CQMS702 Final Exam Practice
Calculus for Business - Toronto Metropolitan University
Complete Worked Solutions Manual
This document gives full step-by-step worked solutions for Questions 1-17 from the uploaded CQMS702 final
exam practice file. The steps are written in a clear exam-solution style, with all major algebra, calculus, and
business-calculus reasoning shown.
Source used: CQMS702 Final Exam Practice with Dr. Wang Questions, Calculus for Business, Toronto
Metropolitan University.
CQMS702 Complete Worked Solutions Page 1
, Question 1 - Consumer Surplus and Producer Surplus
Given demand p = d(x) = -5x + 40 and supply p = s(x) = x^2 + 3x + 20. The market price is set at the
equilibrium price. First find the equilibrium quantity by setting demand equal to supply.
d(x) = s(x)
-5x + 40 = x^2 + 3x + 20
Move all terms to one side.
0 = x^2 + 3x + 20 + 5x - 40
0 = x^2 + 8x - 20
Solve the quadratic equation.
x^2 + 8x - 20 = 0
(x + 10)(x - 2) = 0
x = -10 or x = 2
Quantity cannot be negative, so the equilibrium quantity is x = 2. Now find the equilibrium price using either
demand or supply.
p = d(2) = -5(2) + 40 = -10 + 40 = 30
Thus the equilibrium point is (2, 30).
Consumer surplus is the area between the demand curve and the equilibrium price from x = 0 to x = 2.
CS = integral from 0 to 2 of [d(x) - 30] dx
CS = integral from 0 to 2 of [(-5x + 40) - 30] dx
CS = integral from 0 to 2 of (-5x + 10) dx
CS = [(-5/2)x^2 + 10x] from 0 to 2
CS = [(-5/2)(2)^2 + 10(2)] - [0]
CS = [-10 + 20]
CS = 10
Producer surplus is the area between the equilibrium price and the supply curve from x = 0 to x = 2.
PS = integral from 0 to 2 of [30 - s(x)] dx
PS = integral from 0 to 2 of [30 - (x^2 + 3x + 20)] dx
PS = integral from 0 to 2 of (10 - 3x - x^2) dx
PS = [10x - (3/2)x^2 - (1/3)x^3] from 0 to 2
PS = [10(2) - (3/2)(2)^2 - (1/3)(2)^3] - 0
PS = 20 - 6 - 8/3
PS = 14 - 8/3
PS = 42/3 - 8/3
PS = 34/3
Final Answer: Consumer surplus = 10. Producer surplus = 34/3.
Question 2 - Elasticity of Demand
The demand relation is p^2 = -0.01x^2 - 0.2x + 8.76, where p is price and x is quantity demanded in
thousands. Find the elasticity when p = 2.4.
First find the corresponding x-value by substituting p = 2.4.
(2.4)^2 = -0.01x^2 - 0.2x + 8.76
5.76 = -0.01x^2 - 0.2x + 8.76
CQMS702 Complete Worked Solutions Page 2
Calculus for Business - Toronto Metropolitan University
Complete Worked Solutions Manual
This document gives full step-by-step worked solutions for Questions 1-17 from the uploaded CQMS702 final
exam practice file. The steps are written in a clear exam-solution style, with all major algebra, calculus, and
business-calculus reasoning shown.
Source used: CQMS702 Final Exam Practice with Dr. Wang Questions, Calculus for Business, Toronto
Metropolitan University.
CQMS702 Complete Worked Solutions Page 1
, Question 1 - Consumer Surplus and Producer Surplus
Given demand p = d(x) = -5x + 40 and supply p = s(x) = x^2 + 3x + 20. The market price is set at the
equilibrium price. First find the equilibrium quantity by setting demand equal to supply.
d(x) = s(x)
-5x + 40 = x^2 + 3x + 20
Move all terms to one side.
0 = x^2 + 3x + 20 + 5x - 40
0 = x^2 + 8x - 20
Solve the quadratic equation.
x^2 + 8x - 20 = 0
(x + 10)(x - 2) = 0
x = -10 or x = 2
Quantity cannot be negative, so the equilibrium quantity is x = 2. Now find the equilibrium price using either
demand or supply.
p = d(2) = -5(2) + 40 = -10 + 40 = 30
Thus the equilibrium point is (2, 30).
Consumer surplus is the area between the demand curve and the equilibrium price from x = 0 to x = 2.
CS = integral from 0 to 2 of [d(x) - 30] dx
CS = integral from 0 to 2 of [(-5x + 40) - 30] dx
CS = integral from 0 to 2 of (-5x + 10) dx
CS = [(-5/2)x^2 + 10x] from 0 to 2
CS = [(-5/2)(2)^2 + 10(2)] - [0]
CS = [-10 + 20]
CS = 10
Producer surplus is the area between the equilibrium price and the supply curve from x = 0 to x = 2.
PS = integral from 0 to 2 of [30 - s(x)] dx
PS = integral from 0 to 2 of [30 - (x^2 + 3x + 20)] dx
PS = integral from 0 to 2 of (10 - 3x - x^2) dx
PS = [10x - (3/2)x^2 - (1/3)x^3] from 0 to 2
PS = [10(2) - (3/2)(2)^2 - (1/3)(2)^3] - 0
PS = 20 - 6 - 8/3
PS = 14 - 8/3
PS = 42/3 - 8/3
PS = 34/3
Final Answer: Consumer surplus = 10. Producer surplus = 34/3.
Question 2 - Elasticity of Demand
The demand relation is p^2 = -0.01x^2 - 0.2x + 8.76, where p is price and x is quantity demanded in
thousands. Find the elasticity when p = 2.4.
First find the corresponding x-value by substituting p = 2.4.
(2.4)^2 = -0.01x^2 - 0.2x + 8.76
5.76 = -0.01x^2 - 0.2x + 8.76
CQMS702 Complete Worked Solutions Page 2
