For the functions w=xy+yz+xz, x=u+2v, y=u-2v, and z=uv, express ζw/ζu and ζw/ζv using the chain rule and by expressing w directly in terms of u and v before differentiating. Then evaluate ζw/ζu and ζw/ζv at the point (u, v)=(2/3,0)

i) See explanations below

ii) partial derivative of w with respect to u = 2u + 4uv

partial derivative of w with respect to v = -4v + 2u²

The partial derivative of w with respect to u = 4/3

The partial derivative of w with respect to v = 8/9

Step-by-step explanation:

w=xy+yz+xz

x=u+2v

y=u-2v

z=uv

First we are to express partial derivative of w with respect to u and v respectively using chain rule.

See 1st attachment.

Inserting the value of x, y and z with respect to u and v in w:

w = (u+2v)(u-2v) + (u-2v)(uv) + (u+2v)(uv)

w = u² -2uv +2uv - 2v² +u²v -2uv² + u²v + 2uv²

w = u² - 2v² + 2u²v

We would find the partial derivative of w with respect to u and v respectively.

Partial derivative is done by differentiating a function with one variable while keeping the other variable constant.

At the point (2/3, 0) given, we get the value for the partial derivative of w with respect to u and v respectively.

See 2nd attachment for more explanations.

∠acb = 8°

step-by-step explanation:

∠xba lies on the circle and is half the measure of the intercepted arc ax

hence

measure of arc ax = 2 × 42° = 84°

∠acb is outside the circle and is

∠acb = (arcab - arcax ), hence

∠acb = 0.5 (100 - 84) = 0.5 × 16° = 8°