A function f and a point P are given. Let θ correspond to the direction of the directional derivative. Complete parts (a) through (e).
f(x, y) = 16 - 4x² - 3y², P(-3,4)

(a) Find the gradient and evaluate it at P.
(b.1) What angle(s) is/are associated with the direction of maximum increase?
(b.2) What angle(s) is/are associated with the direction of maximum decrease?
(b.3) What angle(s) is/are associated with the direction of zero change?
(c) Write the directional derivative at P as a function of θ, call this function g(θ).
(d) Find the maximum value of g(θ). What value of θ maximizes g(θ)?
(e) Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part (d) consistent with the values from parts (a) and (b)?

1. a right triangle is graphed on a coordinate plane. find the length of the hypotenuse. round your answer to the nearest tenth. 2. use the angle relationship in the figure below to solve for the value of x. assume that lines a and b are parallel and line c is a transversal.

In δabc shown below, ∠bac is congruent to ∠bca: triangle abc, where angles a and c are congruent given: base ∠bac and ∠acb are congruent. prove: δabc is an isosceles triangle. when completed (fill in the blanks), the following paragraph proves that line segment ab is congruent to line segment bc making δabc an isosceles triangle. (4 points) construct a perpendicular bisector from point b to line segment ac . label the point of intersection between this perpendicular bisector and line segment ac as point d: m∠bda and m∠bdc is 90° by the definition of a perpendicular bisector. ∠bda is congruent to ∠bdc by the definition of congruent angles. line segment ad is congruent to line segment dc by by the definition of a perpendicular bisector. δbad is congruent to δbcd by the line segment ab is congruent to line segment bc because consequently, δabc is isosceles by definition of an isosceles triangle. 1. corresponding parts of congruent triangles are congruent (cpctc) 2. the definition of a perpendicular bisector 1. the definition of a perpendicular bisector 2. the definition of congruent angles 1. the definition of congruent angles 2. the definition of a perpendicular bisector 1. angle-side-angle (asa) postulate 2. corresponding parts of congruent triangles are congruent (cpctc)

The length of a rectangle plus its width is 24 cm. the area is 143 square cm. what are the length and width of the rectangle?