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Problem 10

# Let $g(u, v, w)=\left(u e^{v w}+v e^{w w}+w e^{u t}\right) /\left(u^{2}+v^{2}+w^{2}\right)$. Compute $$g(1,2,3)$$ and $$g(3,2,1)$$.

Expert verified
$$g(1,2,3)=\frac{1\cdot e^{6} + 2\cdot e^{9} + 3\cdot e^{t}}{14}$$ and $$g(3,2,1)=\frac{3\cdot e^{2} + 2\cdot e^{1} + 1\cdot e^{3t}}{14}$$.
See the step by step solution

## Step 1: Understanding the function g(u, v, w)

The function $$g(u, v, w)$$ is given as: $$g(u, v, w)=\frac{ue^{vw} + ve^{ww} + we^{ut}}{u^2+v^2+w^2}$$ Let's break down the expression. The numerator is the sum of three product terms containing exponentials, and the denominator is the sum of squares of $$u$$, $$v$$, and $$w$$.

## Step 2: Compute g(1,2,3)

We are asked to find $$g(1,2,3)$$. That means substituting $$u=1$$, $$v=2$$, and $$w=3$$ in the function: $$g(1,2,3)=\frac{1\cdot e^{(2)(3)} + 2\cdot e^{(3)(3)} + 3\cdot e^{(1)(t)}}{1^2+2^2+3^2}$$ Now we will simplify the expression: 1. Calculate exponentials: $$g(1,2,3)=\frac{1\cdot e^{6} + 2\cdot e^{9} + 3\cdot e^{t}}{1^2+2^2+3^2}$$ 2. Simplify denominator: $$g(1,2,3)=\frac{1\cdot e^{6} + 2\cdot e^{9} + 3\cdot e^{t}}{1+4+9}$$ 3. Final simplification: $$g(1,2,3)=\frac{1\cdot e^{6} + 2\cdot e^{9} + 3\cdot e^{t}}{14}$$ So, $$g(1,2,3)=\frac{1\cdot e^{6} + 2\cdot e^{9} + 3\cdot e^{t}}{14}$$.

## Step 3: Compute g(3,2,1)

Now, we are asked to find $$g(3,2,1)$$. That means substituting $$u=3$$, $$v=2$$, and $$w=1$$ in the function: $$g(3,2,1)=\frac{3\cdot e^{(2)(1)} + 2\cdot e^{(1)(1)} + 1\cdot e^{(3)(t)}}{3^2+2^2+1^2}$$ Now we will simplify the expression: 1. Calculate exponentials: $$g(3,2,1)=\frac{3\cdot e^{2} + 2\cdot e^{1} + 1\cdot e^{3t}}{3^2+2^2+1^2}$$ 2. Simplify denominator: $$g(3,2,1)=\frac{3\cdot e^{2} + 2\cdot e^{1} + 1\cdot e^{3t}}{9+4+1}$$ 3. Final simplification: $$g(3,2,1)=\frac{3\cdot e^{2} + 2\cdot e^{1} + 1\cdot e^{3t}}{14}$$ So, $$g(3,2,1)=\frac{3\cdot e^{2} + 2\cdot e^{1} + 1\cdot e^{3t}}{14}$$. To summarize our results: - $$g(1,2,3)=\frac{1\cdot e^{6} + 2\cdot e^{9} + 3\cdot e^{t}}{14}$$ - $$g(3,2,1)=\frac{3\cdot e^{2} + 2\cdot e^{1} + 1\cdot e^{3t}}{14}$$

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