partial Deratives {cal3,exm2}

Partial deratives

• $f(x,y)=4-2x-2y$
• P= (1,2)
• Create some r(t) by $r(t)=t<1,0>+<1,2>$
• So we take $\frac{d}{dt}f(r(t)),t_0\cdot \frac{1}{|r^{\prime }(t_0)|}$

We can do the shortcut!

• Treat y like a const, and derive for the rest of the things. Then treat x as a const

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Deratives! (WOO) {Formally}

Derative of func $f(x,y)$ along some cruve r(t) @ pt P is:

$$\frac{d}{dt}(f(r(t)))\cdot \frac{1}{|r^{\prime }(t)|},r(t)=t_0$$

Eg. Let P = (1,1), then let $r(t)=<t,t>$, hence $t_0=1$

• Derative of f along r @ P is:
• Applying the formula we get:

$$\frac{d}{dt}(4-2t-2t)-\frac{1}{\sqrt{2}}=-\frac{4}{\sqrt{2}}$$

In 1d calc, derative means slope of line, then in 2d, says as approach some pt along cruve, slope of that curve is such value (the derative)

Question 2!

• Let $f(x,y)=4-2x-2y$
  • $r(t)=<-t,-t>$
  • $P=(1,1)$
• a) What derative along r @ P?
• b) Derative of f along $r_2$ @ (-3,4), $r_2=<t^2-3,t^2+4>$

Esentially this question asks us what happens when example above is in oop dir?

• We find $r^{\prime }(t)=<-1,-1>$ (given that mag of r(t) = $sqrt2$, and we combined the derative of f(r(t)) @ $t_0=4$
  • This is from
• Answer $\frac{4}{\sqrt{2}}$

for 2)

• $r^{\prime }(t)=<2t,2t>$, $r(t_0=(0,0)$, but we can’t do 1/0 for the EQ!

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@ params

• @param surface?
  • $r(s,t)=<x(s,t),y(s,t),...>$
• @param plane?
  • Cartesian: $r_1(s,t)=<s,t>$
  • Polar: $r_1(s,t)=<scos(t,ssin(t)>$
• Have your 3 planes: cartesian, xyplane, polar cords (should know all 3 of these, where s=0 or t=0)
• Learn how to integrate in a different cord system? (its js integrate in cartesian cords * some area strectch factor to convert)
• What it means 2 take derative along a curve (key idea)

Derative representations

• $f^{\prime }(x_0)$ rep slope going right
• r’(t) = tan vect as t inc
• f(x,y) (partial derative {pd}): slope as x inc, y fixed (or OPA)
• r(s,t) (pd): Tan vect as s inc + t fixed
  • f(x,y) 2D ? approximated by tan plane ? if tan plane given by differentiable plane as TTP: pd(r_s), pd(r_t) , $f(x_0,y_0)
  • Then normal vect

Question 1 (hint use sys of EQ 2 solve)

Let P = (2,2), f(x,y) = $2x^{2}y+y$

• Find some r(t) that passes thru P?
  • Just need any pt whr $t_0=P$
• Compute derative along r(t).
  • Setup SoE, solve then obtain P

Domains of multi var funcs

• $f(x,y)=e^{x^2+1}-e^{-y}$
  • What is the domain?: All reals
  • let $r(t)=<t^2+1,t^2>$, what is shape?
    • Let p = (2,1), then what time cords intersect @ this?, $t_0={\pm }_1$
  • Derative @ $t=t_0$? Then $\frac{d}{dt}f(r(t))$ (Follow same formula frm partial Deratives {cal3,exm2})
    • = $\frac{d}{dt}(e^{(t^2+1{)}^2+1}-e^{-(t^2)}{)}^{t_0}\cdot \frac{1}{r^{\prime }(t)}$ .