How to Find the Equation of a Curve Given Its Gradient and a Point

Understanding how to derive the equation of a curve based on its gradient function and a point it passes through is a fundamental skill in calculus. This article will walk you through the process with a detailed example and provide insights into the relevant mathematical concepts.

Step-by-Step Guide: Deriving the Curve's Equation

To find the equation of the curve whose gradient at any point is given by the expression (6x^2 - 7), we need to follow several steps. Let's break down the process:

Step 1: Integrate the Gradient Function

The gradient function is given as:

(frac{dy}{dx} 6x^2 - 7)

To find the equation of the curve, we need to integrate this function with respect to (x):

(y int (6x^2 - 7) dx)

Step 2: Perform the Integration

Let's perform the integration step-by-step:

(y int 6x^2 dx - 7 int dx)

(y 6 cdot frac{x^3}{3} - 7 cdot x C)

(y 2x^3 - 7x C)

Note that (C) is the constant of integration.

Step 3: Solve for the Constant of Integration (C)

We are given a point through which the curve passes: ((-3, 7)). Substituting (x -3) and (y 7) into the equation:

(7 2(-3)^3 - 7(-3) C)

Calculate each term:

(2(-3)^3 2(-27) -54)

(-7(-3) 21)

Substitute these values back into the equation:

(7 -54 21 C)

Combine like terms:

(7 -33 C)

Solving for (C):

(C 7 33 40)

Step 4: Write the Final Equation of the Curve

Substitute the value of (C) back into the equation:

(y 2x^3 - 7x 40)

This is the equation of the curve that passes through the point ((-3, 7)) and has the given gradient function (y' 6x^2 - 7).

Understanding the Concepts

Gradient Function: The gradient function, also known as the derivative, provides the slope of the curve at any given point. Here, we have (frac{dy}{dx} 6x^2 - 7), which tells us how the curve changes at each point.

Integration: Integration is the reverse process of differentiation. When we integrate the gradient function, we are finding the original function, which is the equation of the curve. Remember, when integrating, we always add a constant of integration, which is determined by the given point.

Conclusion

By integrating the gradient function and using the given point to solve for the constant of integration, we can derive the equation of the curve. This process is essential in many areas of calculus and applied mathematics.

Additional Tips

1. Practice integrating different gradient functions to become more comfortable with the process.

2. Double-check your calculations, especially when dealing with exponents and constants.

3. Always verify your final answer by substituting the given point back into the equation to ensure it holds true.

Mastering these steps can help you solve similar problems efficiently and accurately.