Problem 22 - Calculus Single Variable

Question

Compute \(F(c)\) from the given information. $$ F^{\prime}(x)=2 x+3, F(1)=2, c=-1 $$

Step-by-Step Solution

Answer

\( F(-1) = -4 \)

Step 1: Understand the Problem

We need to find the function value \( F(c) \) given that the derivative \( F'(x) = 2x + 3 \). Additionally, we know \( F(1) = 2 \). The goal is to determine \( F(-1) \).

Step 2: Integrate the Derivative

To find \( F(x) \), we need to integrate \( F'(x) = 2x + 3 \). The indefinite integral is computed as follows:\[F(x) = \int (2x + 3) \, dx = x^2 + 3x + C\]where \( C \) is the constant of integration.

Step 3: Use Initial Condition to Find Constant

Given \( F(1) = 2 \), substitute \( x = 1 \) into the function to solve for \( C \):\[F(1) = 1^2 + 3(1) + C = 2\]\[1 + 3 + C = 2\]\[C = 2 - 4 = -2\]

Step 4: Write the Function Including the Constant

Substitute the constant \( C = -2 \) back into the function:\[F(x) = x^2 + 3x - 2\]

Step 5: Evaluate the Function at \( c = -1 \)

Substitute \( x = -1 \) into the function to find \( F(-1) \):\[F(-1) = (-1)^2 + 3(-1) - 2 = 1 - 3 - 2 = -4\]

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus. It is used to calculate the net area under a curve between two points on the x-axis. This principle is a bit different from the indefinite integral, since it focuses on a specific interval. In our problem, the integration process concerning the derivative \( F'(x) = 2x + 3 \) did not result in a definite integral, but rather an indefinite one. However, by knowing specific values, we adapted the indefinite integral, \( x^2 + 3x + C \), into a function matching initial conditions given.

• The definite integral provides an exact area under a curve.
• For definite integrals, limits of integration are crucial for precise calculations.

In practical problems, definite integrals help evaluate accumulations over an interval, such as total distance traveled or total growth over time.

Constant of Integration

The constant of integration arises when calculating indefinite integrals. An indefinite integral represents a family of functions, meaning many functions share the same derivative. The constant adjusts the indefinite integral to fit particular conditions or additional data points. In our problem, after integrating \( F'(x) = 2x + 3 \), we obtain:\[ F(x) = x^2 + 3x + C \]

• \( C \) is crucial for pinning down the specific solution.
• Without \( C \), the general solution would not specifically satisfy the initial condition.

Once \( F(1) = 2 \) was given, we solved for \( C \) to find that it equaled \(-2\), making the function fully defined.

Initial Condition

Initial conditions are used to solve for unknown constants that occur during integration. In our task, the initial condition provided was \( F(1) = 2 \). This information was pivotal in determining the constant of integration \( C \). Without an initial condition, we would have an infinite number of potential solutions for \( F(x) \).

• Initial conditions make it possible to transform a general solution into a specific one.
• They are essential in differential equations to obtain unique solutions.

In essence, providing one point \((x, F(x))\), allowed us to fix the exact value of \( C \), tailoring the solution to fit the given data.

Derivatives

Derivatives measure how a function changes as its input changes. In calculus, they are the foundation for determining rates of change and understanding the behavior of functions. For example, the derivative \( F'(x) = 2x + 3 \) represents the slope or rate of change of the original function \( F(x) \) at any point \( x \).

• Derivatives can help predict graphical behavior such as increasing or decreasing trends.
• The derivative process is reversible; integration is its reverse operation.

By knowing \( F'(x) \), we used integration to derive \( F(x) \). Understanding the role of derivatives is essential for solving these kinds of problems, as they form the link between motions and positions, rates, and totals.