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Semester 1 Review Unit vector =

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A line that contains the point P(x 0,y 0,z 0 ) and direction vector : parametric symmetric

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The equation of the plane containing point P(x 0,y 0,z 0 ) with normal vector, Ex. Let θ be the angle between the planes 5y + 6z = 2 and 2x + 8y – 3z = 1. Find cos θ.

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Sketching 3-D Ex. Sketch

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Cylind RectRect Cylind Spher RectRect Spher

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Parameterizing curves: Derivative and integral of vector functions Ex. Find the parametric equations of the line tangent to at (1,4,4).

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Ex. Find if and

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If r(t) = displacement vector, then r(t) = velocity v(t) r (t) = accelerationa(t) |r(t)| = speedv(t)

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First and second partial derivatives Implicit differentiation: Total Differential: dz = f x dx + f y dy

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Let w = f (x,y), with x = g(t) and y = h(t): Gradient is orthogonal to level curve of f Directional derivative of f in the direction of unit vector u is

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Ex. Find the parametric equations of the line tangent to the curve of intersection of and at (-1,1,2).

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Critical point = where f x and f y are both zero, or where one is undefined d = f xx (a,b) f yy (a,b) – [ f xy (a,b)] 2 If d > 0 and f xx (a,b) < 0, then (a,b) is a rel. max. If d > 0 and f xx (a,b) > 0, then (a,b) is a rel. min. If d < 0, then (a,b) is a saddle point. If d = 0, then the test fails.

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Ex. Find the critical points of and classify them as relative maximum, relative minimum, or saddle point.

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Ex. Use Lagrange multipliers to find the maximum value of subject to the constraint

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Multiple Integrals – changing order Ex.

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Ex. Find the volume bounded by x 2 + y 2 = 64, under the plane z = y, and in the first octant.

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where ρ is density Surface area: Jacobian:

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Ex. Evaluate, where R is the region bounded by

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Bring your textbooks and ID cards on the first day of next semester so that we can exchanged them

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Lines and Planes In three dimensions, we use vectors to indicate the direction of a line. as a direction vector would indicate that Δx = 7, Δy = 6, and.

Lines and Planes In three dimensions, we use vectors to indicate the direction of a line. as a direction vector would indicate that Δx = 7, Δy = 6, and.

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