Finding Tangent Parallel To A Line On A Curve Y=2x^2-x+1
In the realm of calculus, a fascinating problem arises when we seek to find points on a curve where the tangent line exhibits a specific characteristic – parallelism to a given line. This exploration delves into the intricate relationship between derivatives, slopes, and the geometric properties of curves. In this article, we will embark on a journey to unravel the solution to the problem of finding the coordinates of the point on the curve y = 2x² - x + 1 at which the tangent is parallel to the line y = 3x + 4. This problem exemplifies the power of calculus in connecting algebraic equations with geometric interpretations.
Understanding the Core Concepts
Before we delve into the step-by-step solution, it's crucial to solidify our understanding of the fundamental concepts that underpin this problem. These concepts include derivatives, tangent lines, slopes, and the condition for parallelism. Each concept plays a vital role in unraveling the problem and arriving at the correct solution.
Derivatives and Tangent Lines
The derivative of a function at a particular point provides us with the slope of the tangent line to the curve at that point. This slope, often denoted as dy/dx or f'(x), represents the instantaneous rate of change of the function at that specific x-value. The tangent line, a straight line that touches the curve at only one point (in the limit), provides a linear approximation of the curve's behavior in the immediate vicinity of that point. Understanding the relationship between derivatives and tangent lines is paramount in solving problems involving tangents and curves. At its core, differential calculus provides a suite of tools for analyzing the behavior of functions and curves. It provides a means to calculate the instantaneous rate of change of a function, which translates geometrically to the slope of the tangent line to the curve at a given point. This connection between algebra and geometry is central to understanding the problem at hand.
Slopes and Parallel Lines
The slope of a line is a measure of its steepness or inclination. It quantifies the rate at which the line rises or falls as we move along the x-axis. In the equation of a line, y = mx + b, the coefficient m represents the slope. Parallel lines, by definition, have the same slope. This condition forms the cornerstone of our problem-solving approach. We can rewrite the equation y=3x+4 in slope-intercept form, where the coefficient of x represents the slope. In this case, the slope of the line is 3. Parallel lines share the same slope, which means that the tangent line to the curve must also have a slope of 3. This sets up the fundamental equation we need to solve.
Condition for Parallelism
Two lines are parallel if and only if they have the same slope. This seemingly simple condition is a powerful tool in geometry and calculus. In our problem, we will leverage this condition to equate the slope of the tangent line to the curve with the slope of the given line. This equation will allow us to solve for the x-coordinate of the point where the tangent is parallel.
Step-by-Step Solution
Now that we have a firm grasp of the underlying concepts, let's embark on the step-by-step solution to our problem. This methodical approach will guide us through the necessary calculations and logical deductions to arrive at the desired coordinates.
1. Find the Derivative
The first step in our solution is to find the derivative of the given curve, y = 2x² - x + 1. The derivative, dy/dx, will give us the slope of the tangent line at any point on the curve. We can find the derivative using the power rule of differentiation, which states that if y = ax^n, then dy/dx = nax^(n-1). Applying this rule to each term in the equation, we get:
- d/dx (2x²) = 4x
- d/dx (-x) = -1
- d/dx (1) = 0
Combining these results, we find that the derivative of the curve is:
dy/dx = 4x - 1
This derivative represents the slope of the tangent line at any point on the curve. It establishes a crucial link between the position on the curve (represented by x) and the slope of the tangent at that point.
2. Determine the Slope of the Given Line
The given line is y = 3x + 4. This equation is in slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept. By inspection, we can see that the slope of the line is 3. This value will serve as our target slope for the tangent line on the curve. We can rewrite the equation y=3x+4 in slope-intercept form, where the coefficient of x represents the slope. In this case, the slope of the line is 3. Parallel lines share the same slope, which means that the tangent line to the curve must also have a slope of 3. This sets up the fundamental equation we need to solve.
3. Equate the Slopes
Since we want the tangent line to be parallel to the given line, the slope of the tangent line must be equal to the slope of the given line. Therefore, we set the derivative dy/dx equal to 3:
4x - 1 = 3
This equation encapsulates the core condition for parallelism. It mathematically expresses the requirement that the instantaneous rate of change of the curve (represented by the derivative) must match the constant rate of change of the given line.
4. Solve for x
Now, we solve the equation 4x - 1 = 3 for x. Adding 1 to both sides, we get:
4x = 4
Dividing both sides by 4, we find:
x = 1
This value of x represents the x-coordinate of the point on the curve where the tangent line is parallel to the given line. It is a crucial piece of the puzzle that brings us closer to the final solution.
5. Find the y-coordinate
To find the y-coordinate of the point, we substitute the value of x we just found (x = 1) into the equation of the curve, y = 2x² - x + 1:
y = 2(1)² - 1 + 1
Simplifying, we get:
y = 2 - 1 + 1
y = 2
Therefore, the y-coordinate of the point is 2. This completes the coordinate pair, giving us the precise location of the point on the curve where the tangent line fulfills the parallelism condition.
6. State the Coordinates
Finally, we state the coordinates of the point. The point on the curve y = 2x² - x + 1 at which the tangent is parallel to the line y = 3x + 4 is (1, 2). This concludes our step-by-step solution, providing the answer in a clear and concise manner.
Visualizing the Solution
To enhance our understanding, it's beneficial to visualize the solution. Imagine the curve y = 2x² - x + 1 as a parabola opening upwards. The line y = 3x + 4 is a straight line with a positive slope. The point (1, 2) lies on the curve, and at this point, the tangent line to the parabola has the same slope as the given line. This visual representation provides an intuitive grasp of the problem and its solution.
Graphing the curve, the line, and the tangent at the point (1, 2) would further solidify the understanding of the solution. The tangent line would visually demonstrate its parallelism to the given line, reinforcing the geometric interpretation of the problem.
Generalizing the Approach
The approach we've employed to solve this problem can be generalized to find points on any curve where the tangent is parallel to a given line. The key steps remain the same:
- Find the derivative of the curve.
- Determine the slope of the given line.
- Equate the slopes.
- Solve for x.
- Find the corresponding y-coordinate.
This general approach highlights the versatility of calculus in solving a wide range of problems involving curves, tangents, and slopes. It is a powerful tool applicable in various contexts within mathematics and its applications.
Applications and Extensions
The concept of finding points where a tangent is parallel to a line has applications in various fields, including physics, engineering, and economics. For instance, in physics, it can be used to find the point where the velocity of a particle is parallel to a certain direction. In economics, it can be used to find the point where the marginal cost of production is equal to a certain price.
This problem can also be extended in several ways. For example, we could ask to find the points where the tangent is perpendicular to a given line, or where the tangent makes a specific angle with a given line. These extensions build upon the same core concepts but introduce additional complexities and require a deeper understanding of calculus and geometry.
Conclusion
In conclusion, we have successfully found the coordinates of the point on the curve y = 2x² - x + 1 at which the tangent is parallel to the line y = 3x + 4. The solution, (1, 2), was obtained by leveraging the fundamental concepts of derivatives, slopes, and the condition for parallelism. This problem serves as a testament to the power of calculus in bridging the gap between algebraic equations and geometric interpretations. By understanding the underlying principles and applying a systematic approach, we can unravel complex problems and gain valuable insights into the behavior of curves and functions.
The ability to find points where tangents have specific properties is a cornerstone of calculus and has wide-ranging applications in various fields. This problem provides a solid foundation for further exploration of calculus concepts and their applications in the real world.
