# Is a vector orthogonal to the tangent plane facing

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) is a vector orthogonal to the tangent plane facing ”outward” – moving in the directionof this vector leads to the fastest increase off. Forz=f(x, y), this means the gradient points to the next level curve.f(x, y, z) =hfx, fy, fziWhen you want to find the change infalong a specific direction vector, you multiply the change infcaused by a changeinx(which isfx) by the direction vector’sx-component, and you add it to the same foryandz.Du=fx(xcomponent of vector) +fy(ycomponent of vector) +fz(zcomponent of vector)Du=f(x, y)·uWhereuis a unit vector.MaximumDuis|∇f(x, y)|– which is whenuhas the same direction asf(x, y).Maxima and Minima:z=f(x, y) has a relative max or min wherefx= 0 andfy= 0.To find what exactly the point is, take the secondderivatives:D=fxxfyy-(fxy)2IfD >0, the point is a max iffxx<0 or a min iffxx>0; ifD <0, saddle point; ifD= 0, the test fails.For absolute mins/maxes, find all relative min/max and then checkcornersandboundary linesfor critical points andfinally compare all the values to see which is the biggest and which is the smallest.Lagrange Multipliers:To find the maximum and minimum values of a function on at least one constraint; where the level curve is tangent tothe constraint; when the gradient offis parallel to the gradient of the constraint.FunctionF(x, y), constraintsC1(x, y) and sometimesC2(x, y), random constants you solve for:λandμ.You solve the following ugly system and finally compare all values to see which is the biggest and the smallest.F(x, y) =λC1(x, y) +μC2(x, y)5
Unit 4, Part 1(Chapter 12, Part 1)Estimating Volume:We use 2-D Riemann sums. Instead ofnfor number of rectangles, we havemandnfor rectangular prisms (mis thenumber of columns (x),nis the number of rows (y)). Multiply the area of each rectangle by the sum of the heights at eachrectangle (at the middle or at the corners).If you have a chart, your rectangles will ”overlap” exactly one unit, so eachrectangle’s edge is at the same value.Volume:Evaluating a function (the height) across an area; because you’re adding up every height across the base’s area, the resultis its volume.Volume =ZZRf(x, y)dAwheref(x, y) is the height.Region can be a rectangle, the area between two curves, or a circularish area.Always draw your region of integration.Two curves:ZZf(x, y)dydx=ZZf(x, y)dxdyCircularish:ZZf(rcosθ, rsinθ)rdrdθAverage value =volumearea of the base(average height)Center of Mass:Balance point, expected value. You don’t have to understand what moment of inertia means.massm=ZZDρ(x, y)dAMomentMx=ZZD(x, y)dAcenter of mass ¯x=MymFor theyversions, just replaceywithx.Joint Density Functions:A function is a joint-density function if the total area under the curve is exactly one and if the area in each region ispositive (it’s always positive area).To verify that a function is joint-density, show that its volume equals 1 (and state that it’s always positive or something).6
Unit 4, Part 2(Chapter 12, Part 2)Surface Area:For a parametrically defined function