If `y=x^(4)+10` and x change from 2 to 1.99, find the approximate change in y.

Answer

To find the approximate change in \( y \) when \( x \) changes from \( 2 \) to \( 1.99 \), we can use the concept of derivatives. Let's go through the steps:

### Step 1: Define the function
We are given the function:
\[ y = x^4 + 10 \]

### Step 2: Determine the change in \( x \)
The initial value of \( x \) is \( 2 \) and the new value is \( 1.99 \). Thus, the change in \( x \) (denoted as \( \Delta x \)) is:
\[ \Delta x = 1.99 - 2 = -0.01 \]

### Step 3: Find the derivative of \( y \)
To find the approximate change in \( y \), we need to compute the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^4 + 10) = 4x^3 \]

### Step 4: Evaluate the derivative at \( x = 2 \)
Now, we substitute \( x = 2 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=2} = 4(2^3) = 4 \times 8 = 32 \]

### Step 5: Calculate the approximate change in \( y \)
Using the formula for the change in \( y \):
\[ \Delta y \approx \frac{dy}{dx} \cdot \Delta x \]
Substituting the values we found:
\[ \Delta y \approx 32 \cdot (-0.01) = -0.32 \]

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