MAT 241- Calculus 3- Prof.
Santilli
Toughloves Chapter 14
1.) Level curves (contour curves) are the cross sections of the surface z = z(x, y) projected onto
the x-y plane, i.e., z = z(x, y) = constant .
2.) Level surfaces (contour surfaces) are the cross surfaces of the 4-D function, h = h(x, y, z)
projected onto the x-yz space, i.e., h = h(x, y, z) = constant .
3.) A function is continuous at (xo , y o ) if f (xo , yo ) = lim f (x, y) for all paths.
(x,y)(x o,yo )
f f (x, y) z
4.) Definition of Partial Derivatives: fx = fx (x, y) = = = = f1 = D1 f = Dx f
x x x
f (x + x, y) f (x, y) f (x, y + y) f (x, y)
fx (x, y) = lim and fy (x, y) = lim
x0 x y0 y
f (x, y)
5.) is the slope of the tangent line to the surface f (x, y) in the x-direction.
x
f (x, y)
6.) is the slope of the tangent line to the surface f (x, y) in the y-direction.
y
7.) Chain Rule:
s
s x t x t
y x t y x y y
t u t u
s
t
dy dy dx y dy x dy y dx y du y y x y u
= = = + = +
dt dx dt t dx t dt x dt u dt t x t u t
s t
x r t
y u s t
r t
s t
w t
r
dy y x dr y x ds y u dr y u ds y w dr y w ds
= + + + + +
dt x r dt x s dt u r dt u s dt w r dt w s dt
8.) Higher Order Partials:
2 f 2 f (x, y) 2 z
fxx = fxx (x, y) = 2 = = 2 = f11 = D11 f = Dxx f
x x 2 x
f f (x, y) 2 z
2 2
fyy = f yy (x, y) = 2 = = 2 = f22 = D22 f = D22 f
y y 2 y
2 f 2 f (x, y) 2 z
fxy = f xy (x, y) = = = = f12 = D12 f = Dxy f
yx yx yx
9.) If the function has continuous second partials, fxy = f yx
10.) Implicit Function Theorem:
dy f x
= for function of 2 variables
dx fy
z fx
= for function of 3 variables
x fz
11.) Equation of a tangent plane is:
z z o = f x (x o , yo )(x x o ) + f y (x o , yo )( y yo )
f f
12.) Total Differential: dz = dx + dy
x y
f f
13.) Gradient Vector: f = i+ j = fx , fy in plane
x y
f f j + f k = fx , f y , fz in space
f = i +
x y z
f f
14.) Directional Derivative: Du f = cos + sin = f u which is the slope of the
x y
tangent line to the function in the direction of the unit vector u = cos ,sin .
15.) Du fmax = f is the max value of the directional derivative and it occurs in the direction
of the gradient vector- STEEPEST ASCENT.
16.) Du fmin = f is the min. value of the directional derivative and it occurs in the
opposite direction of the gradient vector- STEEPEST DECENT.
17.) f of a function is always normal to the level curves or level surfaces of the function at
any point on the level curve or level .
18.) Given the function z = f (x, y), then F(x, y, z) = f (x, y) z = 0 is a level surface of
F(x, y, z).
19.) Equation of tangent plane to a surface z = f (x, y) at P(x o , yo , zo ):
F F F
(xo , y o , z o )(x x o ) + (x o , yo , z o )( y y o ) + (x , y , z )(z zo ) = 0 where
x y z o o o
F(x, y, z) = f (x, y) z
20.) Equation of the normal line to the surface z = f (x, y) at P(x o , yo , zo ):
x xo y yo z zo
= = where F(x, y, z) = f (x, y) z
Fx (x o , yo , zo ) Fy (x o , yo , z o ) Fz (xo , y o , z o )
F F F
21.) Normal vector the a surface:n = n x i + ny j + nz k = i+ j+ k where
x y z
F(x, y, z) = f (x, y) z
22.) Angle of inclination of a tangent plane to a surface:
n F
cos = z = z where F(x, y, z) = f (x, y) z
n F
23.) Relative extrema in f (x, y) occur at the critical points.
24.) Absolute extrema in f (x, y) occur at the critical points or at the boundary points of the
domain.
25.) Local extrema- critical points at point (a,b): f = 0 or DNE.
26.) 3 possible types of critical points: Local max, local min, and saddle.
fxx (a, b) f xy (a, b)
27.) Second Partial Test- To determine the type of CP at (a,b): Find d = ,
fyx (a, b) f yy (a, b)
If d > 0 and fxx (a, b) > 0 then (a,b) is a local minimum
If d > 0 and fxx (a, b) < 0 then (a,b) is a local maximum
If d < 0 then (a, b, f(a,b)) is a saddle point
If d = 0 then test inconclusive.
28.) Constrained optimization with Lagrange Multiplier: optimization occurs where the
objective level curves = constraint level curves so A(x, y) = g(x, y), where A(x, y) is the
objective level curve, g(x, y) is the constraint level curve and is the Larange Multiplier.
29.) Constrained optimization with 2 constraints: A(x, y) = g(x, y) + h(x, y), where
A(x, y) is the objective level curve, g(x, y) and h(x, y) is the constraint level curve and
and are the Lagrange Multipliers.
