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MATH 203 Lecture Notes - Lecture 1: Binary Logarithm

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Concordia University
Department
Mathematics
Course
MATH 203
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All

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Document Summary

Class 1_203: this are graphs of y = f (x) = subtraction and y = g (x) = p10x - addition and x2. For example: adding (and subtracting) of graphs (x; f (x)) =(cid:18)x; (x; g (x)) =(cid:0)x; p10x(cid:1) gives the graph (x; (g + f ) (x)) =(cid:18)x; ((x; (g (cid:0) f ) (x)) =(cid:18)x; p10x(cid:19)). x2. 10: these are graphs of y = f (x) = and division and y = g (x) = p10x - multiplication x2. For example: multiplying (and dividing) of graphs (x; f (x)) = (cid:18)x; and (x; g (x)) =(cid:0)x; p10x(cid:1) gives the graph (x; (g (cid:2) f ) (x)) = x; ((x; (g (cid:4) f ) (x)) = x; 10p10x x2 : graph: y = f (x) = x2. Multiplication by a real number y y=(af)(x) y=f(x) y=(bf)(x) y=(cf)(x) x. Multiplication by: 0 < b < 1 ! blue, a > 1 ! red, c < 0 ! green.

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Related Questions

Q18. A size 6 dress in Country C is size 52 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 20). Find a formula for the inverse of the function described.
a. f -1(x) = (x - 20)/2
b. f -1(x) = x - 20
c. f -1(x) = (x/2) - 20
d. f -1(x) = (x/2) + 20

Q19. If f(x) = 4x and g(x) = 12, find the point of intersection of the graphs of f and g by solving f(x) = g(x). Give an exact answer.
a. (log 4 12, 12)
b. (log 4 12, 4)
c. (log 4 12, 0)
d. (12, 12)

Q20. Find functions f and g so that f ∘ g = H(x) = (5 - 2x3)2.
a. f(x) = 5 - 2x3 ; g(x) = x2
b. f(x) = (5 - 2x)3 ; g(x) = x2
c. f(x) = x2 ; g(x) = 5 - 2x3
d. f(x) = x3 ; g(x) = (5 - 2x)2

Q29. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = 9, y = 2, z = 5; (9, 2, 5)
b. x = 8, y = 4, z = 5; (8, 4, 5)
c. x = 4, y = 5, z = 4; (4, 5, 4)
d. x = 8, y = -4, z = -5; (8, -4, -5)

Q30. Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.


a. x = 2, y = 3, z = -7; (2, 3, -7)
b. x = 1, y = 3, z = -4; (1, 3, -4)
c. x = 5, y = 13/2, z = -3; (5, 13/2, -3)
d. x = 1, y = -3, z = -16; (1, -3, -16)

Q33. Write the partial fraction decomposition of the rational expression (4x3 + 4x2)/(x2 + 5)2.
a. (4x - 4)/(x2 + 5) + (-20x + 20)/(x2 + 5)2
b. (4x + 4)/(x2 + 5) + (-20x - 20)/(x2 + 5)2
c. (4x + 4)/(x2 + 5) + (20x - 20)/(x2 + 5)2
d. (4x + 4)/(x2 + 5) + (20x + 20)/(x2 + 5)2

Q34. Solve the linear programming problem. Minimize z = 11x + 6y + 7 subject to: x = 0, y = 0, x + y = 1.
a. minimum: 7
b. minimum: 18
c. minimum: 24
d. minimum: 13

Q35. Solve the system of equations by elimination.


a. x = 10, y = 0; (10, 0)
b. x = 10, y = 10; (10, 10)
c. x = 0, y = 10; (0, 10)
d. x = 0, y = 0; (0, 0)

Q36. Perform the row operations on the given augmented matrix.


a.
b.
c.
d.

Q37. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = -4, y = 3; (-4, 3)
b. x = -3, y = -4; (-3, -4)
c. x = 4, y = 3; (4, 3)
d. x = 3, y = 4; (3, 4)

Q38. Write the partial fraction decomposition of the rational expression x/(x2 - 9x + 20).
a. -4/(x - 4) + 5/(x - 5)
b. -5/(x - 4) + 4/(x - 5)
c. 4/(x - 4) + -5/(x - 5)
d. -4/(x - 4) + -5/(x - 5)

Q39. Solve the system of equations.


a. x = -4, y = 4, z = -3; (-4, 4, -3)
b. x = -3, y = 4, z = -4; (-3, 4, -4)
c. x = -4, y = -3, z = 4; (-4, -3, 4)
d. inconsistent

Q40. Find the value of the determinant.


a. 224
b. 24
c. -72
d. 72

Show transcribed image text

2 (iv)


I need help with these as soon as possible Please. Thank You!!

Q15. In 1990, the population of a country was estimated at 4 million. For any subsequent year the population, P(t) (in millions), can be modeled by the equation P(t) = 240/(5 + 54.99e-0.0208t), where t is the number of years since 1990. Estimate the year when the population will be 21 million.
a. approximately the year 2093
b. approximately the year 2041
c. approximately the year 2088
d. approximately the year 2016

Q18. A size 6 dress in Country C is size 52 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 20). Find a formula for the inverse of the function described.
a. f -1(x) = (x - 20)/2
b. f -1(x) = x - 20
c. f -1(x) = (x/2) - 20
d. f -1(x) = (x/2) + 20

Q19. If f(x) = 4x and g(x) = 12, find the point of intersection of the graphs of f and g by solving f(x) = g(x). Give an exact answer.
a. (log 4 12, 12)
b. (log 4 12, 4)
c. (log 4 12, 0)
d. (12, 12)

Q20. Find functions f and g so that f ∘ g = H(x) = (5 - 2x3)2.
a. f(x) = 5 - 2x3 ; g(x) = x2
b. f(x) = (5 - 2x)3 ; g(x) = x2
c. f(x) = x2 ; g(x) = 5 - 2x3
d. f(x) = x3 ; g(x) = (5 - 2x)2

Q29. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = 9, y = 2, z = 5; (9, 2, 5)
b. x = 8, y = 4, z = 5; (8, 4, 5)
c. x = 4, y = 5, z = 4; (4, 5, 4)
d. x = 8, y = -4, z = -5; (8, -4, -5)

Q30. Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.


a. x = 2, y = 3, z = -7; (2, 3, -7)
b. x = 1, y = 3, z = -4; (1, 3, -4)
c. x = 5, y = 13/2, z = -3; (5, 13/2, -3)
d. x = 1, y = -3, z = -16; (1, -3, -16)

Q33. Write the partial fraction decomposition of the rational expression (4x3 + 4x2)/(x2 + 5)2.
a. (4x - 4)/(x2 + 5) + (-20x + 20)/(x2 + 5)2
b. (4x + 4)/(x2 + 5) + (-20x - 20)/(x2 + 5)2
c. (4x + 4)/(x2 + 5) + (20x - 20)/(x2 + 5)2
d. (4x + 4)/(x2 + 5) + (20x + 20)/(x2 + 5)2

Q34. Solve the linear programming problem. Minimize z = 11x + 6y + 7 subject to: x = 0, y = 0, x + y = 1.
a. minimum: 7
b. minimum: 18
c. minimum: 24
d. minimum: 13

Q35. Solve the system of equations by elimination.


a. x = 10, y = 0; (10, 0)
b. x = 10, y = 10; (10, 10)
c. x = 0, y = 10; (0, 10)
d. x = 0, y = 0; (0, 0)

Q36. Perform the row operations on the given augmented matrix.


a.
b.
c.
d.

Q37. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = -4, y = 3; (-4, 3)
b. x = -3, y = -4; (-3, -4)
c. x = 4, y = 3; (4, 3)
d. x = 3, y = 4; (3, 4)

Q38. Write the partial fraction decomposition of the rational expression x/(x2 - 9x + 20).
a. -4/(x - 4) + 5/(x - 5)
b. -5/(x - 4) + 4/(x - 5)
c. 4/(x - 4) + -5/(x - 5)
d. -4/(x - 4) + -5/(x - 5)

Q39. Solve the system of equations.


a. x = -4, y = 4, z = -3; (-4, 4, -3)
b. x = -3, y = 4, z = -4; (-3, 4, -4)
c. x = -4, y = -3, z = 4; (-4, -3, 4)
d. inconsistent

Q40. Find the value of the determinant.


a. 224
b. 24
c. -72
d. 72

Show transcribed image text