[pic 1]
MATH001 Calculus
Homework 1
Student Full Name (as shown on LMS e-Learn [10 marks]) |
Teng Ying Swam |
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HW1.Q1. [10 marks]. Solve the following equation, for 𝒙.
𝟐𝒙 + 𝟗 = 𝟓(𝒙−𝟕)
SOLUTION:
2x + 9 = 5x – 35
9 + 35 = 5x – 2x
44 = 3x
x = [pic 2]
HW1.Q2. [10 marks]. Solve the following equation, for 𝒙.
𝟑𝒙𝟐 −𝒙 = 𝟐𝟐
SOLUTION:
0 = 3x2 – x – 22
Using this formula to solve quadratic equation [pic 3]
[pic 4]
[pic 5]
[pic 6]
[pic 7][pic 8]
HW1.Q3. [10 marks, Textbook, p. 101, P65, modified]. Solve the following equation, for 𝒕.
𝟏. 𝟎𝟎𝟓𝟏𝟐𝒕 −𝟑 = 𝟎
SOLUTION:
1.00512t = 3
12t = log1.0053
t = [pic 9]
t = [pic 10]
t = 18. 35594227 ≈ 18.36
HW1.Q4. [10 marks, Textbook, p. 100. P49]. Solve the following equation.
𝐥og𝟏𝟎(𝒙−𝟏) −𝐥og𝟏𝟎(𝒙 + 𝟏) = 𝟏
SOLUTION:
Log10 = 1[pic 11]
= 10[pic 12]
x -1 = 10(x+1)
x – 1 = 10x + 10
-1 -10 = 10x -x
-11 = 9x
x = (no solution, log cannot be negative)[pic 13]
HW1.Q5. [ , Textbook, p. 65, P69]. At a price of $2.28 per bushel, the supply of barley is 7,500 million bushels and the demand is 7,900 million bushels. At a price of $2.37 per bushel, the supply is 7,900 million bushels and the demand is 7,800 million bushels. Let 𝒑 denote the price in dollars and let 𝒙 denote the number of bushels (in millions).
- Find a price-supply equation of the form 𝒑 = 𝒎𝒙 + 𝒃.
- Find a price-demand equation of the form 𝒑 = 𝒎𝒙 + 𝒃.
- Find the equilibrium point.
SOLUTION:
a) p = mx + b
m = = [pic 14][pic 15]
m = 0.000225
if m = 0.000225, b = 2.37 – (0.000225 x 7900)
b = 0.5925
price-supply equation = p = 0.000225x + 0.5925
b) p = mx + b
m = = [pic 16][pic 17]
m = -0.0009
if m = -0.0009, b = 2.37 – (-0.0009 x 7800)
b = 9.39
price-supply equation = p = -0.0009x + 9.39
c) 0.000225x + 0.5925 = -0.0009x + 9.39
0.0011225x = 8.7975
X = 7820
P = 0.000225(7820) + 0.5925
P = 2.352
Equilibrium point = (7820, 2.352)
HW1.Q6. [10 marks]. Use the following revenue function and cost function found by the marketing research department for a company that manufactures and sells memory chips for microcomputers,
𝑹(𝒙) = 𝒙(𝟕𝟓−𝟑𝒙), 𝑪(𝒙)= 𝟏𝟐𝟓 + 𝟏6𝒙,
where 𝒙 is in millions of chips, and 𝑹(𝒙) and 𝑪(𝒙) are in millions of dollars. Both functions have domain 𝟏 ≤ 𝒙 ≤ 𝟐𝟎. Find the break-even point(s) to the nearest thousands chips.
SOLUTION:
Breakeven point = R(x) = C(x)
x(75 -3x) = 125 + 16x
75x – 3x2 = 125 + 16x
0 = 3x2 - 59x + 125 Using this formula to solve quadratic equation [pic 18]
[pic 19]
[pic 20]
x ≈ 17,251,000 or 2,415,000 (nearest thousand)
HW1.Q7. [10 marks, Textbook, p. 90, P61]. From the dawn of humanity to 1830, world population grew to one billion people. In 100 more years (by 1930) it grew to two billion, and 3 billion more were added in only 60 years (by 1990). In 2013, the estimated world population was 7.1 billion with a relative growth rate of 1.1%.