Chain Rule Case 1: Where z Depends Only on t With Graphs
Page 1:
Discusses the concept where z depends on t through x and y. For example, x = 2t, y = t², z = x³ + 2y.
To find the rate at which z changes with respect to t, a dependence tree is created. The formula for dz/dt is derived as 3x² + 4t.
For t=1, x=2, y=1, the rate of change on the surface x³ + 2y at t=1 along the curve x = 2t, y = t² is 28.
Page 2:
Discusses another example where z = x² + y² along x = cos(t), y = sin(t).
The derivative dz/dt is derived as -2sin(t)cos(t).
For t = π/4, dz/dt = -2.