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# MIdterm 2 Study Guide Math 2419

UTD

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This 8 page Study Guide was uploaded by Saul Cervantes on Monday November 2, 2015. The Study Guide belongs to Math 2419 at University of Texas at Dallas taught by Anotoly Ezlydon in Fall 2015. Since its upload, it has received 36 views. For similar materials see Accelerated Calculus II in Mathematics (M) at University of Texas at Dallas.

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Date Created: 11/02/15

MIDTERM 2 STUDY GUIDE: CALCULUS II Equations ( ) Partial Derivative relative to x: fx∆????→∞im???? ????????+∆????,???????? −????(????????,????????) ∆???? Partial Derivative relative to y: fy = lim???? ????????,????????+∆???? −????(????????,????????) ∆????→∞ ∆???? Gradient: ∇???? = < ????????,????????,???????? > Full differential: ∇???? ∙< ∆????,∆???? > Implicit Differentiation: Given z = f(x, y), and F (x, y, z) = 0, then ???????? −???????? ???????? −???????? ???????? = ????????, and???????? = ???????? ???? ????+???????? −????(????) Directional Derivative u f(X=lim ????→0 Where f is a function ???? of n variables, X =1,X2X ,… n ), and u is a unit vector1u 2 <u n u ,…u >. |????∙????| To find angle of inclination: ????????????????| ???? | Lagrange Multipliers: If f and g are differentiable, then: ∇???? = ????∇???? → ???? = ???? fx = ???????????? fy = ???????????? Volume under a surface: ????(????,????)???????? ∬ ???? Area of domain D: ∬ 1???????? ???? Mass: m = ∬???? ????(????,????)???????? First moment with respect to x:yM ∬ ????????????(????,????)???????? ( ) First moment with respect to y:xM ∬ ???????????? ????,???? ???????? ???????? ???????? Center of mass at (x, y) where x????= and y = ???? Concepts If f is a function of two or more variables, then the gradient of f (???????????????????????????? ???????? ∇????) is the vector of the partial derivatives. Gradient is the derivative of a multivariable function. Theorem: Sufficient Condition of Differentiability—If f is the function of x and y, and fx and fy are continuous in an open region, then f is differentiable in this region. Theorem: If f is differentiable at (xo, yo), then it is also continuous at (xo, yo). Chain Rule: Derivative of Composite Function = Derivative of Outer Function + Derivative of Inner Function. Second Derivative Test: d = f fxx yy ) xy2 o If d > 0 and fxxa, b) > 0, then f has a relative minimum at the point (a, b) o If d > 0 and f (a, b) < 0, then f has a relative maximum at the point xx (a, b) o If d < 0, then point (a, b, f(a, b)) is a saddle point. o If d = 0, the test is inconclusive. Theorem: Suppose f(x, y) is continuous on ????,???? × [????,????], then ???? ???? ???? ???? ∫ ∫ ???? ????,???? ???????????????? = ∫ ∫ ???? ????,???? ???????????????? = ∬ ????(????,????)???????? ???? ???? ???? ???? ????,???? × ????,???? o Note: This theorem only works on functions defined over a rectangle. Examples:

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