If `y=x^4-12` and if `x` changes from 2 to 1.99. what is the appoinmate change in `y.`

Options

A. `0.32`
B. `0.032`
C. `5.68`
D. `5.968`

Explanation

To find the approximate change in \( y \) when \( x \) changes from 2 to 1.99, we can follow these steps:

### Step 1: Write down the given function
The function given is:
\[ y = x^4 - 12 \]

### Step 2: Differentiate the function
To find the rate of change of \( y \) with respect to \( x \), we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^4 - 12) \]
Using the power rule of differentiation, we find:
\[ \frac{dy}{dx} = 4x^3 \]

### Step 3: Calculate the change in \( x \)
The change in \( x \) when \( x \) changes from 2 to 1.99 is:
\[ \Delta x = 1.99 - 2 = -0.01 \]

### Step 4: Substitute \( x = 2 \) into the derivative
Now we substitute \( x = 2 \) into the derivative to find \( \frac{dy}{dx} \) at that point:
\[ \frac{dy}{dx} \bigg|_{x=2} = 4(2^3) = 4(8) = 32 \]

### Step 5: Use the formula for approximate change in \( y \)
The approximate change in \( y \) can be calculated using:
\[ \Delta y \approx \frac{dy}{dx} \cdot \Delta x \]
Substituting the values we have:
\[ \Delta y \approx 32 \cdot (-0.01) = -0.32 \]

### Final Answer
The approximate change in \( y \) when \( x \) changes from 2 to 1.99 is:
\[ \Delta y \approx -0.32 \]

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