How to Find the Equation of the Tangent Line to a Given Curve at a Specific Point

Introduction

Understanding how to find the equation of the tangent line to a given curve at a specific point is a crucial skill in calculus and analytic geometry. This procedure is fundamental in various applications, from physics to engineering. In this article, we will walk through the steps to find the tangent line equation for two different functions - fx 3x^4 - 2x^3 - 7x and fx 3x - 5x - 1. We will use these examples to explain the theory and practical methods.

Step-by-Step Guide to Finding the Tangent Line

Step 1: Given Function and Point of Tangency

The first step in finding the equation of the tangent line is to identify the given function and the point at which the tangent line intersects the curve. For this article, we have the following functions:

Step 2: Calculate the Function Value at the Given Point

The next step is to calculate the function value at the given point, which gives us the y-coordinate of the point of tangency.

Function 1:

fx  3x^4 - 2x^3 - 7xf(-1)  3(-1)^4 - 2(-1)^3 - 7(-1)      3 - (-2)   7      3   2   7      12

The point of tangency is (-1, 12).

Function 2:

fx  3x - 5x - 1f(-1)  3(-1) - 5(-1) - 1      -3   5 - 1      1

The point of tangency is (-1, 1).

Step 3: Find the Derivative of the Function

The derivative of the function gives us the slope of the tangent line at any point on the curve.

Function 1:

fx  3x^4 - 2x^3 - 7xf'(x)  12x^3 - 6x^2 - 7

Evaluating at x -1:

f'(-1)  12(-1)^3 - 6(-1)^2 - 7       12(-1) - 6(1) - 7       -12 - 6 - 7       -25

The slope of the tangent line at x -1 is -25.

Function 2:

fx  3x - 5x - 1f'(x)  3 - 5       -2

Evaluating at x -1:

f'(-1)  -2

The slope of the tangent line at x -1 is -2.

Step 4: Write the Equation of the Tangent Line

Using the point-slope form of a line, we can write the equation of the tangent line. The point-slope form is given by:

y - y1 m(x - x1)

where (x1, y1) is the point of tangency and m is the slope.

Function 1:

y - 12  -25(x   1)y - 12  -25x - 25y  -25x - 25   12y  -25x - 13

The equation of the tangent line for fx 3x^4 - 2x^3 - 7x at x -1 is y -25x - 13.

Function 2:

y - 1  -2(x   1)y - 1  -2x - 2y  -2x - 2   1y  -2x - 1

The equation of the tangent line for fx 3x - 5x - 1 at x -1 is y -2x - 1.

Conclusion

This article has provided a step-by-step guide on how to find the equation of the tangent line to a given curve at a specific point. We've demonstrated the process using two different functions, showing the importance of correctly identifying the derivative to determine the slope of the tangent line. Understanding these concepts is vital for further studies in calculus and applications in various fields.