We are given the function:

$y = x^2$

Step 1: Find $\frac{dy}{dx}$

To differentiate $y = x^2$ with respect to $x$, we apply the power rule for differentiation:

$\frac{dy}{dx} = 2x$

Step 2: Find $\frac{dy}{dx}$ when $x = 4$

Now, we substitute $x = 4$ into the derivative:

$\frac{dy}{dx} = 2(4) = 8$

So, the slope of the curve at $x = 4$ is 8.

Step 3: Approximate $y$ for $x = 3.998$

Now, to approximate the value of $y$ when $x = 3.998$, we can use the tangent line approximation at $x = 4$. The formula for the tangent line at $x = 4$ is:

$y = y_0 + \frac{dy}{dx}(x - x_0)$

Where:

• $x_0 = 4$
• $y_0 = x_0^2 = 16$ (since $x_0 = 4$)
• $\frac{dy}{dx} = 8$

Now, substitute into the tangent line equation:

$y = 16 + 8(3.998 - 4) = 16 + 8(-0.002) = 16 - 0.016 = 15.984$

Approximate value of $y$ when $x = 3.998$ is $y \approx 15.984$.

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Would you like further details or have any questions? Here are some related questions:

1. What is the tangent line approximation for other values of $x$?
2. How can you apply this method to functions other than $y = x^2$?
3. What is the difference between tangent line approximation and linearization?
4. How do you interpret the slope $\frac{dy}{dx}$ geometrically?
5. How can you refine approximations using higher-order derivatives?

Tip: Tangent line approximation is particularly useful when the change in $x$ is small, making it easier to estimate function values near a given point.