Given the function $$ W(x, y)=4 x^{2}+y^{2} $$, find $$ W(5+h, 2+h) $$. $$ W(5+h, 2+h)=\square $$ (t)

Question

UpStudy AI · Accepted Answer

To find $$ W(5+h, 2+h) $$ for the function $$ W(x, y) = 4x^{2} + y^{2} $$, follow these steps:

1. **Substitute $$ x $$ and $$ y $$ with $$ 5+h $$ and $$ 2+h $$ respectively:**
   $$
   W(5+h, 2+h) = 4(5+h)^{2} + (2+h)^{2}
   $$

2. **Expand the squared terms:**
   $$
   (5+h)^{2} = 25 + 10h + h^{2}
   $$
   $$
   (2+h)^{2} = 4 + 4h + h^{2}
   $$

3. **Substitute the expanded forms back into the equation:**
   $$
   W(5+h, 2+h) = 4(25 + 10h + h^{2}) + (4 + 4h + h^{2})
   $$

4. **Distribute the constants:**
   $$
   = 100 + 40h + 4h^{2} + 4 + 4h + h^{2}
   $$

5. **Combine like terms:**
   $$
   = (100 + 4) + (40h + 4h) + (4h^{2} + h^{2})
   $$
   $$
   = 104 + 44h + 5h^{2}
   $$

**Final Answer:**
$$
W(5+h, 2+h) = 104 + 44\,h + 5\,h^{2}
$$
$$ W(5+h, 2+h) = 104 + 44h + 5h^{2} $$

Given the function \( W(x, y)=4 x^{2}+y^{2} \), find \( W(5+h, 2+h) \). \( W(5+h, 2+h)=\square \) (t)