Write down an expression for the area of each region described by the following boundaries. Use technology to calculate the area. a y = ln x, the x-axis, x = 1, and x = 4 b y = x sin x, the x-axis, x = 1, and x = 2 c y = x2ex, the x-axis, x = 0, and x = 2:8 .
Spencer Kociba MATH 200-005 Lecture Notes Week 6 10/24/2016 Directional Derivatives ● Definition for Directional Derivatives ○ If f(x,y) is a function and u =< u ,u > is a unit vector, then the directonal 1 2 derivative of f in the direction of u at (x ,y ) is denoted D f(x ,y ) and is given o o u o o by D f(x ,y ) = d [f(x + u s, y + u s)]|s = 0 u o o ds o 1 o 2 ○ D f(x ,y ) = lim f(o +1 ho y2+u ho−o(x ,y ) u o o h→0 h ● Notes ○ fx(o ,yo) = Duf(xo,yo) u=<1,0> ○ fy(o ,yo) = Duf(xo,yo) u=<0,1> ○ If u is a unit vector < u 1u ,2 >3, then D u(xo,y oz o = d[f(xo+ u 1, y o u 2,z o u s3|s = 0 ds ● Theorem ○ If f(x,y) is differentiable at (xo,yo) and u =< u ,1 > 2s a unit vector then D u(xo,y o exists and D f(u ,yo) o f (xx,yo)uo+ 1 (x yyo)u o 2 Spencer Kociba MATH 200-005 Lecture Notes Week 6 10/25/2016 Directional Derivatives ● D u(x oy o = f xxo,y ou 1 f yx oy ou 2 ● Definition of gradient ○ If f is a function of x and y, the gradient of f is defined by ∇f(x,y) = < f xx,y),f yx,y) > ○ Similarly, ∇f(x,y,z) = < f (xxy,z),f (xyy,z),f (xzy,z) > ● Note:
