A heavy rope, 60 ft long, weighs 0.8 lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?

For the equation x^2+xy+y^2=0, find the equation of the normal to the tangent line at the point (−3,−1)(-3,-1).

Let s(t)= ?? ? − ??? ? − ???? be the equation for a motion particle.

Find: a. the function for velocity v(t). Explain. [10]

b. where does the velocity equal zero? Explain. [20

] c. the function for the acceleration of the particle [10]

d. Using the example above explain the difference between average velocity and instantaneous velocity. (A Graph will be extremely helpful) [25]

e. What condition of the function for the moving particle needs to be present in order for the function not to have a derivative? Use any one of the conditions studied in this chapter to make your case. Specifically explain why the function would not have a derivative given that particular condition. Please draw a picture of the hypothetical scenario using that condition and use the term “continuity” in your explanation.

[35]

Question1:x=2-t, y=3-2t, z=4-3t

a) Explain why the work done by a force field ?(?, ?, ?) is the line integral ∫c ? ∙ ?? where C is a curve defined by ?(?) = ?(?) ? + ?(?)? + ?(?)?

b) Find the work done by the force field ?(?, ?) = −?? + ?? on a particle moving along the straight line y = 2x + 3 from A(0,3) to B(1,5)

Determine the next number in the sequence: a) 1/4. 4/9. 3/5. 8/11. 5/6. 1/32.

b) How is the relation of the sequence in "a)" different from: 1. 4. 11. 29. 76. 199.

At what point does the normal to y=1−3x−x2y=1−3x−x2 at (1,−3)(1,−3) intersect the parabola a second time?

Hint: The normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals -- i.e. if the slope of the first line is mm then the slope of the second line is −1/m

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 160°.

(a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (3, 1, 4).

(b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points towards the origin.

Consider the functionT:R3→R3defined byT(x,y,z) = (3x,x−y,2x+y+z).(i) Prove thatTis a linear transformation. (T’nin bir lineer d ̈on ̈u ̧s ̈um oldu ̆gunu g ̈osteriniz)(ii) Find the representing matrix ofTrelative to the basisβ={α1= (1,0,0),α2= (1,1,0),α3=(1,1,1)}ofR3.(R3 ̈unβ={α1= (1,0,0),α2= (1,1,0),α3= (1,1,1)}tabanına g ̈oreTnin matris g ̈osterimini bulunuz.)

The function

​f(x,y,z)=4x−9y+7z

has an absolute maximum value and absolute minimum value subject to the constraint

x^2+y^2+z^2=146.

Use Lagrange multipliers to find these values.

A.) Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. z=sin(7x^3 y−3xy^4)

B.)Compute the partial derivative: f(x,y)=sin(x^3−4y)

fy(0,π)=

C.) The equation r=5sinθ represents a circle. Find the cartesian coordinates of the center:

x = ,y = , r =

D.) Convert (xy)^3=8 to an equation in polar coordinates.

E.)Find the slope of the tangent line to the polar curve r=sin(3θ) at θ=π/6

F.)Find the equation (in terms of x and y) of the tangent line to the curve r=2sin5θ at θ=π/4

A box with an open top is to being created with a square piece of cardboard 8 inches wide, by cutting four identical squares in each corner. The sides are being folded as well. Find the dimensions of the box that has the largest volume.

1.2

A large circle on a sphere is a circle that forms the intersection of the sphere with a plane through the center of the sphere. Consider the large circle C that arises the intersection sphere x² + y² + z² = 1 and the plane x + y + z = 0.

(a) Express the equations of the specified large circle C using spherical coordinates.

(b) Express the equations of the large circle C using cylindrical coordinates.

(c) Determine a parameterization of C by writing

r (t) = u cos t + v sin t,

where u and v are two orthogonal unit vectors in the plane that cut out C.

(d) Determine the speed and velocity of a particle traveling along the large circle

according to the parameterization in part (c), when the parameter refers to the time.

A piece of wire of length 50 is​ cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to​ (a) minimize and​ (b) maximize the combined area of the circle and the​ square?

a) To minimize the combined​ area, the wire should be cut so that a length of ____ is used for the circle and a length of ____is used for the square. ​(Round to the nearest thousandth as​ needed.)

b) To maximize the combined​ area, the wire should be cut so that a length of ____is used for the circle and a length of ____is used for the square. (Round to the nearest thousandth as​ needed.)