Section 1.1–1.2
                     Functions

                 V63.0121.006/016, Calculus I

                        New...
Announcements




          Get your WebAssign
          accounts and do the Intro
          assignment. Class Key:
      ...
Objectives: Functions and their Representations


          Understand the definition
          of function.
          Wor...
Objectives: A Catalog of Essential Functions
          Identify different classes of
          algebraic functions,
      ...
What is a function?




Definition
A function f is a relation which assigns to to every element x in a set D
a single elem...
Outline
.
Modeling
Examples of functions
   Functions expressed by formulas
   Functions described numerically
   Function...
The Modeling Process



            .                                                        .
                Real-world
...
Plato's Cave




                                                                 .   .   .     .      .     .

V63.0121.0...
The Modeling Process


            .                                                        .
                Real-world
 ...
Outline
.
Modeling
Examples of functions
   Functions expressed by formulas
   Functions described numerically
   Function...
Functions expressed by formulas




Any expression in a single variable x defines a function. In this case,
the domain is ...
Formula function example
Example
                 x+1
Let f(x) =           . Find the domain and range of f.
             ...
Formula function example
Example
                 x+1
Let f(x) =           . Find the domain and range of f.
             ...
Formula function example
Example
                 x+1
Let f(x) =           . Find the domain and range of f.
             ...
Formula function example
Example
                 x+1
Let f(x) =           . Find the domain and range of f.
             ...
No-no's for expressions




       Cannot have zero in the denominator of an expression
       Cannot have a negative numb...
Piecewise-defined functions
Example
Let                                         {
                                        ...
Piecewise-defined functions
Example
Let                                         {
                                        ...
Functions described numerically




We can just describe a function by a table of values, or a diagram.




              ...
Example


Is this a function? If so, what is the range?



               x f(x)
               1 4
               2 5
   ...
Example


Is this a function? If so, what is the range?


                                            . .
                ...
Example


Is this a function? If so, what is the range?


                                            . .
                ...
Example


Is this a function? If so, what is the range?



               x f(x)
               1 4
               2 4
   ...
Example


Is this a function? If so, what is the range?


                                            . .
                ...
Example


Is this a function? If so, what is the range?


                                            . .
                ...
Example


How about this one?



               x f(x)
               1 4
               1 5
               3 6




      ...
Example


How about this one?


                                            . .
                                          ...
Example


How about this one?


                                            . .
                                          ...
An ideal function




                    .   .   .   .   .   .
Why numerical functions matter




In science, functions are often defined by data. Or, we observe data
and assume that it...
Numerical Function Example

Here is the temperature in Boise, Idaho measured in 15-minute
intervals over the period August...
Functions described graphically

Sometimes all we have is the “picture” of a function, by which we
mean, its graph.




  ...
Functions described graphically

Sometimes all we have is the “picture” of a function, by which we
mean, its graph.




  ...
Functions described verbally




Oftentimes our functions come out of nature and have verbal
descriptions:
       The temp...
Outline
.
Modeling
Examples of functions
   Functions expressed by formulas
   Functions described numerically
   Function...
Monotonicity
Example
Let P(x) be the probability that my income was at least $x last year.
What might a graph of P(x) look...
Monotonicity
Example
Let P(x) be the probability that my income was at least $x last year.
What might a graph of P(x) look...
Monotonicity




Definition

       A function f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for
       any two poin...
Examples




Example
Going back to the burrito function, would you call it increasing?




                               ...
Examples




Example
Going back to the burrito function, would you call it increasing?

Example
Obviously, the temperature...
Symmetry




Example
Let I(x) be the intensity of light x distance from a point.

Example
Let F(x) be the gravitational fo...
Possible Intensity Graph


                                     y
                                     . = I(x)




      ...
Possible Gravity Graph
                                     y
                                     . = F(x)




          ...
Definitions




Definition

       A function f is called even if f(−x) = f(x) for all x in the domain of f.
       A func...
Examples




       Even: constants, even powers, cosine
       Odd: odd powers, sine, tangent
       Neither: exp, log


...
Outline
.
Modeling
Examples of functions
   Functions expressed by formulas
   Functions described numerically
   Function...
Classes of Functions




       linear functions, defined by slope an intercept, point and point, or
       point and slop...
Linear functions

Linear functions have a constant rate of growth and are of the form

                                   ...
Linear functions

Linear functions have a constant rate of growth and are of the form

                                   ...
Linear functions

Linear functions have a constant rate of growth and are of the form

                                   ...
Example
Biologists have noticed that the chirping rate of crickets of a certain
species is related to temperature, and the...
Solution




                                                                 .   .   .      .      .    .

