Mathematics II Quiz

Derivatives

• The derivative of a polynomial function can be found using the power rule: ( f'(x) = n \cdot ax^{n-1} ).
• For ( f(x) = 3x^4 - 5x^3 + 2x - 7 ):
  • Differentiate each term:
    • ( 3x^4 ) becomes ( 12x^3 )
    • ( -5x^3 ) becomes ( -15x^2 )
    • ( 2x ) becomes ( 2 )
    • ( -7 ) (a constant) becomes ( 0 )
• Thus, ( f'(x) = 12x^3 - 15x^2 + 2 ).

Limits

• To evaluate ( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} ), factor the numerator:
  • ( x^2 - 4 = (x - 2)(x + 2) ).
• The expression simplifies to ( \frac{(x - 2)(x + 2)}{(x - 2)} = x + 2 ) for ( x \neq 2 ).
• Computing the limit as ( x ) approaches 2 gives ( 2 + 2 = 4 ).

Geometric Sequences

• A geometric sequence has a first term and a common ratio.
• Given:
  • First term ( a = 3 )
  • Common ratio ( r = 2 )
• The nth term formula: ( a_n = ar^{n-1} ).
• For the 5th term:
  • ( a_5 = 3 \cdot 2^{5-1} = 3 \cdot 16 = 48 ).

Rational Root Theorem

• The Rational Root Theorem suggests potential roots of a polynomial can be the factors of the constant term divided by factors of the leading coefficient.
• For ( 2x^3 - 7x^2 + 4x - 1 ):
  • Constant term: -1
  • Leading coefficient: 2
• Possible rational roots include ( \pm 1, \pm \frac{1}{2} ).

Area of a Circle

• The circumference ( C ) of a circle is given by ( C = 2\pi r ).
• For a circumference of 31.4 units, set ( 2\pi r = 31.4 ).
• Solving for radius ( r ):
  • ( r = \frac{31.4}{2\pi} \approx 5 ).
• Area ( A ) is given by ( A = \pi r^2 ):
  • ( A \approx \pi \cdot 5^2 = 25\pi ) square units.