Solve the problem

43) Find equations for the horizontal and vertical tangent lines to the curve r = 1 - sinθ, 0 ≤ θ < 2π.

Please check if your answer is correct with the following:

Horizontal: y = 1/4 at (1/2, π/6), y = 1/4 at (1/2, 5π/6), y = -2 at (2, 3π/2)

Vertical: x = 0 at (0, π/2), x = -3sqrt(3)/4 at (3/2, 7π/6), x = 3sqrt(3)/4 at (3/2, 11π/6)

Given the equation:

sin(x+y)=x+cos(y)

a) Differentiate y with respect to x

b) Give the equation of the line tangent to the curve of at the point (0,π/4)

A closed rectangular box of volume 324 cubic inches is to be made with a square base. If the material for the bottom costs twice per square inch as much as the material for the sides and top, find the dimensions of the box that minimize the cost of materials.

Consider the following planes. 5x − 3y + z = 2, 3x + y − 5z = 4 (a) Find parametric equations for the line of intersection of the planes. (Use the parameter t.) (x(t), y(t), z(t)) = (b) Find the angle between the planes. (Round your answer to one decimal place.)

1. a.) determine vector and parametric equations for the line through the point A(2, 5) with direction vector = (1, −3).   

b.)Determine a vector equation for the line through the points (-1, 4) and (2, -1).

c.) Determine parametric equations for the line through (-2, 3) and parallel to the line with vector equation = (−2, 1) + t(6, 4).

d .) A line passes through the point (-4, 1) and is perpendicular to the line with parametric equations x = -2 -3t, y = 1 + 2t. Find vector and parametric equations for the line.

e.) Find vector and parametric equations for the line with equation 2x + y + 3 = 0.

f.) Determine a scalar equation for the line that passes through the point (-1, 5) and has direction vector = (1, −3).

g.) Determine parametric equations for the line with scalar equation 4x – y + 5 = 0.

h.) Determine a scalar equation for the line with vector equation = (−3, 1) + t(2, 5).

i.) A line passes through the point (1, -4) and is perpendicular to the line 3x + 2y – 6 = 0. Determine a scalar equation for the line.

j.) Use a vector solution to show that a scalar equation for the line through the points P₁(x₁, y₁) and P₂(x₂, y₂) is

let R be the region bounded by the line y=0, by the upper part of the circle x²+y²=4, and by the upper part of the circle x²+y²=9. Find the circulation of the force F= ( 5cosx-y³)i+ (x³+ 4x+ 5siny)j around the curve C , where C is the boundary curve of the region R , oriented counterclockwise. Draw the region R precisely, and show the orientation of the curve C by putting arrows on C

Determine where the given function is concave up and where it is concave down.

​f(x)= 2x^3-6x^2-90x

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that​ revenue, R(x), and​ cost, C(x), of producing x units are in dollars

R(x)=50x-0.1^2, C(x)=4x+10

Find the number of units that must be produced and sold in order to yield the maximum​ profit, given the equations below for revenue and cost.

R(x)=50x-0.5x^2

C(x)=6x+4

Find the absolute maximum and minimum values of the function over the indicated​ interval, and indicate the​ x-values at which they occur.

f(x)=x^2-6x-2 ; [1,7]

Consider the function ?(?,?)=3?^2−4?+??^2 on the closed region ?={(?,?):−1≤?≤1 and −1≤?≤1}R={(x,y):−1≤x≤1 and −1≤y≤1}.

(a) Find all critical points of ?(?,?)f(x,y) in the region ?R, if any, and classify them (local maximum, local minimum, or saddle point).

(b) Determine the absolute maximum and absolute minimum of ?(?,?)f(x,y) on the closed region ?R, and all points at which they occur.

3. The table below shows the number of hybrid cars sold (in thousands) in the US in the following years.                                                          * Source: Wikipedia

1. Find an exponential regression equation for the data. Let x = 0 represent the year 2000. Round a to two decimal places, and round b to four decimal places.

2. Describe what a and b tell you from your equation above in the context of the problem. Be very specific.

8. [C] The surface of the Earth can be modeled by a unit sphere. The Arctic Circle is the circle on the unit sphere at an angle of φ = 23.5 ◦ from the +z-axis. What percentage of the Earth’s surface lies above the Arctic Circle?

Problem Page

Question

A scientist has two solutions, which she has labeled Solution A and Solution B. Each contains salt. She knows that Solution A is

60%

salt and Solution B is

85%

salt. She wants to obtain

80

ounces of a mixture that is

75%

salt. How many ounces of each solution should she use?

1. In League of Legends, a player’s Effective Health when defending against physical damage is given by E = H(100 + A)/100 where H is health and A is armor. Health costs 2.5 gold per unit and Armor costs 18 gold per unit. Ten minutes into the game, you have 1080 health and 10 armor. You have only 720 gold to spend, and the health armor costs the same as before. Again, the goal is to maximize the effectiveness E. Notice that you don’t want to maximize the effectiveness of what you purchased - you want to maximize the effectiveness E of your resulting health and armor. How much of each should you buy?

2. Thirty minutes into the game, you have 2000 health and 50 armor. You have 1800 gold to spend, and again the costs are the same. You once more want to maximize the effectiveness E of your resulting health and armor. How much of each should you buy?

cosxdx + [7+(2/y)]sinxdy = 0

Find if the equation is exact. If it is exact, solve.

If it is not exact, find an integrating factor to make it exact, verify that it is exact and solve it.

a. Find the two first partials ?? and ?? of the following function. ?(?, ?) = ?³?⁵ + ?^4x sin(?)

b. Find and classify all critical points of ?(?, ?) = ?³ − 3? + ?² − 4? + 7.

c. Maximize and minimize ?(?, ?) = 3? + ? + 33 subject to ?² + ?² = 40.