Use the information to find and compare $$ \Delta y $$ and $$ d y $$. (Round your answers to four decimal places.) $$ \begin{array}{lll} \text { Function } & x \text {-Value } & \text { Differential of } x \ y=x^{4}+6 & x=-2 & \Delta x=d x=0.01 \end{array} $$ $$ \Delta y= $$ $$ \square $$ W Your answer cannot be understood or graded. Mora Information $$ d y= $$ $$ \square $$ Your answer cannot be understood or graded. More Information

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We are given the function
$$
y = x^4 + 6,
$$
with $$ x = -2 $$ and a small change in $$ x $$ given by
$$
\Delta x = dx = 0.01.
$$

### Step 1. Calculate the differential $$ dy $$

The differential $$ dy $$ is given by
$$
dy = f'(x) \, dx.
$$
First, compute the derivative of $$ y $$:
$$
f(x) = x^4 + 6 \quad \Rightarrow \quad f'(x) = 4x^3.
$$
At $$ x = -2 $$:
$$
f'(-2) = 4(-2)^3 = 4(-8) = -32.
$$
Thus,
$$
dy = -32 \times 0.01 = -0.32.
$$

### Step 2. Calculate the change $$ \Delta y $$

The actual change $$ \Delta y $$ is defined as
$$
\Delta y = f(x + \Delta x) - f(x).
$$
We need to compute $$ f(-2 + 0.01) = f(-1.99) $$ and $$ f(-2) $$:

1. Compute $$ f(-1.99) $$:
   $$
   f(-1.99) = (-1.99)^4 + 6.
   $$
   First, compute $$ (-1.99)^4 $$. Since
   $$
   (-1.99)^2 = 3.9601,
   $$
   then
   $$
   (-1.99)^4 = (3.9601)^2 \approx 15.6816.
   $$
   Therefore,
   $$
   f(-1.99) \approx 15.6816 + 6 = 21.6816.
   $$

2. Compute $$ f(-2) $$:
   $$
   f(-2) = (-2)^4 + 6 = 16 + 6 = 22.
   $$

Now, compute $$ \Delta y $$:
$$
\Delta y = f(-1.99) - f(-2) \approx 21.6816 - 22 = -0.3184.
$$

### Step 3. Final Answers

$$
\Delta y \approx -0.3184, \quad dy \approx -0.3200.
$$
$$ \Delta y \approx -0.3184, \quad dy \approx -0.3200. $$

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