Lecture 10 Key Points

Matching Functions to Graphs

• Radial dependence ⇒ look for circular level‐sets
  • f(x,y)=\frac{1}{1+x^2+y^2}: decays from max at origin; bowl‐shaped surface.
  • f(x,y)=\ln(x^2+y^2): \to -\infty at origin, increases outward; deep pit.
  • f(x,y)=\cos\sqrt{x^2+y^2}: concentric ripples (radial cosine).

• Product dependence ⇒ saddle patterns
  • f(x,y)=|xy|: "tent" rising in all four quadrants.
  • f(x,y)=\cos(xy): alternating ridges/valleys along hyperbolas.

Limits & Continuity (3-D)

• Limit exists at (a,b) if \lim_{(x,y)\to(a,b)}f(x,y)=L along every path.

• Continuity at (a,b) when limit equals f(a,b).

• Standard functions (polynomials, trig, exp, log) ⇒ continuous on domains.

Basic Limit Laws (Multivariable)

If \lim f=L and \lim g=G then

• \lim(kf)=kL,

• \lim(f\pm g)=L\pm G,

• \lim(fg)=LG,

• \lim\dfrac{f}{g}=\dfrac{L}{G} (when G\neq0).

Techniques for Evaluating Limits

• Try special paths: y=0, x=0, y=mx, x=y^2, etc.

• Different limits along two paths ⇒ limit DNE.

Squeeze Theorem (2 variables)

If f\le g\le h and \lim f=\lim h=L then \lim g=L.

Continuity of Piecewise Functions

• Verify limit from all directions equals the defined value at junction point.

• Failure of limit ⇒ discontinuity, even if value is assigned.

Partial Derivatives (First Principles)

• fx(a,b)=\displaystyle\lim{h\to0}\dfrac{f(a+h,b)-f(a,b)}{h} (treat y constant).

• fy(a,b)=\displaystyle\lim{h\to0}\dfrac{f(a,b+h)-f(a,b)}{h}.

• Compute by ordinary rules, holding other variables fixed.

Example: g(x,y)=xe^{x-y}

• g_x=(1+x)e^{x-y}

• g_y=-xe^{x-y}

Higher Partial Derivatives & Clairaut

• Second partials: f{xx}=\dfrac{\partial^2 f}{\partial x^2}, f{yy}, f{xy}, f{yx}.

• Clairaut’s theorem: if second partials are continuous, f{xy}=f{yx}.

Tangent Plane & Linearization

At point (a,b):

• Slopes: m1=fx(a,b), m2=fy(a,b).

• Tangent plane: z=m1x+m2y+d with d=f(a,b)-m1a-m2b.

• Linearization (approximation near (a,b)):
  L(x,y)=f(a,b)+fx(a,b)(x-a)+fy(a,b)(y-b).

Differentials & Error Approximation

• Total differential: dz=fx\,dx+fy\,dy.

• Use |dz| to estimate absolute error when dx,dy are measurement tolerances.
  Example (cylinder V=\pi r^2h): dV=2\pi rh\,dr+\pi r^2\,dh.

Multivariate Chain Rule

For z=f(x,y),\;x=g(t),\;y=h(t):
\dfrac{dz}{dt}=fx\,\dfrac{dx}{dt}+fy\,\dfrac{dy}{dt}.
Equivalent to substituting total differential with dependent variables.