MA151 Calculus 1

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MA151 Calculus 1

• 2007-08

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Algebra Review

• We are going to start the course by review the
  basic rules and procedures of algebra that you
  need to know in order to be successful in
  Calculus.

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Arithmetic Operations

• Properties of the Real Numbers
• Commutative Property
• a b b a ab ba
• Associative Property
• (ab)c a(bc) (ab)c a(bc)
• Distributive Property
• a(bc) ab ac

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• If we put a -1 into the Distributive Property,
  we get -1(bc) -b
  c
• If we use the Distributive Property three times,
  we get (ab)(cd)(ab)c(ab)dacbcadbd or
  if we use FOIL (Firsts-Outsides-Insides-Lasts) ,
  we have (ab)(cd)acadbcbd

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• Special cases,
• (ab)2 a2 2ab b2
• (a-b)2 a2 - 2ab b2

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Fractions
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Factoring

• To factor a quadratic of the form x2bxc, we
  need to choose numbers r and s such that rs b
  and rs c so that (xr)(xs) x2(rs)xrs

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Special Instances

• Difference of Squares
• a2-b2(a-b)(ab)
• Difference of Cubes
• a3-b3(a-b)(a2abb2)
• Sum of Cubes
• a3b3(ab)(a2-abb2)

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Completing the Square

• Completing the square means rewriting a
  quadratic ax2bxc in the form a(xp)2q.
• Factor the number a from the terms involving x.
• Add and subtract the square of half the
  coefficient of x.

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Quadratic Formula

• The roots of a quadratic equation ax2bxc0 are

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• The quantity b2-4ac in the quadratic formula is
  called the discriminant.
• If b2-4ac gt 0, the equation has two real roots.
• If b2-4ac 0, the roots are equal.
• If b2-4ac lt 0, the equation has no real roots.
  (The roots are complex.)
• These cases correspond to the number of times the
  parabola y ax2bxc crosses the x-axis.

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Radicals
means
Dont forget
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Exponents
Let a be any positive number and let n be a
positive integer. Then, by definition,
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Rules for Inequalities

• If a lt b, then a c lt b c.
• If altb and cltd, then a c lt b d
• If a lt b and c gt 0, then ac lt bc.
• If a gt b and c gt 0, then ac gt bc.
• If 0 lt a lt b, then 1/a gt 1/b.

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Absolute Value

• a a if a gt 0
• a -a if a lt 0
• ab ab
• b?0
• an an

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Equations with Absolute Value

• x a iff x a
• x lt a iff a lt x lt a
• x gt a iff x gt a or x lt -a