Quiz \#02 (Derivative \& Graphing) ' 1. Find the equation of the tangent line to the curve. $$ y=x^{4}-3 x^{2}+2 \quad $$ at the point when $$ x=1 $$

Question

UpStudy AI · Accepted Answer

To find the equation of the tangent line to the curve $$ y = x^4 - 3x^2 + 2 $$ at the point where $$ x = 1 $$, follow these steps:

### 1. **Find the Point of Tangency**

First, determine the y-coordinate when $$ x = 1 $$:

$$
y = (1)^4 - 3(1)^2 + 2 = 1 - 3 + 2 = 0
$$

So, the point of tangency is **$$ (1, 0) $$**.

### 2. **Find the Derivative**

Next, find the derivative $$ \frac{dy}{dx} $$ to determine the slope of the tangent line:

$$
y = x^4 - 3x^2 + 2
$$

$$
\frac{dy}{dx} = 4x^3 - 6x
$$

### 3. **Evaluate the Derivative at $$ x = 1 $$**

Substitute $$ x = 1 $$ into the derivative to find the slope $$ m $$:

$$
m = 4(1)^3 - 6(1) = 4 - 6 = -2
$$

### 4. **Write the Equation of the Tangent Line**

Using the point-slope form of a line:

$$
y - y_1 = m(x - x_1)
$$

Substitute $$ m = -2 $$ and $$ (x_1, y_1) = (1, 0) $$:

$$
y - 0 = -2(x - 1)
$$

$$
y = -2x + 2
$$

### **Final Answer**

The equation of the tangent line is:

$$
\boxed{y = -2x + 2}
$$
The equation of the tangent line at $$ x = 1 $$ is $$ y = -2x + 2 $$.

Quiz \#02 (Derivative \& Graphing) ' 1. Find the equation of the tangent line to the curve. \( y=x^{4}-3 x^{2}+2 \quad \) at the point when \( x=1 \)