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Problem 1

# Calculate $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}$, and $$\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}$$ when defined. HINT [See Quick Examples page 1098.] $$f(x, y)=10,000-40 x+20 y$$

Expert verified
The partial derivatives of the function $$f(x, y) = 10000 - 40x + 20y$$ are $$\frac{\partial f}{\partial x} = -40$$ and $$\frac{\partial f}{\partial y} = 20$$. Since they are constants and do not depend on the point, their values at $$(1, -1)$$ are $$\left.\frac{\partial f}{\partial x}\right|_{(1,-1)} = -40$$ and $$\left.\frac{\partial f}{\partial y}\right|_{(1,-1)} = 20$$.
See the step by step solution

## Step 1. Finding the partial derivative with respect to x

To find the partial derivative of $$f(x, y)$$ with respect to $$x$$, we treat $$y$$ as a constant and differentiate with respect to $$x$$. So, $\frac{\partial f}{\partial x} = \frac{d}{dx}(10000 - 40x + 20y).$

## Step 2. Calculating the partial derivative with respect to x

Differentiating with respect to $$x$$, we get $\frac{\partial f}{\partial x} = -40.$

## Step 3. Finding the partial derivative with respect to y

Similarly, to find the partial derivative of $$f(x, y)$$ with respect to $$y$$, we treat $$x$$ as a constant and differentiate with respect to $$y$$. So, $\frac{\partial f}{\partial y} = \frac{d}{dy}(10000 - 40x + 20y).$

## Step 4. Calculating the partial derivative with respect to y

Differentiating with respect to $$y$$, we get $\frac{\partial f}{\partial y} = 20.$

## Step 5. Evaluate the partial derivatives at (1, -1)

Now we need to find the values of the partial derivatives at the given point $$(1, -1)$$. However, since the partial derivatives we calculated in previous steps are constants and do not depend on $$x$$ and $$y$$, their values remain unchanged at the point $$(1, -1)$$. Thus, $\left.\frac{\partial f}{\partial x}\right|_{(1,-1)} = \frac{\partial f}{\partial x} = - 40$ and $\left.\frac{\partial f}{\partial y}\right|_{(1,-1)} = \frac{\partial f}{\partial y} = 20.$

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