By Peter V O'Neil

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Calculus: 1,001 Practice Problems For Dummies

1001 Calculus perform difficulties For Dummies takes you past the guide and suggestions provided in Calculus For Dummies, providing you with 1001 possibilities to perform fixing difficulties from the key subject matters on your calculus direction. Plus, a web part will give you a set of calculus difficulties provided in multiple-choice structure to additional assist you try your abilities as you cross.

Sample text

Y Computing ∂ϕ/∂ x is less straightforward, since x occurs in both integrals defining ϕ(x, y). For ∂ϕ/∂ x, use the condition that ∂ M/∂ y = ∂ N /∂ x to write ∂ ∂ϕ = ∂x ∂x x M(ξ, y0 ) dξ + x0 y = M(x, y0 ) + y0 y = M(x, y0 ) + y0 ∂ ∂x y N (x, η) dη y0 ∂N (x, η) dη ∂x ∂M (x, η) dη ∂y = M(x, y0 ) + M(x, y) − M(x, y0 ) = M(x, y). This completes the proof. In the case of y d x + dy = 0, M(x, y) = y and N (x, y) = 1, so ∂N ∂M = 1 and = 0. 1 tells us that this differential equation is not exact on any rectangle in the plane.

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Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4 Homogeneous, Bernoulli, and Riccati Equations 27 or x du = f (u) − u. dx The variables u and x separate as 1 1 du = d x. f (u) − u x We attempt to solve this separable equation and then substitute u = y/x to obtain the solution of the original homogeneous equation.