Calculus Answers

Let a∈R be a constant. The parametrization r(u,v)=⟨asin(u)cos(v),asin(u)sin(v),acos(u)⟩ for (u,v)∈[0,π/2]×[0,2π]

describes
Select one:
a. a cylinder with radius a


b. an ellipse with radius a


c. a circle with radius a


d. a cylinder with height a


e. a sphere with radius a


f. a hemisphere with radius a

Which of the following statements is true about line integrals?
Select one:
a. ∫−C F⋅dr=−2∫C F⋅dr


b. ∫−C F⋅dr=0


c. ∫−C F⋅dr=∫C F⋅dr


d. ∫−C F⋅dr=−∫C F⋅dr

A parameterization of the line segment from (-3,-3,-3) to (0,0,0) is
Select one:
a. r(t)=⟨3t,3t,3t⟩
for t∈[−3,0]
.

b. r(t)=⟨t,−t,t⟩
for t∈[−3,0]
.

c. r(t)=⟨t,t,t⟩
for t∈[−3,0]
.
d. r(t)=⟨t,t⟩
for t∈[0,3]
.

e. r(t)=⟨t,t,t⟩
for t∈[−3,3].

What is the plot of the vector field F(x,y)=⟨0,−y⟩

Let F(x,y,z)=⟨2xy,x^2+z,y⟩ be a conservative vector field. A potential function of F

is
Select one:
a. x^2y+yz+z


b. x^2yz


c. x^2y+yz−87

d. x^6yz


e. ⟨x^2y,yz,z⟩

Find the area bounded by the curve y = |x−1|, the x-axis and the lines x = −7 and x = 11

As an engineer, you have been tasked to design a cooling tower int he shape of a hyperboloid of one sheet.the horizontal cross sections of the cooling tower are circular with 10m. the cooling tower is 40m tall with maximum cross-sectional radius of 15m.
A) Construct a mathematical equation for this cooling tower.
B) If x=a cos(u)cosh(v), y=b sin(u)cosh(v) and z=c sin h(v), show that (x,y,z) lies on your equation in Q1(A).
C) A colleague wants to construct the cooling tower using a hyperbolic cylinder, give reasons for your result in Q1(A) as the best model for the design of cooling tower.

As an engineer,you have been asked to design a cooling tower in the shape of hyperboloid of one sheet. the horizontal cross sections of the cooling tower are circular with 10m. the cooling tower is 40m tall with maximum cross-sectional radius of 15m. construct a mathematical equation for this cooling tower.

What is the limit of cosx-sin2x over x^2 as x approaches to zero.

what is the limit of 3x^2+3ax-2a^2 over x^2-a^2 as x approaches to a.