Question

(1 point) Use the contour diagram for f(x, y) shown below to estimate the directional derivative off in the direction v at the point P. (a) At the point P = (2, 2) in the direction ✓ = 7, the directional derivative is approximately O‘ot 16.0 18.0 12.0 14.0 2.0 (b) At the point P = (3, 2) in the direction ✓ = -1, the directional derivative is approximately 8.0 (c) At the point P = (4,1) in the direction ✓ = (1 +1)/V2, the directional derivative is approximately 2.0 -0.8 4 (d) At the point P = (0,0) in the direction ✓ = 7, the directional derivative is approximately -0. 87 0.8 1.6 2.4 3. 2 (Click on graph to enlarge)

Answer 1

Sol Considen the poirt P- (2,2) 2 1t 23 t is equised to -fird the disectional dai vative of f in the dinection of PQ =T l Pel De-7+0P De -Tt2+23 = 31+2J So o (3,2) The directionl derivative of fin the disection of Vis f(P)f(e)-f(P) fa (2,2)P(3,2) -f (2,2) Foom given aaph fl2,2) 4, f(3,2)-6 Herce 6-4 2,2) 2 Consider the point P- (3,2) 37+ 23 Tt is bequioed to find 7 the dinectional derivative of -f in the dicection of Po --7 It Pol OP --7

T+OP O 7434 2 2 +23 So, e C2,2 The dinectiond davative of f inthe disection of v is P)(e)-f(P) If Por P (3,2) C2,2)-f(3,2) Foom aiven goaph f(2,2) 4f(3,2) 6 Hence f(3,2)- -6 -2 c) Consida the point P- (4,D 443 17 It is quined to find -the dinectional darivotive of p disection of in the Per V= -7 Oe-OP 5it3 So, e(5,2)

The dinectionl doivative. of f in the disection of V is f(o) -f(P) (4,) f(52) f(4,1) Foom given graph. fl4,1)4,PIs,2) = 10. Hence 0411-10-4 6 32

Consider the point P- CO,0) = oi+o3 find the dibectional doivative is sequied -to in the OP dthsection of V PO Poll So, en(o,0) 1 The dinectional derivative of f in the diection f Vis f(P)f(e) -f(P) (0,0)(0,0) -(0,0) P(a,o) O