Other Math Answers

In this section, you are going to think deeper and test further your understanding of

the domain and range of the exponential function.

Give five examples of exponential functions in the form 𝑓(𝑥) = 𝑏

𝑥 and

𝑓(𝑥) = 𝑎 ∙ 𝑏

𝑃(𝑥) + ℎ with its domain and range.

Let G

v1 v2 v3 v5 v4

Determine the following: a. CL(G)− closure of graph G b. Determine whether G is Hamiltonian or not. Justify your answer. c. Determine whether G is Eulerian or not. Justify your answer.

Let X = [1,10] be the universal set and A = [1,4], b = [2,8] and c=[3,6] be the subsets of X. Find each of the following sets and display them on the real line.

1. B' and
2. An(B-C)
3. AnB

The demand function 𝑄 and cost function 𝐶(𝑄) of a commodity are given by the equations \[ Q = 20 - 0{,}01P\]

𝐶(𝑄)=60+6𝑄,

where P and 𝑄 are the price and quantity, respectively.

The total revenue function (𝑇𝑅) in terms of 𝑃 is

a.

TR=20−0,01P.

b.

TR=P(120−0,01P2).

c.

TR=20P−0,01P2.

d.

𝑇𝑅=𝑃2(20−0,01𝑃2).

The demand for seats at a mini soccer match is given by

𝑄=150−𝑃

2

,

Q=150−P2,

where 𝑄

Q is the number of seats and 𝑃

P is the price per seat. Find the price elasticity of demand if seats cost 𝑅4

R4 each. What does this value mean?

a.

𝜀𝑑=−0,24;inelastic since|𝜀𝑑

|<1,a1%price increase will result in0,24%less seats to be sold

εd=−0,24;inelastic since|εd|<1,a1%price increase will result in0,24%less seats to be sold

b.

𝜀𝑑=0,24;elastic since|𝜀𝑑

|>0,a1%price increase will result in0,24%more seats to be sold

εd=0,24;elastic since|εd|>0,a1%price increase will result in0,24%more seats to be sold

c.

𝜀𝑑=−16,75;elastic since|𝜀𝑑

|>1,a1%price increase will result in16,75%less seats to be sold

εd=−16,75;elastic since|εd|>1,a1%price increase will result in16,75%less seats to be sold

d.

𝜀𝑑=16,75;elastic since|𝜀𝑑

|<1,a1%price increase will result in16,75%less seats to be sold

The demand function 𝑄(𝑃)

Q(P) and cost functions 𝐶(𝑄)

C(Q) of a company's are given by the equations:

𝑄=12000−60𝑃

Q=12000−60P

𝐶(𝑄)=10000+4𝑄,

C(Q)=10000+4Q,

where 𝑃

P and 𝑄

Q are the price and quantity, respectively. 

What is the company's profit function?

a.

=−60𝑃−4𝑄+2 000

=−60P−4Q+2 000

b.

=−60𝑃

2

+11 760𝑃−58 000

=−60P2+11 760P−58 000

c.

=−60𝑃

2

+12 240𝑃−58 000

=−60P2+12 240P−58 000

d.

=−60𝑃

2

+12 240𝑃+38 000

Implicit and a parametric representation for the plane containing both 𝑃 and 𝑙

Given the following points: 𝑃0 = (−1, −1,3),𝑃1 = (−1,3, −1),𝑃2 = (3,5,3)𝑃3 = (3,3,5), 𝑃4 = (− 1 2 , 3,3). a) Show that 𝑃0, 𝑃1,𝑃2, 𝑃3 lie on the same plane, H, and find the implicit equation of H. A pyramid is defined by the plane H and the following triangular faces: (𝑃0,𝑃2,𝑃4 ), (𝑃0, 𝑃1,𝑃4 ), (𝑃1,𝑃3,𝑃4 ), (𝑃2,𝑃3, 𝑃4 )

b) Determine the outwards facing unit normal vector of each triangular face. c) Calculate the implicit representation of the planes containing each face of the shape. d) For each of the following points determine if it is inside or outside the shape (hint: the point is inside the shape if it lies on the same side for all the planes) i) (− 1 2 , 1,2) ii) (1,0,1) iii) (3,2,4)

Let 𝑙1 be the line that passes through 𝑝1 = (2,9,8) and 𝑝2 = (1,9,9) and let 𝑙2 be the line that passes through 𝑝3 = (1,1,1) and 𝑝4 = (2,5,4) a) Find out if the two lines intersect and if so, find the intersection point of 𝑙1 and 𝑙2 b) Let 𝑆 be the sphere whose center is the intersection of 𝑙1 and 𝑙2 and whose radius is 𝑟 = 4. Write the implicit representation of the sphere. c) Find the implicit representation of the two planes 𝑃1 and 𝑃2 that are tangent to the sphere 𝑆 at the points of intersections of 𝑙1 towards 𝑝2 and 𝑙2 towards 𝑝4 , respectively.

A 2D light ray is sent from point 𝑃 = (1, −1). It is reflected off a surface (represented by a line) at 𝑅 = (6,11), and reaches a receiver point at 𝑄 = (25,13 2 17) . Note that light rays hitting a surface reflect in a direction which is symmetric according to the normal. a) Find the implicit representation of the surface such that its “up” is towards 𝑃 (i.e. it faces the incoming ray). b) Find the angle between the ray and the surface.