Compute the directional derivative of the following function at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 2 3 f(x,y) = 19 -x" - 3y ; P(3, - 3); 1 13 13 Find the gradient of f(x,y) = 19 - x - 3y . - X 3 Vf(x, y) = 19- x" - 3y 219-x - 3y The directional derivative is (Type an exact answer, using radicals as needed.)1 'l 1 . . = + . _ . = _ _ _ Consider the function flincvz) 2 they; the point P{ 1:11]: and the unIt vector u (E : 1E : J?)- a. Compute the gradient off and evaluate it at P. b. Find the unit vector in the direction of maximum increase off at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction ofthe given vector. at. What is the gradient at the point I:- 1:1:1}? {4.4:4> b. Iv'u'hat is the unit vector in the direction of maximum increase? EDD [Type exact answers. using radicals as needed.) Find an equation ofthe plane tangent to the following surface at the given points. 2: |n|:1+;-:5r]; [5.6.In 31) and {5: 6. |n31j E 5 The tangent plane at [5.5. In 31] is z = In 31+ El): 51+ El\"; E}. The tangent plane at ( 5. Er: In 31} is: = D. The cost of a trip that is L miles long, driving a car that gets m miles per gallon, with gas costs of Sp/gal is C = - m dollars. Suppose you plan a trip of L = 1400 mi in a car that gets m = 34 mi/gal, with gas costs of p = $3.80/gal. Complete parts (a) through (d). a. Explain how the cost function is derived. What is the general cost for the car traveling 1 mile? Type the answer using the symbols L, p, and m as needed. The cost for 1 mile is 0.11176Find an equation of the plane tangent to the following surface at the given points. z = 4 cos (x - y) + 4; 6 -7,6 and 3'3 8 The tangent plane at the point 6 6 6 isz = 8-213 X - + 213 y + (Type an exact answer, using radicals as needed.) tangent