StudentAidUnit 10: Unit 10 — Review Exercise
9th Class Mathematics · Unit 10: Unit 10
2.Plot the graph of the following functions: (i) (i) y = 3^(-x) for x from -2 to 4 (ii) (ii) y = 2/x, x ≠ -0
Given
(i) Function
y = 3-x
Formula
Rewrite
y = 13x
For x = -2
y = 3-(-2) = 32 = 9
For x = -1
y = 3-(-1) = 31 = 3
For x = 0
y = 30 = 1
For x = 1
y = 3-1 = 13 ≈ 0.33
For x = 2
y = 3-2 = 19 ≈ 0.11
For x = 3
y = 3-3 = 127 ≈ 0.03
For x = 4
y = 3-4 = 181 ≈ 0.01
Result
Graph plotted
Exponential decay curve through (−2,9), (−1,3), (0,1), (1,0.33), (2,0.11), (3,0.03), (4,0.01)
Given
(ii) Function
y = 2x, x neq 0
For x = -8
y = 2-8 = -0.25
For x = -6
y = 2-6 ≈ -0.33
For x = -4
y = 2-4 = -0.5
For x = -2
y = 2-2 = -1
For x = -1
y = 2-1 = -2
For x = 0
y = undefined
For x = 1
y = 21 = 2
For x = 2
y = 22 = 1
For x = 4
y = 24 = 0.5
For x = 6
y = 26 ≈ 0.33
For x = 8
y = 28 = 0.25
Result
Graph plotted
Hyperbola with vertical asymptote at x=0
3.(i) Sales for a new magazine are expected to grow according to the equation: S = 200000(1 - e^(-0.05t)), where t is given in weeks. (b) Calculate the number of magazines sold, when t = 5 and t = 35.
Given
(i) Equation
S = 200,000(1 - e-0.05t)
Working
For t = 5
S = 200,000(1 - e-0.05 × 5)
Simplify exponent
S = 200,000(1 - e-0.25)
Calculate e^(-0.25)
e-0.25 ≈ 0.7788
Substitute
S = 200,000(1 - 0.7788) = 200,000(0.2212)
Result
Result for t = 5
S = 44,239.84 ≈ 44,240 magazines
Working
For t = 35
S = 200,000(1 - e-0.05 × 35)
Simplify exponent
S = 200,000(1 - e-1.75)
Calculate e^(-1.75)
e-1.75 ≈ 0.1738
Substitute
S = 200,000(1 - 0.1738) = 200,000(0.8262)
Result
Result for t = 35
S = 165,245.2 ≈ 165,245 magazines
4.Plot the graph of following for x from -5 to 5: (i) (i) y = x^2 - 3 (ii) (ii) y = 15 - x^2
Given
(i) Function
y = x2 - 3
For x = -5
y = (-5)2 - 3 = 25 - 3 = 22
For x = -3
y = (-3)2 - 3 = 9 - 3 = 6
For x = -1
y = (-1)2 - 3 = 1 - 3 = -2
For x = 0
y = 02 - 3 = -3
For x = 1
y = 12 - 3 = 1 - 3 = -2
For x = 3
y = 32 - 3 = 9 - 3 = 6
For x = 5
y = 52 - 3 = 25 - 3 = 22
Result
Graph plotted
Parabola opening upward with vertex at (0,-3)
Given
(ii) Function
y = 15 - x2
For x = -5
y = 15 - (-5)2 = 15 - 25 = -10
For x = -3
y = 15 - (-3)2 = 15 - 9 = 6
For x = -1
y = 15 - (-1)2 = 15 - 1 = 14
For x = 0
y = 15 - 02 = 15
For x = 1
y = 15 - 12 = 15 - 1 = 14
For x = 3
y = 15 - 32 = 15 - 9 = 6
For x = 5
y = 15 - 52 = 15 - 25 = -10
Result
Graph plotted
Parabola opening downward with vertex at (0,15)
5.Plot the graph of y = (1/2)(x+4)(x-1)(x-3) from x = -5 to 4.
Given
Function
y = 12(x+4)(x-1)(x-3)
For x = -4
y = 12(0)(-5)(-7) = 0
For x = -3
y = 12(1)(-4)(-6) = 12(24) = 12
For x = -2
y = 12(2)(-3)(-5) = 12(30) = 15
For x = 0
y = 12(4)(-1)(-3) = 12(12) = 6
For x = 1
y = 12(5)(0)(-2) = 0
For x = 2
y = 12(6)(1)(-1) = -3
For x = 3
y = 12(7)(2)(0) = 0
For x = 4
y = 12(8)(3)(1) = 12
Result
Graph plotted
Cubic polynomial with zeros at x=-4,1,3
6.The supply and demand functions for a particular market are given by the equations: (i) P_s = Q^2 + 5 and P_d = Q^2 - 10Q, where P represents price and Q represents quantity.
Given
Supply function
Ps = Q2 + 5
Demand function
Pd = Q2 - 10Q
For Q = -20
Ps = (-20)2 + 5 = 400 + 5 = 405; quad Pd = (-20)2 - 10(-20) = 400 + 200 = 600
For Q = -10
Ps = (-10)2 + 5 = 100 + 5 = 105; quad Pd = (-10)2 - 10(-10) = 100 + 100 = 200
For Q = 0
Ps = 02 + 5 = 5; quad Pd = 02 - 10(0) = 0
For Q = 5
Ps = 52 + 5 = 25 + 5 = 30; quad Pd = 52 - 10(5) = 25 - 50 = -25
For Q = 10
Ps = 102 + 5 = 100 + 5 = 105; quad Pd = 102 - 10(10) = 100 - 100 = 0
For Q = 20
Ps = 202 + 5 = 400 + 5 = 405; quad Pd = 202 - 10(20) = 400 - 200 = 200
Result
Both curves plotted
Supply and demand curves over Q in [-20, 20]
7.A television manufacturer company make 40 inches LEDs. The cost of manufacturing x LEDs is C(x) = 60,000 + 250x and the revenue from selling x LEDs is R(x) = 1200x. Find the break-even point and find the profit or loss when 100 LEDs are sold.
Given
Cost function
C(x) = 60,000 + 250x
Revenue function
R(x) = 1200x
Formula
Break-even point condition
R(x) = C(x)
Working
Set equal
1200x = 60,000 + 250x
Subtract 250x
1200x - 250x = 60,000
Combine
950x = 60,000
Solve for x
x = 60,000950 ≈ 63.16 ≈ 63
Result
Break-even point
x ≈ 63 LEDs
Formula
Profit function
Profit = R(x) - C(x) = 1200x - (60,000 + 250x)
Simplify
Profit = 1200x - 60,000 - 250x = 950x - 60,000
Working
For x = 100
Profit = 950(100) - 60,000
Calculate
Profit = 95,000 - 60,000
Result
Profit at x = 100
Profit = 35,000