V63.0121.006/0...
Solution

       The point-slope form of the equation for a line is appropriate
       here: If a line passes through (x0 ...
Solution

       The point-slope form of the equation for a line is appropriate
       here: If a line passes through (x0 ...
Solution

       The point-slope form of the equation for a line is appropriate
       here: If a line passes through (x0 ...
Solution

       The point-slope form of the equation for a line is appropriate
       here: If a line passes through (x0 ...
Other Polynomial functions



       Quadratic functions take the form

                                     f(x) = ax2 + ...
Other Polynomial functions



       Quadratic functions take the form

                                        f(x) = ax2...
Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?




                 ...
Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?

Solution
The general...
Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?

Solution
The general...
Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?

Solution
The general...
Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives

                                     −...
Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives

                                     −...
Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives

                                      ...
Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives

                                      ...
Other power functions




       Whole number powers: f(x) = xn .
                                                1
      ...
Rational functions




Definition
A rational function is a quotient of polynomials.

Example
                             ...
Trigonometric Functions




       Sine and cosine
       Tangent and cotangent
       Secant and cosecant




           ...
Exponential and Logarithmic functions




       exponential functions (for example f(x) = 2x )
       logarithmic functio...
Outline
.
Modeling
Examples of functions
   Functions expressed by formulas
   Functions described numerically
   Function...
Transformations of Functions



Take the squaring function and graph these transformations:
       y = (x + 1)2
       y =...
Transformations of Functions



Take the squaring function and graph these transformations:
       y = (x + 1)2
       y =...
Vertical and Horizontal Shifts




Suppose c > 0. To obtain the graph of
       y = f(x) + c, shift the graph of y = f(x) ...
Vertical and Horizontal Shifts




Suppose c > 0. To obtain the graph of
       y = f(x) + c, shift the graph of y = f(x) ...
Vertical and Horizontal Shifts




Suppose c > 0. To obtain the graph of
       y = f(x) + c, shift the graph of y = f(x) ...
Vertical and Horizontal Shifts




Suppose c > 0. To obtain the graph of
       y = f(x) + c, shift the graph of y = f(x) ...
Vertical and Horizontal Shifts




Suppose c > 0. To obtain the graph of
       y = f(x) + c, shift the graph of y = f(x) ...
Now try these




       y = sin (2x)
       y = 2 sin (x)
       y = e−x
       y = −ex




                             ...
Scaling and flipping




To obtain the graph of
       y = f(c · x), scale the graph of f                            by c
...
Scaling and flipping




To obtain the graph of
       y = f(c · x), scale the graph of f horizontally by c
       y = c ·...
Scaling and flipping




To obtain the graph of
       y = f(c · x), scale the graph of f horizontally by c
       y = c ·...
Scaling and flipping




To obtain the graph of
       y = f(c · x), scale the graph of f horizontally by c
       y = c ·...
Scaling and flipping




To obtain the graph of
       y = f(c · x), scale the graph of f horizontally by c
       y = c ·...
Outline
.
Modeling
Examples of functions
   Functions expressed by formulas
   Functions described numerically
   Function...
Composition is a compounding of functions in
succession




                                                   g
         ...
Composing




Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.




                                       ...
Composing




Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

Solution
f ◦ g(x) = sin2 x while g ◦ f(x) ...
Decomposing



Example
        √
Express x2 − 4 as a composition of two functions. What is its
domain?

Solution
         ...
Summary




       The fundamental unit of investigation in calculus is the function.
       Functions can have many repre...
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Lesson 1: Functions

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Functions are the fundamental concept in calculus. We discuss four ways to represent a function and different classes of mathematical expressions for functions.

