Showing that P Lies on the Curve C

Oct 13, 2025 by miodeya.com 126 views

Introduction

In mathematics, curves and points are fundamental concepts that are used to describe various geometric shapes and relationships. In this article, we will explore the curve C, which is defined by the equation y = (2/3)x^3 - x^2 + 4x - 3, and the point P, which has coordinates (3, 18). We will show that P lies on the curve C and then find the equation of the tangent to the curve at point P.

Showing that P Lies on the Curve C

To show that P lies on the curve C, we need to substitute the coordinates of P into the equation of the curve and verify that the equation holds true. The coordinates of P are (3, 18), so we substitute x = 3 and y = 18 into the equation of the curve:

y = (2/3)x^3 - x^2 + 4x - 3 18 = (2/3)(3)^3 - (3)^2 + 4(3) - 3

Now, let's simplify the equation:

18 = (2/3)(27) - 9 + 12 - 3 18 = 18 - 9 + 12 - 3 18 = 18

As we can see, the equation holds true, and therefore, P lies on the curve C.

Finding the Equation of the Tangent to the Curve at Point P

To find the equation of the tangent to the curve at point P, we need to find the derivative of the curve at x = 3. The derivative of the curve is given by:

dy/dx = d((2/3)x^3 - x^2 + 4x - 3)/dx = 2x^2 - 2x + 4

Now, we substitute x = 3 into the derivative to find the slope of the tangent at point P:

dy/dx = 2(3)^2 - 2(3) + 4 = 18 - 6 + 4 = 16

The slope of the tangent at point P is 16. To find the equation of the tangent, we use the point-slope form of a line:

y - y1 = m(x - x1)

where (x1, y1) is the point of tangency, and m is the slope. In this case, (x1, y1) = (3, 18), and m = 16. Substituting these values into the equation, we get:

y - 18 = 16(x - 3)

Now, let's simplify the equation:

y - 18 = 16x - 48 y = 16x - 48 + 18 y = 16x - 30

Therefore, the equation of the tangent to the curve at point P is y = 16x - 30.

Conclusion

In this article, we showed that the point P lies on the curve C and found the equation of the tangent to the curve at point P. We used the equation of the curve and the coordinates of the point to verify that P lies on the curve, and then we found the derivative of the curve to determine the slope of the tangent at point P. Finally, we used the point-slope form of a line to find the equation of the tangent. This article demonstrates the importance of mathematical concepts in understanding the properties of curves and points.

Mathematical Derivations

Derivative of the Curve

The derivative of the curve is given by:

dy/dx = d((2/3)x^3 - x^2 + 4x - 3)/dx = 2x^2 - 2x + 4

Slope of the Tangent at Point P

The slope of the tangent at point P is given by:

dy/dx = 2(3)^2 - 2(3) + 4 = 18 - 6 + 4 = 16

Equation of the Tangent

The equation of the tangent is given by:

y - y1 = m(x - x1)

where (x1, y1) is the point of tangency, and m is the slope. In this case, (x1, y1) = (3, 18), and m = 16. Substituting these values into the equation, we get:

y - 18 = 16(x - 3)

Now, let's simplify the equation:

y - 18 = 16x - 48 y = 16x - 48 + 18 y = 16x - 30

Therefore, the equation of the tangent to the curve at point P is y = 16x - 30.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Differential Equations, 2nd edition, James R. Brannan
[3] Calculus: Early Transcendentals, 2nd edition, James Stewart

Introduction

In our previous article, we explored the curve C, which is defined by the equation y = (2/3)x^3 - x^2 + 4x - 3, and the point P, which has coordinates (3, 18). We showed that P lies on the curve C and found the equation of the tangent to the curve at point P. In this article, we will answer some frequently asked questions related to the curve C and the point P.

Q&A Session

Q: What is the equation of the curve C?

A: The equation of the curve C is y = (2/3)x^3 - x^2 + 4x - 3.

Q: What are the coordinates of the point P?

A: The coordinates of the point P are (3, 18).

Q: How do we show that P lies on the curve C?

A: To show that P lies on the curve C, we substitute the coordinates of P into the equation of the curve and verify that the equation holds true.

Q: What is the derivative of the curve C?

A: The derivative of the curve C is given by:

dy/dx = d((2/3)x^3 - x^2 + 4x - 3)/dx = 2x^2 - 2x + 4

Q: What is the slope of the tangent to the curve at point P?

A: The slope of the tangent to the curve at point P is 16.

Q: What is the equation of the tangent to the curve at point P?

A: The equation of the tangent to the curve at point P is y = 16x - 30.

Q: How do we find the equation of the tangent to the curve at point P?

A: To find the equation of the tangent to the curve at point P, we use the point-slope form of a line:

y - y1 = m(x - x1)

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

Q: What is the significance of the curve C and the point P?

A: The curve C and the point P are used to demonstrate the importance of mathematical concepts in understanding the properties of curves and points.

Conclusion

In this article, we answered some frequently asked questions related to the curve C and the point P. We provided the equation of the curve C, the coordinates of the point P, and the derivative of the curve C. We also explained how to show that P lies on the curve C, find the slope of the tangent to the curve at point P, and determine the equation of the tangent to the curve at point P. This article demonstrates the importance of mathematical concepts in understanding the properties of curves and points.

Additional Resources

[1] Calculus, 3rd edition, Michael Spivak
[2] Differential Equations, 2nd edition, James R. Brannan
[3] Calculus: Early Transcendentals, 2nd edition, James Stewart

Frequently Asked Questions

Q: What is the curve C?

A: The curve C is a cubic curve defined by the equation y = (2/3)x^3 - x^2 + 4x - 3.

Q: What is the point P?

A: The point P is a point on the curve C with coordinates (3, 18).

Q: How do we find the equation of the tangent to the curve at point P?

A: To find the equation of the tangent to the curve at point P, we use the point-slope form of a line:

y - y1 = m(x - x1)

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

Q: What is the significance of the curve C and the point P?

A: The curve C and the point P are used to demonstrate the importance of mathematical concepts in understanding the properties of curves and points.

Q: How do we show that P lies on the curve C?

A: To show that P lies on the curve C, we substitute the coordinates of P into the equation of the curve and verify that the equation holds true.