First Derivative and Its Significance

Oct 13, 2025 by talynt.org 77 views

The given curve is defined by the equation $\frac{x^2}{16} - \frac{y^2}{9} = 1$ . This is a hyperbola, a type of conic section, with its center at the origin (0, 0). The equation can be rewritten as $\frac{x^2}{4^2} - \frac{y^2}{3^2} = 1$ , which is in the standard form of a hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ , where $a = 4$ and $b = 3$ .

First Derivative and Its Significance

The first derivative of the curve, $\frac{d y}{d x} = \frac{9 x}{16 y}$ , represents the slope of the tangent line to the curve at any point $(x, y)$ . This derivative is crucial in understanding the behavior of the curve, including its concavity and inflection points.

Second Derivative and Its Significance

The second derivative of the curve, $\frac{d^2 y}{d x^2} = -\frac{81}{16 y^3}$ , represents the rate of change of the slope of the tangent line to the curve. This derivative is essential in determining the concavity and inflection points of the curve.

Inflection Points and Concavity

Inflection points are the points on the curve where the concavity changes. The concavity of a curve is determined by the sign of the second derivative. If the second derivative is positive, the curve is concave upward, and if it is negative, the curve is concave downward.

To determine the inflection points, we need to find the values of $x$ and $y$ where the second derivative is equal to zero or undefined. Setting the second derivative equal to zero, we get:

-\frac{81}{16 y^3} = 0

Solving for $y$ , we get:

y^3 = \infty

This implies that $y$ is undefined, which means that the second derivative is undefined at all points on the curve. Therefore, the curve does not have any inflection points.

Concavity and Its Significance

The concavity of a curve is determined by the sign of the second derivative. If the second derivative is positive, the curve is concave upward, and if it is negative, the curve is concave downward.

To determine the concavity of the curve, we need to examine the sign of the second derivative. Since the second derivative is always negative, the curve is concave downward at all points.

Conclusion

In conclusion, the given curve is a hyperbola with its center at the origin (0, 0). The first derivative represents the slope of the tangent line to the curve, while the second derivative represents the rate of change of the slope of the tangent line. The curve does not have any inflection points, and it is concave downward at all points.

Key Takeaways

The given curve is a hyperbola with its center at the origin (0, 0).
The first derivative represents the slope of the tangent line to the curve.
The second derivative represents the rate of change of the slope of the tangent line.
The curve does not have any inflection points.
The curve is concave downward at all points.

Final Thoughts

In the previous article, we explored the properties of the given curve, including its equation, first derivative, and second derivative. We also determined the concavity and inflection points of the curve. In this article, we will answer some frequently asked questions about the curve and its derivatives.

Q: What is the equation of the curve?

A: The equation of the curve is $\frac{x^2}{16} - \frac{y^2}{9} = 1$ . This is a hyperbola, a type of conic section, with its center at the origin (0, 0).

Q: What is the first derivative of the curve?

A: The first derivative of the curve is $\frac{d y}{d x} = \frac{9 x}{16 y}$ . This represents the slope of the tangent line to the curve at any point $(x, y)$ .

Q: What is the second derivative of the curve?

A: The second derivative of the curve is $\frac{d^2 y}{d x^2} = -\frac{81}{16 y^3}$ . This represents the rate of change of the slope of the tangent line to the curve.

Q: Does the curve have any inflection points?

A: No, the curve does not have any inflection points. The second derivative is always negative, which means that the curve is concave downward at all points.

Q: Is the curve concave upward or downward?

A: The curve is concave downward at all points. This is because the second derivative is always negative.

Q: What is the significance of the first derivative?

A: The first derivative represents the slope of the tangent line to the curve at any point $(x, y)$ . This is important in understanding the behavior of the curve.

Q: What is the significance of the second derivative?

A: The second derivative represents the rate of change of the slope of the tangent line to the curve. This is important in determining the concavity and inflection points of the curve.

Q: Can you provide an example of how to use the first and second derivatives?

A: Yes, here is an example:

Suppose we want to find the slope of the tangent line to the curve at the point $(4, 3)$ . We can use the first derivative to find the slope:

\frac{d y}{d x} = \frac{9 x}{16 y}

Substituting $x = 4$ and $y = 3$ , we get:

\frac{d y}{d x} = \frac{9(4)}{16(3)} = \frac{36}{48} = \frac{3}{4}

Therefore, the slope of the tangent line to the curve at the point $(4, 3)$ is $\frac{3}{4}$ .

Q: Can you provide an example of how to use the second derivative?

A: Yes, here is an example:

Suppose we want to determine the concavity of the curve at the point $(4, 3)$ . We can use the second derivative to find the rate of change of the slope of the tangent line:

\frac{d^2 y}{d x^2} = -\frac{81}{16 y^3}

Substituting $y = 3$ , we get:

\frac{d^2 y}{d x^2} = -\frac{81}{16(3)^3} = -\frac{81}{16(27)} = -\frac{3}{16}

Since the second derivative is negative, the curve is concave downward at the point $(4, 3)$ .

Conclusion

In this article, we have answered some frequently asked questions about the curve and its derivatives. We have also provided examples of how to use the first and second derivatives to find the slope of the tangent line and determine the concavity of the curve.