Merced College; Don Power

MULTIVARIATE CALCULUS - CH 14, LECTURE

PARTIAL DERIVATIVES

14.1 Functions of Two or More Variables

Find function values X1-5, 13, 17-18

Compositions X5-8, 14-16

Chart reading X9-12

Domain19-24

3D graphs 25-34

Contour maps35-37

Level curves, "isobars," "isotherms."

Level surfaces

"Ant on a hotplate" problem like 57: T=y/x^3, ant starts at (2,–3), wants to keep same temp: what is the temp and what is the path?

14.2 Limits and Continuity

14.3 Partial Derivatives

For y=f(x), def of derivative (both forms) and graphical representation

Partial derivative at a point (x₀,y₀):

Similar expression for the partial with respect to y

Partial derivative in general:

Similar expression for the partial with respect to y

Problem with surface: consider slopes along lines parallel to x-axis and parallel to y-axis

Calculations: hold y constant or x constant

Ex: z=f(x,y)=x²y - x³

Notation

Numerical example: X9b

Second derivatives, mixed partials

Calculations and both notational styles

Note mixed partials are equal

Geometric interpretation of second derivatives

Concavity (unmixed partials)

Rate at which slope changes (twisting of a surface) (mixed partials)

Notice what is happening along the edges of these graphs.

plot3d(x^3*y - y^2,x=-5..5,y=-5..5); plot3d(x^2-3*x*y-y^2,x=-2..2,y=-2..2);

Implicit partial differentiation

For z_x:

z is a function of x: chain rule results in multiplication by z_x

x is a variable

y is a constant

Example #56: Find z_x for ln(2x²+y-z³)=x (in this case it is possible to solve for z and compare)

X98 is like example10; start by expanding the summation

14.4 Differentiability, Local Linearity, and Differentials

Def of differentiability for a function of one variable

Geometric significance: contrast dy vs delta y if delta x (same as dx) is small and non-zero:

Difference between the actual change in a function vs linear approximation of the change

Extension of def to two or three vars:

Denom is distance

Num is diff between actual function vs local linear approx, where...

Local linear approx is f(x₀,y₀,z₀) + f_x(x-x₀)+f_y(y-y₀)+f_z(z-z₀) i.e. z + dz

The dz portion of that last expression can be expressed as the "total differential"

f_xdx + f_ydy +f_zdz

Use def and also use thm 5 to show that a func is differentiable

Ex: Show that f(x,,y) = xz²sin(y) is differentiable by showing that the partials are continuous

Ex: Show that a plane (z=3x-2y+4) is differentiable at (x₀,y₀) using the definition

Examples:

Given a func of two vars, estimate f(2.03,3.98) using differentials

First, find the value of the function at (2,3)

Then, estimate the change in the function

Find the percentage change in the area of an ellipse if the major axis is increased by 3.0% and the minor axis is decreased by 1.0% : a₂ = .030 a₁ and b₂ = .010 b₁

What is the max error in the calculated volume of a pyramid (built on a square base) if the sides are measured at 100 m ± 1 cm and the height is estimated at 50 m ± 10cm

Exercises 25, 40

14.5 Chain Rule

Conical sandpile is being formed, as material is added, dr/dt = 6 in/min, dh/dt = 4 in/min

Find dV/dt when r = 12 in, h = 36 in V=1/3 pr²h

V_t = V_r r_t + V_h h_t

14.6 Directional Derivatives and Gradients

Geometrically, Du = slope of the surface z=f(x,y) in the direction of u at the point (x0,y0,z0)

u must be a unit vector

See def in text and work through example: Note: it's really a limit process

More practical: Du = <fx,fy,fz> dot <u1,u2,u3>

= "del" f dot u

Note u may be written as <cosq,sinq> where q is the angle of inclination of u with the x-axis

Thm 5,6: Important properties

If del f = 0 at some point, then all direc derivs at that point are 0

Otherwise

del f is a vector in the direction of the greatest rate of increase in the function

a vector in the opposite direction to del f is in the direction of the greatest decrease

the max rate of increase is the norm of del f

del f is normal to the level curve of f(x,y) at the point (x0,y0)

del f is normal to the level surface of f(x,y,z) at (x0,y0,z0)

Apps: Find the path of a heat-seeking particle (Example 6)

14.7 Tangent Planes and Normal Vectors

14.8 Maxima and Minima of Functions of Two Variables

Defs:

Interior point: Point P for which there is some disk or ball centered at P that contains only points inside the set. Interior of a set is the set of all interior points.

Boundary point: Point P for which every disk or ball centered at P contains both points inside and points outside the set. Boundary of a set is the set of all boundary points.

Open set: contains none of its boundary points

Closed set: contains all its boundary points

Bounded set: Entire set can be contained in some finite rectangle (2-space) or box (3-space)

Extreme value thm: if f(x,y) is continuous on a closed and bounded set R, then f has both an absolute max and an absolute min on R

Thm: If f has a relative extremum at a point P, and both f_x and f_y are defined at P, then both are 0.

Def of critical point: point for which both f_x and f_y are 0, or either partial derivative does not exist.

Facts:

Relative extrema must occur at critical points (but not all critical points are locations of extrema)

Absolute extrema must occur either

1. In the interior, in which case they are also relative extrema, or

2. On the boundary (So we must restrict the set to the boundary an then find the extrema)

Second Partials Test

Let D = f_xxf_yy-f_xy². For a point P,

If D is positive, f has a relative extremum

If f_xx [or f_yy] is positive, it is a minimum

If f_xx [or f_yy] is negative, it is a maximum

If D is negative, f has a saddle point

Ex: Hyperbolic paraboloid z = x²-y²

[But also consider the cylinder z=y⁴ at the origin]

If D = 0, no conclusion can be drawn from this test.

Ex: Example 6

Discussion before Exercise 45: Deriving equations for a regression line;

Show how to use Cramer's rule for #45c

14.9 Lagrange Multipliers

Typical problem: Maximize a function S = xy+2xz+2yz subject to the constraint xyz-32=0

in general, maximize the function f(x,y,z) subject to the constraint g(x,y,z) = some constant.

Typical application: Find the point on a given surface where the temperature is maximum,

given a function that describes the temperatures throughout the space containing the surface.

Typical application: Find the maximum and min values on the boundary of a region.

Fundamental principle: gradients of functions f and g are scalar multiples of each other

Setup: del f = λ del g: The scalar λ is the "Lagrange multiplier"

This yields simultaneous equations

An additional equation is needed to solve the system: use the constraint equation.

Ex 4: Find the dimensions of a rectangular box, open at the top, having vol=32 ft³,

requiring the least amount of material for its construction. (Eqns: see "typical problem" above)

Ex: minimize dist from x²-yz=5 to the origin; Lagrange multipliers give us two sets of points: (±sqrt5,0,0) and (0,±sqrt5,-+sqrt5). The first set gives the minimum distance; the second set does not give the max distance (which doesn't exist): it does give a minimax point: a sphere centered at the origin would have the same tangent plane as the surface at the second set of points.

Maple views:

> with(plots):

> implicitplot3d(x^2-y*z=5,x=-5..5,y=-5..5,z=-5..5);

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