Real Functions of Several Variables Examples of Basic by Leif Mejlbro

By Leif Mejlbro

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9 Let 1 , w(x, t) = √ 1 − 2xt + t2 (x, t) ∈ A. 1) Sketch the domain A 2) Prove that w fulfils the differential equation (x − t) ∂w ∂w =t . ∂x ∂t A Domain, and a check of a differential equation. D Analyze the condition 1 − 2xt + t2 > 0. Then insert w into the differential equation. 2 1 –2 –1 0 1 2 –1 –2 Figure 15: The domain A lies between the two branches of hyperbola x = 1 2 t+ 1 . t I The domain is described by 1 − 2xt + t2 > 0. e. A lies between the two branches. Obviously, w ∈ C ∞ (D). com 58 Calculus 2c-1 Examinations of functions When (x, y) ∈ A, then w(x, y) = 0.

E. A = A◦ is given by A = A◦ = {(x, y) | x2 < y < 4, −2 < x < 1} ∪ {(x, y) | y > max{4, x2 }, x ∈ R} = {(x, y) | y ∈ R+ \ {4}, x ∈ R, y > x2 }. The closure is A = (x, y) | y ≥ x2 , and the boundary is ∂A = (x, y) | y = x2 } ∪ {(x, 4) | −2 < x < 2 . 12 Sketch the domain A of the function f (x, y) = 4 − x2 − y 2 , x+y−2 and sketch or describe the interior A◦ and the boundary ∂A. A Domain. D Find the zero sets of the numerator and the denominator in the fraction and then examine the signs of these.

Since Ga (x) → −∞ for thus G√ a is increasing in ] − x → ± a2 − 1, the range of Ga is given by 1+ ] − ∞, Ga (0)] = −∞, ln(a − 1) − a −1 . 2 3) It follows from (2) when y = a that dGa (0) 1 1 = − , da a−1 2 Please click the advert Student Discounts + Student Events + Money Saving Advice = Happy Days! com 57 Calculus 2c-1 Examinations of functions hence Ga (0) is increasing for a ∈ ]1, 3[ and decreasing for a ∈ ]3, +∞[. Since Ga (0) → −∞ for a → 1+ and for a → +∞, we see that G3 (0) = f (0, 3) = ln 2 − 5 2 is a global maximum.

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