1. Given that sin 52=s, cos 52 =t, tan 52=u, find the six function values of 38 in terms of s, t, and u. sin 38-0 cos 38= tan 38=0 cot 38 = sec 38= CSC 38 0

2. Find the derivative of xy^2 + yz at (1, 1, 2) in the direction of the vector 2i - j + 2k

Sol81:

To find the derivative of $xy^2+yz$ at $(1,1,2)$ in the direction of the vector $2i-j+2k$, we can use the directional derivative formula:

?Du?f(a,b,c)=?f(a,b,c)?u?u?

where $ abla f(a,b,c)$ is the gradient of $f$ at $(a,b,c)$ and $\\\\vec{u}$ is the direction vector.

First, we need to find the gradient of $f$:

???f=?y2,2xy+z,y?

Evaluated at $(1,1,2)$, we get:

?(1,1,2)=?1,4,1??f(1,1,2)=?1,4,1?

Next, we normalize the direction vector $\\\\vec{u}$:

?22+(-1)2+22=3?u?=22+(-1)2+22?=3

?=?2,-1,2?3=?23,-13,23??u?u?=3?2,-1,2??=?32?,-31?,32??

Finally, we can find the directional derivative:

(1,1,2)=?(1,1,2)??=?1,4,1??23,-13,23?=83-43+23=63=2Du?f(1,1,2)=?f(1,1,2)?u?u?=?1,4,1??32?,-31?,32??=38?-34?+32?=36?=2?