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Lesson 1: Functions

  1. 1. Section 1.1–1.2 Functions V63.0121.006/016, Calculus I New York University May 17, 2010 Announcements Get your WebAssign accounts and do the Intro assignment. Class Key: nyu 0127 7953 Office Hours: TBD . . . . . .
  2. 2. Announcements Get your WebAssign accounts and do the Intro assignment. Class Key: nyu 0127 7953 Office Hours: TBD . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 2 / 54
  3. 3. Objectives: Functions and their Representations Understand the definition of function. Work with functions represented in different ways Work with functions defined piecewise over several intervals. Understand and apply the definition of increasing and decreasing function. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 3 / 54
  4. 4. Objectives: A Catalog of Essential Functions Identify different classes of algebraic functions, including polynomial (linear, quadratic, cubic, etc.), polynomial (especially linear, quadratic, and cubic), rational, power, trigonometric, and exponential functions. Understand the effect of algebraic transformations on the graph of a function. Understand and compute the composition of two functions. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 4 / 54
  5. 5. What is a function? Definition A function f is a relation which assigns to to every element x in a set D a single element f(x) in a set E. The set D is called the domain of f. The set E is called the target of f. The set { y | y = f(x) for some x } is called the range of f. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 5 / 54
  6. 6. Outline . Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry Classes of Functions Linear functions Other Polynomial functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 6 / 54
  7. 7. The Modeling Process . . Real-world . . m . odel Mathematical . Problems Model s . olve .est t . i .nterpret . Real-world . Mathematical . Predictions Conclusions . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 7 / 54
  8. 8. Plato's Cave . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 8 / 54
  9. 9. The Modeling Process . . Real-world . . m . odel Mathematical . Problems Model s . olve .est t . i .nterpret . Real-world . Mathematical . Predictions Conclusions S . hadows F . orms . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 9 / 54
  10. 10. Outline . Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry Classes of Functions Linear functions Other Polynomial functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 10 / 54
  11. 11. Functions expressed by formulas Any expression in a single variable x defines a function. In this case, the domain is understood to be the largest set of x which after substitution, give a real number. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 11 / 54
  12. 12. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 12 / 54
  13. 13. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solution The denominator is zero when x = 2, so the domain is all real numbers except 2. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 12 / 54
  14. 14. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solution The denominator is zero when x = 2, so the domain is all real numbers except 2. As for the range, we can solve x+2 2y + 1 y= =⇒ x = x−1 y−1 So as long as y ̸= 1, there is an x associated to y. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 12 / 54
  15. 15. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solution The denominator is zero when x = 2, so the domain is all real numbers except 2. As for the range, we can solve x+2 2y + 1 y= =⇒ x = x−1 y−1 So as long as y ̸= 1, there is an x associated to y. Therefore domain(f) = { x | x ̸= 2 } range(f) = { y | y ̸= 1 } . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 12 / 54
  16. 16. No-no's for expressions Cannot have zero in the denominator of an expression Cannot have a negative number under an even root (e.g., square root) Cannot have the logarithm of a negative number . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 13 / 54
  17. 17. Piecewise-defined functions Example Let { x2 0 ≤ x ≤ 1; f(x) = 3−x 1 < x ≤ 2. Find the domain and range of f and graph the function. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 14 / 54
  18. 18. Piecewise-defined functions Example Let { x2 0 ≤ x ≤ 1; f(x) = 3−x 1 < x ≤ 2. Find the domain and range of f and graph the function. Solution The domain is [0, 2]. The range is [0, 2). The graph is piecewise. . . 2 . . . 1 . . . . . 0 . 1 . 2 . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 14 / 54
  19. 19. Functions described numerically We can just describe a function by a table of values, or a diagram. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 15 / 54
  20. 20. Example Is this a function? If so, what is the range? x f(x) 1 4 2 5 3 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 16 / 54
  21. 21. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 5 3 6 . . 3 .. 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 16 / 54
  22. 22. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 5 3 6 . . 3 .. 6 Yes, the range is {4, 5, 6}. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 16 / 54
  23. 23. Example Is this a function? If so, what is the range? x f(x) 1 4 2 4 3 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 17 / 54
  24. 24. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 4 3 6 . . 3 .. 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 17 / 54
  25. 25. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 4 3 6 . . 3 .. 6 Yes, the range is {4, 6}. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 17 / 54
  26. 26. Example How about this one? x f(x) 1 4 1 5 3 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 18 / 54
  27. 27. Example How about this one? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 1 5 3 6 . . 3 .. 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 18 / 54
  28. 28. Example How about this one? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 1 5 3 6 . . 3 .. 6 No, that one’s not “deterministic.” . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 18 / 54
  29. 29. An ideal function . . . . . .
