Download Test #2 Review

MAC 2311 – Calculus III
REVIEW FOR TEST #2
FLORIDA INTERNATIONAL UNIV.
FALL 2011
REMEMBER TO BRING AN 8x11 BLUE EXAM BOOKLET FOR THE TEST
Relevant sections of the textbook: [Ch.13: Sec 13.1-13.9 and Ch.14: Sec. 14.1-14.5]
MAIN PROBLEM SOLVING TECHNIQUES:
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How to tell if the limit of f(x,y) as x, y tends to a, b exists and how to find its value.
How to find partial derivatives, and second order partial & mixed partial derivatives; and
how to use the chain rules.
How to find total differentials and how to use them to estimate percentage errors in
measurements and to find the local linear approximation of a function.
How to find the gradient and directional derivatives of a function.
How to find the normal to z = f(x,y) & to a parametric surface r = x(u,v), y(u,v), z(u,v) .
How to find the equation of the tangent plane to the surface z = f(x,y) at the point
x0, y0, f(x0,y0) .
How to find the gradient-zero points of f(x,y) & test if they are local max. or local min.
How to find global max. & min., critical points, and to solve related word problems.
How to find constrained maximum & minimum by using Lagrange multipliers.
How to evaluate double integrals over type I & II regions and use them to find volumes.
How to evaluate double integrals in polar coordinates & use them to find volumes of solids.
How to find the surface area of z = f(x,y) and of a parametric surface.
How to find triple integrals & use them to find the mass of a solid body with variable density.
KEY CONCEPTS AND MAIN DEFINITIONS:
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Limit of f(x,y) along a specified curve as x, y tends to a, b and in an arbitrary way &
continuity of functions of 2 or 3 variables.
The gradient of a function and the differentiability of functions of 2 or 3 variables.
The total differential of a function & the local linear approximation of a function.
The gradient and directional derivatives of a function of 2 or 3 variables.
The tangent plane to a surface z = f(x,y) at the point x0, y0, f(x0,y0) .
The normal to z = f(x,y) and to a parametric surface r(u,v) = x(u,v), y(u,v), z(u,v) .
Local extrema of a function, gradient-zero points & the discriminant.
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Global extrema (maxima & minima) of a function & critical points.
Constrained global extrema of a function and Lagrange multipliers.
Double integrals interpreted as the volume under z = f(x,y) and above the region R.
Iterated integrals & double integrals over type I & type II regions of the x-y plane.
Double integrals in polar coordinates and simple polar regions.
Surface area of z = f(x,y) and of a parametric surface r(u,v) = x(u,v), y(u,v), z(u,v) .
The triple integral interpreted as the mass of a solid region G with variable density.
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REVIEW FOR TEST #2 - MAIN FORMULAS:
FALL 2011
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fx(x,y) = lim h0 [f(x+h,y) – f(x,y)] / h ; fy(x,y) = lim k0 [f(x,y+k) – f(x,y)] / k.
f/x = fx , f/y = fy , 2f/x2 = fxx , 2f/y2 = fyy , 2f/xy = fyx .
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f (x,y) =  f/x, f/y  ; f (x,y,z) =  f/x, f/y, f/z  .
f (x,y) is normal to level curve f(x,y) = k; f (x,y,z) is normal to level surface f(x,y,z) = k.
Duf (x,y,z) = f (x,y,z) . (u / ||u||) = f (x,y,z) . û
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f(x,y) is differentiable at the point x0,y0 if the limit as ∆x, ∆y tends to 0, 0 of
[ {f (x0+∆x, y0 +∆y) - f(x0, y0)} - {f (x0, y0) . ∆x, ∆y} ] / ||∆x, ∆y|| = 0 .
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Differentials: dw = ( w/x).dx + (w/y).dy + (w/z).dz .
∆w ≈ ( w/x). ∆x + (w/y).∆y + (w/z).∆z .
f (x0+∆x, y0 +∆y) ≈ f(x0, y0) + fx(x0, y0).∆x + fy(x0, y0).∆y
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Chain rules:
(dw/dt) = ( w/x). (dx/dt) + (w/y). (dy/dt) + (w/z) (dz/dt).
(w/u) = ( w/x). (x/u) + (w/y). (y/u) + (w/z)
(z/u).
(w/v) = ( w/x). (x/v) + (w/y). (y/v) + (w/z)
(z/v).
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Normal to the surface z = f(x,y) at x0, y0, f(x0,y0)  is -fx(x0, y0) , -fy(x0, y0) , 1.
Normal to the surface r(u,v) = x(u,v), y(u,v), z(u,v)  is (r/u)×(r/v).
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At a gradient–zero point x0, y0 we have f (x0, y0) = 0, i.e., f/x = 0 & f/y = 0.
If x0, y0 is a gradient-zero point and D = {fxx(x0, y0). fxx(x0, y0)} - {fxy(x0, y0)}2 ,
(a) D > 0 and fxx(x0, y0) > 0 will tell us we have a local minimum at x0, y0,
(b) D > 0 and fxx(x0, y0) < 0 will tell us we have a local maximum at x0, y0,
(c) D < 0 will tell us we have a local saddle point at x0, y0, and
(d) D = 0 will tell us nothing because the point x0, y0 could be any of the above.
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x0, y0 is a critical point of f(x,y) if f (x0,y0) = 0 or if f (x,y) does not exist at x0,y0.
The global maximum and global minimum of f(x,y) will be achieved at a critical point or at a
point on the boundary of the domain of f(x,y).
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The constrained extrema of f(x,y) subject to the constraint g(x,y) = 0 will occur at a point
where f (x0, y0) = λ .g(x0, y0) provided g(x0, y0) is not 0. We must also check the
endpoints of g(x,y) = 0; and at the points where g=0 or f =0, if these lie on g(x,y) = 0.
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∫∫R f(x,y) dA = ∫[a,b] {∫[g(x), h(x)] f(x,y).dy} dx = ∫[c,d] {∫[p(y), q(y)] f(x,y).dx} dy
∫∫R f(x,y) dA = ∫[α,β] {∫[g(θ), h(θ)] f(r,θ) . r.dr} dθ
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Volume of the solid determined by z1(x,y) & z2(x,y) over the plane region R is given by
Volume(G) = ∫∫R {z2(x,y) - z1(x,y)}.dA .
Area (R) = ∫∫R 1.dA .
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Surface Area of the surface z = f(x,y) is given by:
S = ∫∫R √{(fx)2 + (fy)2 + 1}. dA
Surface Area of the surface r =  x(u,v), y(u,v), z(u,v)  is: ∫∫R ||(r/u)×(r/v)|| . dA
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Mass of the solid region G with density f(x,y,z) at x, y, z is given by.
Mass(G) = ∫∫∫G f(x,y,z).dV .
Volume (G) = ∫∫∫G 1.dV .