initial Conditions in Calculus

Understanding initial conditions in calculus is critical for determining the specific solution to an antiderivative problem. An initial condition helps to identify the particular constant of integration in the solution of an indefinite integral. Without this, the solution could vary infinitely based on the value of this constant.
In the provided exercise, we had:

• The antiderivative \( F(x) = 4x - 3 \tan^{-1}(x) + C \).
• The given initial condition \( F(1) = 0 \) allowed us to solve for \( C \).
• By substituting \( x = 1 \) and equating \( F(1) \) to zero, we found \( C = \frac{3\pi}{4} - 4 \).

The initial condition crucially pinpoints the unique value of \( C \) and thus selects the precise family member from the infinite antiderivatives.

Graphing Functions

Graphing functions is an invaluable tool for visualizing the relationships between functions and their antiderivatives. A graph provides insight into the behavior of the function over a range of values and is essential in verifying solutions, like in the exercise where \( f(x) \) and its antiderivative \( F(x) \) were compared.
When checking your solution or understanding the relationship between functions:

• The derivative \( f(x) \) should represent the slope or rate of change of the antiderivative \( F(x) \).
• For example, observe where \( F(x) \) crosses the x-axis. In this exercise, the antiderivative must cross at the point given by the initial condition: \( x = 1 \).
• Make sure that inflection points, which are locations on the graph where the curve changes direction, are coherent between \( f(x) \) and \( F'(x) \).

Graphing can confirm our analytical solutions' accuracy by displaying them visually, ensuring they adhere to the mathematical constraints and initial conditions.