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    Calculus I Essentials
    Calculus I Essentials
    Calculus I Essentials
    Ebook135 pages59 minutes

    Calculus I Essentials

    By Editors of REA

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    About this ebook

    REA’s Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Calculus I covers functions, limits, basic derivatives, and integrals.
    LanguageEnglish
    PublisherResearch & Education Association
    Release dateJan 1, 2013
    ISBN9780738670393
    Calculus I Essentials

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      CHAPTER 1

      FUNDAMENTALS

      1.1 NUMBER SYSTEMS

      The real number system can be broken down into several parts and each of these parts have certain operations which can be performed on them. First, let us define the components of the real number system.

      The natural numbers, denoted N, are 1, 2, 3, 4, .... The integers, denoted Z, are ...-3,-2,-1,0,1,2,3,.... The rational numbers, denoted Q, are all numbers of the form p/q where p and q are integers and q#0. A real number x is a non-terminating decimal (with a sign + or -).

      Six basic algebraic properties of rational, numbers:

      The closure property: If x and y are rational numbers, then x+y and x·y are also rational numbers.

      Additive and multiplicative identity elements: If x is a rational number, then x+0=x and x · 1=x .

      Associative property: If x, y and z are rational numbers, then x+(y+z)=(x+y)+z, x(y·z)=(x·y)z.

      Additive and multiplicative inverses: For each rational number, x, such that x+(-x)=0; if x#0, there exists a rational number x-1 such that x·x-1=1.

      Commutative property: If x and y are rational numbers, then x+y=y+x , x·y=y·x.

      Distributive property: If x, y, and, z are rational numbers, then

      x · (y+z) = (x · y) + (x · z)

      If q and p are rational numbers and p-q is negative, then q is greater than p, (q>p) or p is less than q, (p

      1.1.1 PROPERTIES OF RATIONAL NUMBERS

      Trichotomy property: If p and q are rational numbers, then one and only one of the relations q=p, q>p or q

      Transitive property: If p, q, and r are rational numbers, and if p

      If p, q, and r are rational numbers and p

      If p, q, and r are rational numbers and if p

      1.2 INEQUALITIES

      To solve a linear inequality

      ax+b>0 or x>–b/a, where a>0

      draw a number line, dashed for x<-b/a and solid for x>-b/a.

      To solve (ax+b)(cx+d)>0 graphically, where a>0 and c>0:

      The solution lies in the interval where both lines are dashed and both lines are solid.

      is the solution to the above inequality.

      1.3 ABSOLUTE VALUE

      Definition: The absolute value of a real number x is defined as

      For real numbers a and b:

      |a| = |-a|

      |ab| = |a|·|b|

      -|a| ≤ a ≤ |a|

      ab ≤ |a| |b|

      |a+b| ² = (a+b)²

      |a+b| ≤ |a| + |b| (Triangle Inequality)

      |a-b| ≥ | |a| - |b | |

      For positive values of b

      |a| < b if and only if -b < a < b

      |a| > b if and only if a > b or a < -b

      |a| = b if and only if a = b or a = -b

      1.4 SET NOTATION

      A set is a collection of objects called elements. Let A and B be sets.

      x ∈ A : x is an element of A

      x ∉ B : x is not an element of B

      A is a subset of B , (A ⊂ B) , means that A is contained in another set B and each element of A is also an element of

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