  30. 30. Why numerical functions matter In science, functions are often defined by data. Or, we observe data and assume that it’s close to some nice continuous function. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 20 / 54
  31. 31. Numerical Function Example Here is the temperature in Boise, Idaho measured in 15-minute intervals over the period August 22–29, 2008. . 1 . 00 . 9 .0. 8 .0. 7 .0. 6 .0. 5 .0. 4 .0. 3 .0. 2 .0. 1 .0. . . . . . . . 8 . /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29 8 8 8 8 8 8 8 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 21 / 54
  32. 32. Functions described graphically Sometimes all we have is the “picture” of a function, by which we mean, its graph. . . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 22 / 54
  33. 33. Functions described graphically Sometimes all we have is the “picture” of a function, by which we mean, its graph. . . The one on the right is a relation but not a function. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 22 / 54
  34. 34. Functions described verbally Oftentimes our functions come out of nature and have verbal descriptions: The temperature T(t) in this room at time t. The elevation h(θ) of the point on the equator at longitude θ. The utility u(x) I derive by consuming x burritos. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 23 / 54
  35. 35. Outline . Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry Classes of Functions Linear functions Other Polynomial functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 24 / 54
  36. 36. Monotonicity Example Let P(x) be the probability that my income was at least $x last year. What might a graph of P(x) look like? . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 25 / 54
  37. 37. Monotonicity Example Let P(x) be the probability that my income was at least $x last year. What might a graph of P(x) look like? . . 1 . .5 . 0 . . . $ .0 $ . 52,115 $ . 100K . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 25 / 54
  38. 38. Monotonicity Definition A function f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f. A function f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 26 / 54
  39. 39. Examples Example Going back to the burrito function, would you call it increasing? . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 27 / 54
  40. 40. Examples Example Going back to the burrito function, would you call it increasing? Example Obviously, the temperature in Boise is neither increasing nor decreasing. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 27 / 54
  41. 41. Symmetry Example Let I(x) be the intensity of light x distance from a point. Example Let F(x) be the gravitational force at a point x distance from a black hole. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 28 / 54
  42. 42. Possible Intensity Graph y . = I(x) . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 29 / 54
  43. 43. Possible Gravity Graph y . = F(x) . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 30 / 54
  44. 44. Definitions Definition A function f is called even if f(−x) = f(x) for all x in the domain of f. A function f is called odd if f(−x) = −f(x) for all x in the domain of f. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 31 / 54
  45. 45. Examples Even: constants, even powers, cosine Odd: odd powers, sine, tangent Neither: exp, log . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 32 / 54
  46. 46. Outline . Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry Classes of Functions Linear functions Other Polynomial functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 33 / 54
  47. 47. Classes of Functions linear functions, defined by slope an intercept, point and point, or point and slope. quadratic functions, cubic functions, power functions, polynomials rational functions trigonometric functions exponential/logarithmic functions . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 34 / 54
  48. 48. Linear functions Linear functions have a constant rate of growth and are of the form f(x) = mx + b. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 35 / 54
  49. 49. Linear functions Linear functions have a constant rate of growth and are of the form f(x) = mx + b. Example In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write the fare f(x) as a function of distance x traveled. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 35 / 54
  50. 50. Linear functions Linear functions have a constant rate of growth and are of the form f(x) = mx + b. Example In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write the fare f(x) as a function of distance x traveled. Answer If x is in miles and f(x) in dollars, f(x) = 2.5 + 2x . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 35 / 54
  51. 51. Example Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F and 173 chirps per minute at 80 ◦ F. (a) Write a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 36 / 54
  52. 52. Solution . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
  53. 53. Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
  54. 54. Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
  55. 55. Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6 So an equation for T and N is 1 1 113 T − 70 = (N − 113) =⇒ T = N − + 70 6 6 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
  56. 56. Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6 So an equation for T and N is 1 1 113 T − 70 = (N − 113) =⇒ T = N − + 70 6 6 6 37 If N = 150, then T = + 70 = 76 1 ◦ F 6 6 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
  57. 57. Other Polynomial functions Quadratic functions take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 38 / 54
  58. 58. Other Polynomial functions Quadratic functions take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0. Cubic functions take the form f(x) = ax3 + bx2 + cx + d . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 38 / 54
  59. 59. Example A parabola passes through (0, 3), (3, 0), and (2, −1). What is the equation of the parabola? . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 39 / 54
  60. 60. Example A parabola passes through (0, 3), (3, 0), and (2, −1). What is the equation of the parabola? Solution The general equation is y = ax2 + bx + c. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 39 / 54
