1.1: Functions, Calculus Style

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1.1 Functions, Calculus Style
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What is Calculus?

• The branch of mathematics involving derivatives
  and integrals. The study of motion in which
  changing values are studied.math.about.com/librar
  y/blc.htm

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• The mathematics of change and motion. The main
  concepts of calculus are limits, derivatives, and
  areas under curves.www.doe.mass.edu/frameworks/ma
  th/1996/gloss.html

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• The study of functions.

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What is a function?

• A function is a procedure for assigning a unique
  output to any acceptable input.
• The set of inputs is the domain and the set of
  outputs is the range.

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• f(x) x5 4x3 -2x2 3
• g(x) 3x2 4x 6

x 2
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Is this a function?
Time (in hours) 0 1 2 3 4
of cells 16 48 144 432 1296
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Is this a function?
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• Not all functions have convenient algebraic
  representations (formulas). Many times it is
  helpful to approximate a formula if one does not
  already exist.

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• This process is called curve fitting. The
  constants that we use in curve fitting are called
  the parameters.

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Is this a function?

• For any real number h, let f(h) be the weight of
  an average person with height h (in cm).

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What do we know from the table?
x -3 -2 -1 0 1 2
f(x) -12 8 4 -3 -4 -7
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What dont we know from the table?
x -3 -2 -1 0 1 2
f(x) -12 8 4 -3 -4 -7
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Piecewise-Defined Functions

• A piecewise function is a function that has
  different rules for different subsets of the
  domain.

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A piecewise function

• 3x if x lt -4
• f(x) x 3 if -4 x 6
• 2x 1 if x gt 6

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Ex 5
What is the balloons vertical velocity at t
.5? t 7?
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• The vertical velocity, V(t), in the previous
  graph is nothing more than the slope of the
  graph at a particular point.

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• Most real world problems require you to translate
  from a verbal problem to a mathematical
  expression or function.

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• A sheet of paper is 14 inches long and 11 inches
  wide. To make a box from this paper, squares are
  cut from each corner and then the sides are
  folded up. What are the dimensions of the box
  with the largest volume?

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14
11
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We can graph our function to determine the
largest volume.
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Example 7

• Given f(x) x 2.
• Let D(x) be the distance from the origin to (x,
  f(x)) on the graph of f(x).
• Determine a formula for D and then graph your
  formula.

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• Another use of functions is to approximate
  another function. For example
• p(x)
• is an approximation of the function f(x) . We
  will worry about such functions later in the
  course.

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Assignment

• Page 11,
• 12, 14, 18 and 19