https://file.aiquickdraw.com/mathbot/file/1726615527821z50h4h67.jpg

To find the equation of the tangent to the curve at a specific point, follow these steps:

The given equation is:
\[
y = x^2(2x - 1)
\]
We are also given that \( x = -1 \). 

### Step 1: Differentiate the equation to find the slope of the tangent
We need to differentiate \( y \) with respect to \( x \) to find the slope of the tangent line. The product rule will be applied to differentiate \( y = x^2(2x - 1) \).

\[
y' = \frac{d}{dx} \left( x^2(2x - 1) \right)
\]

Using the product rule, we get:
\[
y' = \frac{d}{dx}(x^2) \cdot (2x - 1) + x^2 \cdot \frac{d}{dx}(2x - 1)
\]

\[
y' = 2x(2x - 1) + x^2(2)
\]

\[
y' = 2x(2x - 1) + 2x^2
\]

Simplifying:
\[
y' = 4x^2 - 2x + 2x^2 = 6x^2 - 2x
\]

### Step 2: Substitute \( x = -1 \) to find the slope at this point
Now, substitute \( x = -1 \) into the derivative to find the slope of the tangent line.

\[
y' = 6(-1)^2 - 2(-1) = 6(1) + 2 = 6 + 2 = 8
\]

So, the slope of the tangent at \( x = -1 \) is 8.

### Step 3: Find the point on the curve at \( x = -1 \)
Substitute \( x = -1 \) into the original equation to find the \( y \)-coordinate.

\[
y = (-1)^2(2(-1) - 1) = 1(-2 - 1) = 1(-3) = -3
\]

Thus, the point on the curve is \( (-1, -3) \).

### Step 4: Write the equation of the tangent line
Now that we have the slope \( m = 8 \) and the point \( (-1, -3) \), we can use the point-slope form of the equation of a line:

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

Substitute \( m = 8 \), \( x_1 = -1 \), and \( y_1 = -3 \):

\[
y + 3 = 8(x + 1)
\]

Simplify:

\[
y + 3 = 8x + 8
\]

\[
y = 8x + 5
\]

### Final Answer:
The equation of the tangent line is:
\[
y = 8x + 5
\]

Would you like any further details or clarifications?

Here are some related questions:
1. How do you apply the product rule in differentiation?
2. How do you find the equation of a tangent line given a curve?
3. What is the significance of the slope in a tangent line equation?
4. How do you interpret the point-slope form of a line equation?
5. What other methods can be used to find the derivative of a function?

**Tip:** When differentiating a product of functions, always use the product rule to ensure accuracy.

Find equation of tangent for y = x^2(2x - 1) at x = -1

Find the equation of the tangent to the curve y = x^2(2x - 1) at x = -1 using differentiation and the product rule. This detailed solution covers finding the slope and the point-slope form of the tangent line equation. Ideal for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation