https://file.aiquickdraw.com/mathbot/file/1728770293489wbw0ojrx.jpg

what is the equation of the tangent line at (pi^2, pi)

To find the equation of the tangent line at the given point \((\pi^2, \pi)\) for the curve \(3 \sin y + x = y^2\), we will follow these steps:

### 1. **Verify the point lies on the curve**:
The point is \((\pi^2, \pi)\). We substitute \(x = \pi^2\) and \(y = \pi\) into the given equation:

\[
3 \sin(\pi) + \pi^2 = \pi^2.
\]

Since \(\sin(\pi) = 0\), this simplifies to:

\[
0 + \pi^2 = \pi^2,
\]

which is true, so the point \((\pi^2, \pi)\) does lie on the curve.

### 2. **Implicit Differentiation**:
We need to find the slope of the tangent line by differentiating the given equation implicitly:

\[
3 \sin y + x = y^2.
\]

Taking the derivative with respect to \(x\) on both sides:

\[
\frac{d}{dx}(3 \sin y) + \frac{d}{dx}(x) = \frac{d}{dx}(y^2),
\]

which becomes:

\[
3 \cos y \cdot \frac{dy}{dx} + 1 = 2y \cdot \frac{dy}{dx}.
\]

Now, solve for \(\frac{dy}{dx}\) (which is the slope of the tangent line):

\[
3 \cos y \cdot \frac{dy}{dx} + 1 = 2y \cdot \frac{dy}{dx}.
\]

Group the \(\frac{dy}{dx}\) terms together:

\[
\frac{dy}{dx}(3 \cos y - 2y) = -1.
\]

Thus, the slope is:

\[
\frac{dy}{dx} = \frac{-1}{3 \cos y - 2y}.
\]

### 3. **Evaluate the slope at \(y = \pi\)**:
Substitute \(y = \pi\) into the slope equation:

\[
\frac{dy}{dx} = \frac{-1}{3 \cos(\pi) - 2\pi}.
\]

Since \(\cos(\pi) = -1\), this simplifies to:

\[
\frac{dy}{dx} = \frac{-1}{3(-1) - 2\pi} = \frac{-1}{-3 - 2\pi} = \frac{1}{3 + 2\pi}.
\]

### 4. **Equation of the Tangent Line**:
The tangent line at \((\pi^2, \pi)\) has the form:

\[
y - y_1 = m(x - x_1),
\]

where \(m = \frac{1}{3 + 2\pi}\), \(x_1 = \pi^2\), and \(y_1 = \pi\).

Substitute these values into the equation:

\[
y - \pi = \frac{1}{3 + 2\pi}(x - \pi^2).
\]

Finally, solving for \(y\) in slope-intercept form:

\[
y = \frac{1}{3 + 2\pi}(x - \pi^2) + \pi.
\]

This is the equation of the tangent line at \((\pi^2, \pi)\).

---

Let me know if you'd like further clarifications or details! 

Here are five related questions for further practice:
1. How would the equation of the tangent line change if the point were \((4, 2)\) on the same curve?
2. How do you verify that a point lies on a curve in general?
3. What is the geometric meaning of the derivative in the context of tangent lines?
4. Can you find the second derivative \(\frac{d^2y}{dx^2}\) at \((\pi^2, \pi)\)?
5. How does implicit differentiation differ from explicit differentiation?

**Tip**: When dealing with implicit functions, always remember to apply the chain rule when differentiating terms that involve the dependent variable!

What is the equation of the tangent line at (π^2, π) on the curve 3 sin(y) + x = y^2?

Explore how to find the equation of the tangent line to the curve 3sin(y) + x = y^2 at the point (π^2, π). This solution uses implicit differentiation to determine the slope and applies the tangent line formula. Ideal for advanced calculus learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation