Mathematics Exam Review Notes

Population Growth

  • Exponential Growth Function:
    • Given: A = 24e^{kt}
    • Population in 1982 (t=0): A(0) = 24 million
    • Population in 1992 (t=10): A(10) = A(0) + 4 million = 28 million
    • Setting up the equation:
    • 28 = 24e^{10k}
    • e^{10k} = \frac{28}{24} = \frac{7}{6}
    • Taking natural logs: 10k = ln(\frac{7}{6})
    • Solve for k: [ k = \frac{1}{10} ln(\frac{7}{6}) \approx 0.025 ]

Half-Life Problem

  • Half-Life of Plutonium-234: 9 hours
    • Calculation for 3 Days (72 hours):
    • Using the formula: [ A = A0(\frac{1}{2})^{t/T{1/2}} ]
    • Here, A0 = 10 mg and T{1/2} = 9 hours,
    • Total time = 72 hours = 8 half-lives, hence: [ 10(\frac{1}{2})^8 \approx 0.039 \text{mg} ]

Newton's Law of Cooling

  • Formula: [ U = T + (U_0 - T)e^{-kt} ]
    • Initial temperature: U_0 = 194°, Room temp: T = 72°
    • Condition after 11 mins (U = 140°): [ 140 = 72 + (194 - 72)e^{-11k} ]
    • Solve for k and find time to cool to 102°:
    • Setting up: [ 102 = 72 + 122e^{-kt} ]

Logistic Growth Function

  • Function Given: [ f(t) = \frac{27000}{1 + 674.0 e^{-1.9t}} ]
    • Limiting size of population: 27000 people.

Remainder Theorem

  • Function: [ f(x) = 5x^6 - 3x^3 + 8 ] divided by [ x + 1 ]
    • Remainder is f(-1): [ R = 10 ]

Factor Theorem

  • Function: [ f(x) = 4x^3 + 5x^2 - 5x + 2 ]
    • Check for factor x + 2, evaluate f(-2).

Rational Zeros Theorem

  • Function: [ f(x) = x^5 - 2x^2 + 2x + 10 ]
    • Possible rational zeros: ±1, ±5, ±2, ±10

Finding all zeros and factors

  • Given Polynomial: [ f(x) = 2x^4 - 16x^3 + 33x^2 - 8x + 16 ]
    • Identify zeros and factor accordingly.

Intercepts of Functions

  • Function: [ f(x) = (x - 3)^2(x^2 - 25) ]
    • Find x-intercepts and y-intercepts through substitution.

System Solutions Using Matrices and Cramer’s Rule

  • Write augmented matrices from given systems and solve accordingly.
  • Cramer's Rule applicable when matrix determinant is non-zero.

Sequence and Series Calculations

  • Arithmetic Sequence properties and nth term derivation.
  • Geometric Sequence properties and computations.

Stay organized and detailed for each concept to aid preparation effectively!