Math 375 - 11/30

Partial Derivative • Infinitesimal Interpretation of Derivative ○ • Definition ○ ∂f/(∂x_k ) (x_1,…,x_n )=lim_(h0)⁡〖(⏞(f(x_1,…,x_k+h…,x_n ) )┴(only x_k changes)−f(x_1,…,x_n ))/h ○ = The derivative of f(x_1,…x_n ) with respect to x_k, with all other variables fixed • Other Notations ○ ∂f/(∂x_k ) (x_1,…,x_n )=f_(x_k )=f^′ (x;e_k ) • Example ○ f(x,y,z)=x^2+xy3 ○ ∂f/∂x=∂/∂x (x^2+xy3 )=2x+y^3 ○ ∂f/∂y=∂/∂y (x^2+xy3 )=3xy^2 ○ ∂f/∂z=∂/∂z (x^2+xy3 )=0 (Because x^2+xy3 does not depend on z) • Second Derivative ○ f_xx=(∂^2 f)/(∂x^2 )=∂/∂x (∂f/∂x)=∂/∂x (2x+y^3 )=2 ○ f_xy=(∂^2 f)/∂y∂x=∂/∂y (∂f/∂x)=∂/∂y (2x+y^3 )=3y^2 ○ f_yx=(∂^2 f)/∂x∂y=∂/∂x (∂f/∂y)=∂/∂x (3xy^2 )=3y^2 ○ f_yy=(∂^2 f)/(∂y^2 )=∂/∂y (∂f/∂y)=∂/∂y (3xy^2 )=6xy • Clairaut’s Theorem ○ If ∂f/∂x,∂f/∂y,(∂^2 f)/∂x∂y exists and (∂^2 f)/∂x∂y is continuous at (a,b)∈R2 ○ Then (∂^2 f)/∂y∂x also exists and (∂^2 f)/∂x∂y=(∂^2 f)/∂y∂x • Example of f_xy≠f_yx ○ f(x,y)={■8(1&x 0@0&x≤0)┤ ○ ∂f/∂x={■8(0&x≠0@Does Not Exist&x=0)┤ ○ see the graph below (horizontal axis: x, vertical axis: f(x,y)) ○ ∂f/∂y=0 for all (x,y) ○ (∂^2 f)/∂x∂y=∂/∂x (∂f/∂y)=∂/∂x (0)=0 ○ (∂^2 f)/∂y∂x=∂/∂y (∂f/∂x)={■8(0&x≠0@Does Not Exist&x=0)┤ ○ Therefore (∂^2 f)/∂x∂y≠(∂^2 f)/∂y∂x Total Derivative & Linear Approximation Formula • Illumination ○ f(x+Δx,y+Δy)−f(x,y) ○ =f(x+Δx,y+Δy)−f(x+Δx,y)+f(x+Δx,y)−f(x,y) ○ =[f(x+Δx,y)−f(x,y)]+[f(x+Δx,y+Δy)−f(x+Δx,y)] ○ =(f(x+Δx,y)−f(x,y))/Δx×Δx+(f(x+Δx,y+Δy)−f(x+Δx,y))/Δy×Δy ○ ≈∂f/∂x×Δx+∂f/∂y×Δy • Theorem ○ If f_x and f_y are continuous, then there exist functions ε_x and ε_y ○ f(x+Δx,y+Δy)=f(x,y)+∂f/∂x (x,y)Δx+∂f/∂y (x,y)Δy+ε_x Δx+ε_y Δy ○ Where ε_x,ε_y→0 as Δx,Δy→0 ○ Note § (f(x+Δx,y+Δy)−f(x+Δx,y))/Δy=∂f/∂y (x,y)+ε_y § (f(x+Δx,y)−f(x,y))/Δx=∂f/∂x (x,y)+ε_x • Linear Approximation ○ f(x_1+Δx_1,…,x_n+Δx_n ) ○ =f(x_1,…,x_n )+f_(x_1 ) (x_1,…,x_n )Δx_1+…+f_(x_n ) (x_1,…,x_n )Δx_n+ε_1 Δx_1+…+ε_n Δx_n ○ Where ε_k→0 as Δx_1,…,Δx_n→0 • Linear Approximation (Vector Notation) ○ x=(x_1,…,x_n )∈Rn ○ Δx=(Δx_1,…,Δx_n )∈Rn ○ ε=(ε_1,…,ε_n )∈Rn ○ f(x+Δx)=f(x)+∇ ⃗f(x)⋅Δx+ε⋅Δx ○ Where § ∇ ⃗f(x)=(∂f/(∂x_1 ) (x),…,∂f/(∂x_n ) (x)) is called the gradient of f § ∇ ⃗f(x)⋅Δx=f_(x_1 ) (x_1,…,x_n )Δx_1+…+f_(x_n ) (x_1,…,x_n )Δx_n § ε⋅Δx=ε_1 Δx_1+…+ε_n Δx_n • Example ○ f(x,y)=x^2+xy3 ○ Find the linear approximation at (x,y)=(1,2) ○ Calculate f(1,2),f_x (1,2),f_y (1,2) § f(1,2)=1^2+1⋅23=9 § f_x (1,2)=[2x+y^3 ]_█(x=1@y=2)=2+2^3=10 § f_y (1,2)=[3xy^2 ]_█(x=1@y=2)=3⋅1⋅2^2=12 § ∇ ⃗f(1,2)=[█(10@12)] ○ f(1+Δx,2+Δy) § =f(1,2)+f_x (1,2)Δx+f_y (1,2)Δy+ε_x Δx+ε_y Δy § =⏟(9+10Δx+12Δy)┬approximation+⏟(ε_x Δx+ε_y Δy)┬error ○ f(1.01,1.99)=f(1+0.01,2−0.01)≈9+10⋅0.01−12⋅0.01=8.89 ○ Tangent plane at (1,2)