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

Question

UpStudy AI · Accepted Answer

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

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

Use the information to find and compare and . (Round your answers to four decimal places.)

W Your answer cannot be understood or graded. Mora Information
Your answer cannot be understood or graded. More Information

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Calculus Questions

Use the information to find and compare and . (Round your answers to four decimal places.) W Your answer cannot be understood or graded. Mora Information Your answer cannot be understood or graded. More Information