  61. 61. Example A parabola passes through (0, 3), (3, 0), and (2, −1). What is the equation of the parabola? Solution The general equation is y = ax2 + bx + c. Each point gives an equation relating a, b, and c: 3 = a · 02 + b · 0 + c −1 = a · 22 + b · 2 + c 0 = a · 32 + b · 3 + c . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 39 / 54
  62. 62. Example A parabola passes through (0, 3), (3, 0), and (2, −1). What is the equation of the parabola? Solution The general equation is y = ax2 + bx + c. Each point gives an equation relating a, b, and c: 3 = a · 02 + b · 0 + c −1 = a · 22 + b · 2 + c 0 = a · 32 + b · 3 + c Right away we see c = 3. The other two equations become −4 = 4a + 2b −3 = 9a + 3b . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 39 / 54
  63. 63. Solution (Continued) Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6b . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 40 / 54
  64. 64. Solution (Continued) Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6b Subtract these two and we have −6 = −6a =⇒ a = 1. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 40 / 54
  65. 65. Solution (Continued) Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6b Subtract these two and we have −6 = −6a =⇒ a = 1. Substitute a = 1 into the first equation and we have −12 = 12 + 6b =⇒ b = −4 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 40 / 54
  66. 66. Solution (Continued) Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6b Subtract these two and we have −6 = −6a =⇒ a = 1. Substitute a = 1 into the first equation and we have −12 = 12 + 6b =⇒ b = −4 So our equation is y = x2 − 4x + 3 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 40 / 54
  67. 67. Other power functions Whole number powers: f(x) = xn . 1 negative powers are reciprocals: x−3 = 3 . x √ fractional powers are roots: x1/3 = 3 x. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 41 / 54
  68. 68. Rational functions Definition A rational function is a quotient of polynomials. Example x3 (x + 3) The function f(x) = is rational. (x + 2)(x − 1) . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 42 / 54
  69. 69. Trigonometric Functions Sine and cosine Tangent and cotangent Secant and cosecant . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 43 / 54
  70. 70. Exponential and Logarithmic functions exponential functions (for example f(x) = 2x ) logarithmic functions are their inverses (for example f(x) = log2 (x)) . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 44 / 54
  71. 71. Outline . Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry Classes of Functions Linear functions Other Polynomial functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 45 / 54
  72. 72. Transformations of Functions Take the squaring function and graph these transformations: y = (x + 1)2 y = (x − 1)2 y = x2 + 1 y = x2 − 1 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 46 / 54
  73. 73. Transformations of Functions Take the squaring function and graph these transformations: y = (x + 1)2 y = (x − 1)2 y = x2 + 1 y = x2 − 1 Observe that if the fiddling occurs within the function, a transformation is applied on the x-axis. After the function, to the y-axis. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 46 / 54
  74. 74. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units y = f(x) − c, shift the graph of y = f(x) a distance c units y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
  75. 75. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
  76. 76. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
  77. 77. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units to the right y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
  78. 78. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units to the right y = f(x + c), shift the graph of y = f(x) a distance c units to the left . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
  79. 79. Now try these y = sin (2x) y = 2 sin (x) y = e−x y = −ex . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 48 / 54
  80. 80. Scaling and flipping To obtain the graph of y = f(c · x), scale the graph of f by c y = c · f(x), scale the graph of f by c If |c| < 1, the scaling is a If c < 0, the scaling includes a . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
  81. 81. Scaling and flipping To obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f by c If |c| < 1, the scaling is a If c < 0, the scaling includes a . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
  82. 82. Scaling and flipping To obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f vertically by c If |c| < 1, the scaling is a If c < 0, the scaling includes a . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
  83. 83. Scaling and flipping To obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f vertically by c If |c| < 1, the scaling is a compression If c < 0, the scaling includes a . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
  84. 84. Scaling and flipping To obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f vertically by c If |c| < 1, the scaling is a compression If c < 0, the scaling includes a flip . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
  85. 85. Outline . Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry Classes of Functions Linear functions Other Polynomial functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 50 / 54
  86. 86. Composition is a compounding of functions in succession g . ◦f . x . f . . g . . g ◦ f)(x) ( f .(x) . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 51 / 54
  87. 87. Composing Example Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 52 / 54
  88. 88. Composing Example Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f. Solution f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 52 / 54
  89. 89. Decomposing Example √ Express x2 − 4 as a composition of two functions. What is its domain? Solution √ We can write the expression as f ◦ g, where f(u) = u and g(x) = x2 − 4. The range of g needs to be within the domain of f. To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 53 / 54
  90. 90. Summary The fundamental unit of investigation in calculus is the function. Functions can have many representations There are many classes of algebraic functions Algebraic rules can be used to sketch graphs . . . . . . V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 54 / 54
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