Tangent Line Equation Intercepts And Length Calculation For Y=x³-2x²+3x+7
Introduction
In this article, we delve into the process of finding the equation of the tangent line to the curve y = x³ - 2x² + 3x + 7 at the point (1, a). This involves utilizing differential calculus to determine the slope of the tangent and subsequently employing the point-slope form to establish the tangent's equation. Furthermore, we aim to identify the points where this tangent intersects the x and y axes, denoted as A and B respectively. Finally, we will calculate the distance between these intercepts, effectively determining the length of the line segment AB. This exploration will provide a comprehensive understanding of how calculus can be applied to analyze curves and their tangent lines, offering valuable insights into the geometrical properties of functions.
1. Determining the Value of 'a' and the Point of Tangency
To begin, we need to find the value of 'a', which represents the y-coordinate of the point on the curve where the tangent line is drawn. Given that the point lies on the curve y = x³ - 2x² + 3x + 7, we can substitute the x-coordinate, which is 1, into the equation to find the corresponding y-coordinate. This substitution yields:
y = (1)³ - 2(1)² + 3(1) + 7
y = 1 - 2 + 3 + 7
y = 9
Therefore, the value of 'a' is 9, and the point of tangency is (1, 9). This point is crucial as it anchors the tangent line to the curve. Understanding this initial step is essential because it sets the foundation for all subsequent calculations. The accuracy of this step directly impacts the accuracy of the tangent line equation and the determination of the intercepts. We are now certain of the specific location on the curve where we will construct the tangent line, allowing us to proceed with finding the slope of the curve at this point.
2. Calculating the Derivative to Find the Slope
The next crucial step in finding the equation of the tangent line is to determine the slope of the curve at the point (1, 9). In calculus, the slope of a curve at a particular point is given by the derivative of the function at that point. The derivative of a function represents the instantaneous rate of change of the function. For the curve y = x³ - 2x² + 3x + 7, we need to find its derivative, denoted as dy/dx or y'. Applying the power rule of differentiation, which states that the derivative of xⁿ is nxⁿ⁻¹, we differentiate each term of the equation:
y = x³ - 2x² + 3x + 7
dy/dx = d(x³)/dx - 2 * d(x²)/dx + 3 * d(x)/dx + d(7)/dx
dy/dx = 3x² - 4x + 3 + 0
Thus, the derivative of the function is dy/dx = 3x² - 4x + 3. This derivative gives us a formula to calculate the slope at any point on the curve. Now, we need to find the slope at the specific point (1, 9). We substitute x = 1 into the derivative:
dy/dx |_(x=1) = 3(1)² - 4(1) + 3
= 3 - 4 + 3
= 2
Therefore, the slope of the tangent line at the point (1, 9) is 2. This value is a key component in determining the equation of the tangent line. The derivative's calculation is a cornerstone of calculus, providing the necessary tool to understand the curve's behavior at any given point. With the slope now known, we can move forward to constructing the equation of the tangent line using the point-slope form.
3. Determining the Tangent Line Equation
With the slope of the tangent line calculated as 2 and the point of tangency established as (1, 9), we can now construct the equation of the tangent line. The point-slope form of a line's equation is given by:
y - y₁ = m(x - x₁)
where m is the slope of the line, and (x₁, y₁) is a point on the line. In our case, m = 2, x₁ = 1, and y₁ = 9. Substituting these values into the point-slope form, we get:
y - 9 = 2(x - 1)
Now, we simplify this equation to obtain the slope-intercept form, y = mx + b, which is a more standard form for representing linear equations:
y - 9 = 2x - 2
y = 2x - 2 + 9
y = 2x + 7
Therefore, the equation of the tangent line to the curve y = x³ - 2x² + 3x + 7 at the point (1, 9) is y = 2x + 7. This equation provides a clear and concise representation of the tangent line, allowing us to further analyze its properties, such as its intercepts with the x and y axes. The process of converting the point-slope form to the slope-intercept form ensures that the equation is in a readily usable format for subsequent calculations and analysis.
4. Finding the Intercepts A and B
To find the points where the tangent line intersects the x and y axes, we need to calculate the x and y intercepts. These intercepts are the points where the line crosses the respective axes, and they provide valuable information about the line's position and orientation in the coordinate plane. The tangent line equation we derived is y = 2x + 7. Let's start by finding the x-intercept, which is the point A where the line intersects the x-axis. At the x-intercept, the y-coordinate is always 0. Therefore, we set y = 0 in the tangent line equation and solve for x:
0 = 2x + 7
2x = -7
x = -7/2
So, the x-intercept A has coordinates (-7/2, 0). This means the tangent line crosses the x-axis at the point where x is -7/2 and y is 0.
Next, we find the y-intercept, which is the point B where the line intersects the y-axis. At the y-intercept, the x-coordinate is always 0. Therefore, we set x = 0 in the tangent line equation and solve for y:
y = 2(0) + 7
y = 7
Thus, the y-intercept B has coordinates (0, 7). This indicates that the tangent line crosses the y-axis at the point where x is 0 and y is 7.
In summary, the x-intercept A is at (-7/2, 0), and the y-intercept B is at (0, 7). These intercepts are critical points on the tangent line, and knowing their coordinates allows us to calculate the length of the line segment AB, which is the distance between these two points.
5. Calculating the Length of AB
Now that we have the coordinates of the points A and B, which are the x and y intercepts of the tangent line, we can calculate the length of the line segment AB. The coordinates of A are (-7/2, 0), and the coordinates of B are (0, 7). To find the distance between two points in a coordinate plane, we use the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
where d is the distance between the points (x₁, y₁) and (x₂, y₂). In our case, let A = (x₁, y₁) = (-7/2, 0) and B = (x₂, y₂) = (0, 7). Substituting these values into the distance formula, we get:
d = √[(0 - (-7/2))² + (7 - 0)²]
Now, we simplify the expression:
d = √[(7/2)² + (7)²]
d = √[49/4 + 49]
To add the fractions, we need a common denominator, which is 4. So, we rewrite 49 as 196/4:
d = √[49/4 + 196/4]
d = √[245/4]
Now, we simplify the square root. We can factor 245 as 49 * 5, so:
d = √[49 * 5 / 4]
d = √(49/4 * 5)
d = √(49/4) * √5
d = 7/2 * √5
Therefore, the length of the line segment AB is (7√5) / 2 units. This result provides a precise measure of the distance between the x and y intercepts of the tangent line, completing our geometrical analysis of the tangent line and its relationship to the coordinate axes.
Conclusion
In this comprehensive exploration, we have successfully found the equation of the tangent line to the curve y = x³ - 2x² + 3x + 7 at the point (1, 9), which is y = 2x + 7. We then determined the coordinates of the x-intercept A as (-7/2, 0) and the y-intercept B as (0, 7). Finally, we calculated the length of the line segment AB using the distance formula, arriving at the result (7√5) / 2 units. This exercise demonstrates a practical application of differential calculus in finding tangent lines and analyzing their geometrical properties. The process involved several key steps, including finding the derivative of the function to determine the slope, using the point-slope form to construct the tangent line equation, and applying the distance formula to calculate the length of the line segment between the intercepts. These concepts are fundamental in calculus and provide valuable tools for understanding the behavior of curves and their tangent lines. This detailed analysis not only answers the specific questions posed but also reinforces the broader principles of calculus and analytical geometry.
