-a) Find the equation of the line in its vector, parametric and symmetrical shapes that passes through the point.

P (5, -2, 4) and it's perpendicular to the plane 4x-3y+5z=0.

b) Find the equation of the plane (Cartesian and vector) that passes through the points. A (1, -1, -1), B (3, 5, 7), C (1, 6, 8).

Use power series to solve the initial value problem

x^2y''+xy'+x^2y=0, y(0)=1, y'(0)=0

f(x,y)=sin(2x)sin(y)

intervals for x and y:

-π/2 ≤ x ≤ π/2 and -π ≤ y ≤ π

find extrema and saddle points

In the solution, I mainly interested how to findcritical points in case of the system of trigonometric equations (fx=0 and fy=0).
,

h(y)= 9/y^2-9

Algebraically determine where ℎ is increasing/decreasing and where ℎ is concave up/down, writing these in interval notation. Also, find all local extrema and inflection points of ℎ, writing these as ordered pairs.

increasing: _____________ local maxima: _______________________ decreasing: _________________________ local minima: ________________________ concave up: _____ inflection points: ___________________ concave down: ______________________

b)Find the equations for all asymptotes for ℎ, and justify each one.

vertical: _________________________ horizontal: ________________

Multivariable calculus

Evaluate: ∮ 3? 2 ?? + 2???? using two different methods. C is the boundary of the graphs C y = x2 from (3, 9) to (0, 0) followed by the line segment from (0, 0) to (3, 9).

2. Evaluate: ∮(8? − ? 2 ) ?? + [2? − 3? 2 + ?]?? using one method. C is the boundary of the graph of a circle of radius 4 oriented counterclockwise

Use elementary row or column operations to evaluate the following determinant. You may use a calculator to do the multiplications. 1 -9 6 -9 -5 2 -17 6 -21 -11 2 -13 4 -17 -12 0 6 4 2 0 0 -2 16 8 2

A manufacturing company produces two models of an HDTV per week, x units of model A and y units of model B at a cost (in dollars) of

f (x, y) =(8x^2+8y^2)/4 .

If it is necessary (because of shipping considerations) that4x + 5y = 20 ,

how many of each type of set should be manufactured peer week to minimize cost? What is the minimum cost?

Consider a population of antelope, with P0=2,750, carrying capacity K=3,875, and growth rate r=0.073 (or 7.3%) per year. Assuming that this population follows logistic growth, find P(t), then use the solution to predict the population after 8 years. Round your final answer to the nearest integer value and do not include units.

for each of the five functions f1-(x,1/y) ,f2-(0,y) ,f3 (x+2y,y) ,f4-(x-2,y+1) ,f5 (x,y^3-y) prove or disprove that fn is distance-preserving

Find the distances:

A) Between ?1=〈2+2?,−1+?,−3?〉and ?2=〈4,−5−3?,1+4?〉 .

B) Between the planes 2?−?+5?=0 and 2?−?+5?=5 .

C) From the point (1,2,3) to the line ?=〈−?,4−?,1+4?〉 .

You are designing the top of a pencil pouch that consists of a rectangle to which a semicircle has been attached to both ends. The semicircle will be made from material that costs 1ct per square centimeter. The material of the rectangular part costs 4ct per square centimeter. The perimeter of the pouch must be 80cm. How do you have to choose the radius and the 'open' side of the rectangle so the cost is minimal? Before you start, label your sketch! Please write down strategy and final result.

find the volume of the largest rectangular what's in the first octant with three faces on the co-ordinate planes and one vertex on the plane
8x+y+3z=9

(1 point) Is the point (−4,−5,3) visible from the point (4,5,0) if there is an opaque ball of radius 1 centred at the origin? Suppose that you stand at the point (4,5,0) and look in the direction of a point that is not visible because it is behind the ball. You will then be looking at a point on the sphere. If (−4,−5,3) is not visible from (4,5,0), find the point on the sphere at which you are looking if you look in the direction of (−4,−5,3). Otherwise, find the point on the sphere at which you look if you are looking in the direction of (−4,−5,2). Point (?,?,?)=