Math 375 - 12/6

Question ○ Use gradient to approximate|■8(sin⁡(π/2+0.1)&ln⁡(1.1)@3&√4)| ○ Let f(x,y,z)=|■8(sin⁡x&ln⁡y@3&√z)|, then ○ f(π/2,1,4)=2 ○ Note ○ Near x=a, f(x)≈f(a)+∇f(a)(x−a) ○ Calculate Gradient ○ ∇f(x,y,z)=[█(f_x@f_y@f_z )]=[█(√z cos⁡x@−3∕y@sin⁡x∕(2√z) )] ○ ∇f(π/2,1,4)=[█(0@−3@1∕4)] ○ Approximation ○ f((π/2+0.1), (1+0.1), (4+0.1)) ○ ≈f(π/2,1,4)+∇f(π/2,1,4)(x−a) ○ =2+[█(0@−3@1∕4)]⋅[█(0.1@0.1@0.1)] ○ =2−0.3+0.025 ○ =1.725