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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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$### Contour Diagrams and Function Identification Below are four contour diagrams labeled I to IV. Each diagram represents the contour lines of a different function. Your task is to identify which contour diagram corresponds to each given function. **Contour Diagrams**: - **I)** - The contour lines form a symmetrical shape around the origin with lines curving outward and upward symmetrically, creating a cross-like pattern. - **II)** - This diagram features elongated contours running primarily from left to right with a noticeable curvature, indicating a function that is not symmetrical in terms of both x and y. - **III)** - The contours form concentric circles centered around the origin, indicating radial symmetry. - **IV)** - The contours here create a saddle-shaped pattern with contours sweeping from top left to bottom right and bottom left to top right. **Functions**: - **a)** $ z = f(x,y) = x^2 + y^2 $ - **b)** $ z = f(x,y) = x^2 - y^2 $ - **c)** $ z = f(x,y) = x - y^2 $ - **d)** $ z = f(x,y) = xy $ **Questions**: a) Which sketch is a contour diagram of the function $ z = f(x,y) = x^2 + y^2 $? b) Which sketch is a contour diagram of the function $ z = f(x,y) = x^2 - y^2 $? c) Which sketch is a contour diagram of the function $ z = f(x,y) = x - y^2 $? d) Which sketch is a contour diagram of the function $ z = f(x,y) = xy $? ### Explanation: - **Function $ z = x^2 + y^2 $** produces concentric circles representing a peak or valley at the origin. Thus, **contour diagram III** matches this function. - **Function $ z = x^2 - y^2 $** creates a saddle surface, typically appearing as a hyperbolic pattern. Hence, **contour diagram I** fits this profile. - **Function $ z = x - y^2 $** results in contours that don't share symmetry about the origin and are more linear and elongated. Therefore, **contour diagram II** corresponds to this function. - **Function $ z = xy $** should$

Transcribed Image Text:### Contour Diagrams and Function Identification Below are four contour diagrams labeled I to IV. Each diagram represents the contour lines of a different function. Your task is to identify which contour diagram corresponds to each given function. **Contour Diagrams**: - **I)** - The contour lines form a symmetrical shape around the origin with lines curving outward and upward symmetrically, creating a cross-like pattern. - **II)** - This diagram features elongated contours running primarily from left to right with a noticeable curvature, indicating a function that is not symmetrical in terms of both x and y. - **III)** - The contours form concentric circles centered around the origin, indicating radial symmetry. - **IV)** - The contours here create a saddle-shaped pattern with contours sweeping from top left to bottom right and bottom left to top right. **Functions**: - **a)** $ z = f(x,y) = x^2 + y^2 $ - **b)** $ z = f(x,y) = x^2 - y^2 $ - **c)** $ z = f(x,y) = x - y^2 $ - **d)** $ z = f(x,y) = xy $ **Questions**: a) Which sketch is a contour diagram of the function $ z = f(x,y) = x^2 + y^2 $? b) Which sketch is a contour diagram of the function $ z = f(x,y) = x^2 - y^2 $? c) Which sketch is a contour diagram of the function $ z = f(x,y) = x - y^2 $? d) Which sketch is a contour diagram of the function $ z = f(x,y) = xy $? ### Explanation: - **Function $ z = x^2 + y^2 $** produces concentric circles representing a peak or valley at the origin. Thus, **contour diagram III** matches this function. - **Function $ z = x^2 - y^2 $** creates a saddle surface, typically appearing as a hyperbolic pattern. Hence, **contour diagram I** fits this profile. - **Function $ z = x - y^2 $** results in contours that don't share symmetry about the origin and are more linear and elongated. Therefore, **contour diagram II** corresponds to this function. - **Function $ z = xy $** should

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