Let's solve each part step by step for the function: \[ f(x, y) = \frac{x}{4} e^{\cos(y)} \] --- ### (a) Cross-section at \( y = \frac{\pi}{4} \) Substitute \( y = \frac{\pi}{4} \): \[ f(x, \frac{\pi}{4}) = \frac{x}{4} e^{\cos(\frac{\pi}{4})} \] Recall that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \): \[ f(x, \frac{\pi}{4}) = \frac{x}{4} e^{\frac{\sqrt{2}}{2}} \] This is a linear function in \( x \), scaled by the constant \( \frac{1}{4} e^{\frac{\sqrt{2}}{2}} \). **Graph:** This is a straight line through the origin with slope \( \frac{1}{4} e^{\frac{\sqrt{2}}{2}} \). --- ### (b) Slope of tangent at \( x = 0 \) The slope is simply the coefficient of \( x \) in the function above: \[ \text{Slope at } x = 0 = \frac{1}{4} e^{\frac{\sqrt{2}}{2}} \] --- ### (c) Express answer in terms of partial derivatives The slope of the tangent to \( f(x, y) \) at \( x = 0 \) for \( y = \frac{\pi}{4} \) is given by the partial derivative of \( f \) with respect to \( x \): \[ \frac{\partial f}{\partial x} = \frac{1}{4} e^{\cos(y)} \] At \( (0, \frac{\pi}{4}) \): \[ \frac{\partial f}{\partial x} \bigg|_{(x = 0, y = \frac{\pi}{4})} = \frac{1}{4} e^{\cos(\frac{\pi}{4})} = \frac{1}{4} e^{\frac{\sqrt{2}}{2}} \] This matches the answer in (b). --- ### (d) Should your answer to 3b be the same as to 2b? Why or why not? Yes, the answers should be **the same**. **Reason:** The slope of the tangent to the cross-section at a fixed \( y \) is exactly the partial derivative of \( f \) with respect to \( x \) at that \( y \). This is because the partial derivative measures the rate of change of \( f \) in the \( x \)-direction while holding \( y \) constant, which is what the cross-section does. --- ## **Summary Table** | Part | Solution | |------|-----------| | (a) | \( f(x, \frac{\pi}{4}) = \frac{x}{4} e^{\frac{\sqrt{2}}{2}} \) (straight line) | | (b) | Slope at \( x=0 \) is \( \frac{1}{4} e^{\frac{\sqrt{2}}{2}} \) | | (c) | Partial derivative: \( \frac{\partial f}{\partial x} = \frac{1}{4} e^{\cos(y)} \) evaluated at \( y = \frac{\pi}{4} \) gives the same value as (b) | | (d) | Yes, they should be the same; both represent the rate of change of \( f \) with respect to \( x \) at fixed \( y \) | --- **If you need the actual graph, let me know!**