Question

Graph several functions that satisfy the following differential equations. Then find and graph the particular function that satisfies the given initial condition. $$f^{\prime}(x)=3 x^{2}-1 ; f(1)=2$$

Short answer

Answer: The particular function is \(f(x) = x^3 - x + 2\).

Step-by-step solution

Integrate the given differential equation

Integrate both sides of the equation \(f'(x)=3x^2-1\) with respect to x: $$\int f'(x) \,dx = \int (3x^2 - 1) \,dx$$

Add a constant of integration, C

Integrate and add a constant of integration on the right-hand side: $$f(x) = \int (3x^2 - 1) \,dx = x^3 - x + C$$ Here, C is a constant.

Use the initial condition to find the value of C

We are given the initial condition \(f(1)=2\). Plug in the values into the equation we found in Step 2: $$2 = (1)^3 - (1) + C$$ Solve for C: $$C=2$$

Write the particular function that satisfies both the given differential equation and the initial condition

Replace C with the value we found in Step 3: $$f(x) = x^3 - x + 2$$

Graph the particular function

To graph the function, we can plot key points and use graphing software. Here are some key points: x | f(x) ------- -1 | 0 0 | 2 1 | 2 2 | 6 Now, plot these points and draw a smooth curve through them to draw the graph of the function \(f(x)=x^3-x+2\). To better visualize the curve and its characteristics, use graphing software or graphing calculators as well.

Key concepts

Initial Conditions

In the context of differential equations, initial conditions are specific values provided for the solution of the equations. They help us determine the unique solution among potentially infinite general solutions.

When dealing with differential equations like the one in the problem, the initial conditions, represented here as \( f(1) = 2 \), act as a guiding parameter. Essentially, they tell us that when \( x = 1 \), the function \( f(x) \) should return a value of 2. This condition is crucial because it allows us to find a specific value for the "constant of integration," which we will explore next.

• Pinpoints the exact solution.
• Used to solve for unknown constants.
• Confirms the correctness of the solution.

Integration

Integration is a fundamental technique in calculus used to reverse the process of differentiation. When we're given a differential equation, like \( f'(x)=3x^2 - 1 \), integration allows us to find the original function \( f(x) \).

In this exercise, the integration process begins by integrating both sides of the equation with respect to \( x \). This step involves calculating the antiderivative of the expression on the right-hand side.

• Reverses differentiation.
• Helps to find original functions.
• Essential for solving differential equations.

After integrating, you'll result in an expression that includes a constant (because integration is indefinite unless initial conditions are given), expressing all the potential forms of the original function prior to finding the constant of integration.

Constant of Integration

The constant of integration, often symbolized as \( C \), emerges naturally when performing indefinite integration. This constant accounts for any vertical shifts of the resulting graph.

In our step-by-step solution, after integrating the differential equation \( 3x^2 - 1 \), we obtained \( f(x) = x^3 - x + C \). The \( C \) represents an unknown yet to be determined without further conditions. That's where the initial condition \( f(1) = 2 \) becomes crucial.

• Appears in indefinite integrals.
• Represents an overall shift in the function's value.
• Determined using initial conditions.

This constant is determined by substituting the initial condition into the integrated function, thanks to which we find the specific value of \( C \) that fits both the original differential equation and the initial condition provided.

Graphing Functions

Graphing functions is a way to visually interpret data or mathematical relationships. In calculus, graphing functions can help us understand the nature and behavior of solutions to differential equations.

In this case, the function \( f(x) = x^3 - x + 2 \) represents the particular solution derived from initial conditions. By plotting points like \((-1, 0), (0, 2), (1, 2), (2, 6)\), and connecting them with a smooth continuous curve, we can observe the growth and trends of the function.

• Provides visual understanding.
• Highlights function behavior.
• Useful for spotting trends or anomalies.

Graphing software or calculators are often utilized for accuracy and to better visualize the intricacies that may not be apparent from mere equations. They are tools to help students see the real-world applications and implications of solutions to mathematical problems.