Math 114 Exam 2 Review

Trigonometry Study Guide

• Review problems from the provided study guide on Canvas.

Optimization Problems

• Maximize Product: If two numbers add to 12, find the largest product.

  • Let the two numbers be ( x ) and ( 12-x ).
  • Product: ( P = x(12 - x) = 12x - x^2 )
  • This is a quadratic function that opens downwards. To find maximum:
    • Vertex: ( x = \frac{-b}{2a} = \frac{12}{2} = 6 )
    • Maximum product: ( P = 6(12 - 6) = 36 )

• Rectangular Pigsty Problem: Amy has 600 feet of fencing.

  • Divide into two rectangles:
    • Let length = ( l ) and width = ( w ).
    • Perimeter: ( 2l + 3w = 600 )
    • Area: ( A = l \times w )
    • Express ( l ) in terms of ( w ): ( l = 300 - 1.5w ). Use this to express the area in terms of ( w ).

Polynomial Functions

• Finding Zeros and Multiplicity:

  • For ( m(t) = t^5 - 10t^4 + 25t^3 ):

    • Factor: ( t^3(t^2 - 10t + 25) = t^3(t - 5)^2 );
    • Zeros: ( t = 0 ) (multiplicity 3), ( t = 5 ) (multiplicity 2).

  • For ( j(x) = 5x^6(x - 5)^2(2x + 9)(x - \sqrt{3})(x + \sqrt{3}) ):

    • Zeros: ( x = 0 ) (multiplicity 6), ( x = 5 ) (multiplicity 2), ( x = -\frac{9}{2} ) (multiplicity 1), ( x = \sqrt{3} ) (multiplicity 1), ( x = -\sqrt{3} ) (multiplicity 1).

• End Behavior:

  • For ( f(x) = x^4 - 4x^2 + x ):

    • As ( x \to -\infty, f(x) \to \infty ), As ( x \to \infty, f(x) \to \infty )

  • For polynomial example: ( f(x) = -4x^5 - 3x^3 + 2 ):

    • As ( x \to -\infty, f(x) \to -\infty ), As ( x \to \infty, f(x) \to -\infty )

Rational Functions

• Finding Domain, Intercepts, Asymptotes:

  • For ( k(x) = \frac{x^2 + 5x + 8}{x + 3} ):
    • Domain: All ( x ) except ( x = -3 );
    • Vertical asymptote at ( x = -3 ).

• Similarly analyze:\n - ( l(x) = \frac{x^2 + x - 2}{(x + 2)(x^2 - 2x - 15)} ) for intercepts and discontinuities.

  • ( g(x) = \frac{x^2 - 4}{x^2 + 4} )

Logarithmic Functions

• Solve Logarithmic Equations:

  • Example: ( \ln(3x + 14) = \ln(6x - 4) )
    • Set arguments equal: ( 3x + 14 = 6x - 4 ) → ( x = 6.

• Graphing Functions: Graph functions while indicating any intercepts and asymptotes.

  • For example: For ( f(x) = 2^{x-1} - 3 ): Set up intercepts by solving ( 2^{x-1} = 3 ).

Inverse Functions

• Finding the Inverse: Example:
  • For ( f(x) = \log_{10}(x) )
  • Swap x and y to find the inverse: ( f^{-1}(x) = 10^x )
  • Domain of inverse: All real numbers; Range: All positive real numbers.

Properties of Logarithms

• Expand and combine logarithmic expressions using properties such as product, quotient, and power laws.
  • Example: ( \log3(8x^2) = \log3(8) + 2\log_3(x) )

Special Values of Logarithms

• Evaluate commonly known logarithmic values:
  • ( \log_{10}(100000) = 5 )
  • ( \log_3(\frac{1}{81}) = -4 )
  • ( \ln(1) = 0 )
  • ( \ln(e^{-0.234}) = -0.234 )