per. 4 Suh, Michael

183 days ago by michaelsuh

First Semester

  • Functions

    • Definition of a function

      • Domain
      • Range
      • Composition
      • Inverse

    • Types of Functions

      • Polynomial
        • Degree
        • Leading coefficient
        • Degree of polynomial using constant difference
      • Quadratic Functions
        • Minimum and maximum
        • Roots
        • Word Problems
      • Exponential Functions
        • Graph
        • Domain
        • Range
      • Square Root
        • Domain
        • Range
        • Composition
      • Cubic functions
      • Even/odd functions
      • Ceiling/floor function
        • Evaluate
        • Word problems

    • Transformations

      • Scale factors of points
      • Translation of points
      • Translations of functions

    • Sequence and series

      • Convergence and divergence
      • Limit
      • Recursive
      • Summation notations
      • Arithmetic series
      • Sum of infinite geometric series

The domain is the input of the function.

The range is the output of the function.
[factorial (x) for x in [0..10]] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800\right]
f(x) = x^2 - 9 [f(x) for x in [0..10]] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[-9, -8, -5, 0, 7, 16, 27, 40, 55, 72, 91\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[-9, -8, -5, 0, 7, 16, 27, 40, 55, 72, 91\right]
domain = [-5..5] [f(x) for x in domain] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[16, 7, 0, -5, -8, -9, -8, -5, 0, 7, 16\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[16, 7, 0, -5, -8, -9, -8, -5, 0, 7, 16\right]
[(x,f(x)) for x in domain] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(-5, 16\right), \left(-4, 7\right), \left(-3, 0\right), \left(-2, -5\right), \left(-1, -8\right), \left(0, -9\right), \left(1, -8\right), \left(2, -5\right), \left(3, 0\right), \left(4, 7\right), \left(5, 16\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(-5, 16\right), \left(-4, 7\right), \left(-3, 0\right), \left(-2, -5\right), \left(-1, -8\right), \left(0, -9\right), \left(1, -8\right), \left(2, -5\right), \left(3, 0\right), \left(4, 7\right), \left(5, 16\right)\right]
points([(x,f(x)) for x in domain]) 
       

The inverse is the equation is the mirrored image arcoss the x-axis.
F = [(x,f(x)) for x in domain] [(y,x) for (x,y) in F] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(16, -5\right), \left(7, -4\right), \left(0, -3\right), \left(-5, -2\right), \left(-8, -1\right), \left(-9, 0\right), \left(-8, 1\right), \left(-5, 2\right), \left(0, 3\right), \left(7, 4\right), \left(16, 5\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(16, -5\right), \left(7, -4\right), \left(0, -3\right), \left(-5, -2\right), \left(-8, -1\right), \left(-9, 0\right), \left(-8, 1\right), \left(-5, 2\right), \left(0, 3\right), \left(7, 4\right), \left(16, 5\right)\right]
points([(y,x) for (x,y) in F]) 
       
f(x) = 4*x + 5 f 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 4 \, x + 5
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 4 \, x + 5
var('x y') solve(x == 4*y+5, y) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = \frac{1}{4} \, x - \frac{5}{4}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = \frac{1}{4} \, x - \frac{5}{4}\right]

The compostition of a function is if for expample f and g are two functions then the composition of f and g is f(g(x)) = g(f(x)).
g(x) = 1/4*x + 5 g 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{4} \, x + 5
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{4} \, x + 5
f(g(x)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x + 25
\newcommand{\Bold}[1]{\mathbf{#1}}x + 25
g(f(x)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x + \frac{25}{4}
\newcommand{\Bold}[1]{\mathbf{#1}}x + \frac{25}{4}
[f(x) for x in [0..10]] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45\right]
[g(x) for x in _] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{25}{4}, \frac{29}{4}, \frac{33}{4}, \frac{37}{4}, \frac{41}{4}, \frac{45}{4}, \frac{49}{4}, \frac{53}{4}, \frac{57}{4}, \frac{61}{4}, \frac{65}{4}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{25}{4}, \frac{29}{4}, \frac{33}{4}, \frac{37}{4}, \frac{41}{4}, \frac{45}{4}, \frac{49}{4}, \frac{53}{4}, \frac{57}{4}, \frac{61}{4}, \frac{65}{4}\right]

List comprehension is a very useful way to express mathematical ideas. It is a way of creating lists from other lists.

It has the structure [f(x) for x in [some list]] where f(x) can be a previously defined function or just some general expression:
[factorial(x) for x in [0..10]] 
       
f(x) = x^2 + 2*x - 5 [f(x) for x in [0..10]] 
       
domain = [-10..10] [f(x) for x in domain] 
       
[(x,f(x)) for x in domain] 
       
points([(x,f(x)) for x in domain]) 
       

Here's an idea for illustrating inverses:
F = [(x,f(x)) for x in domain] [(y,x) for (x,y) in F] 
       
points([(y,x) for (x,y) in F]) 
       
f(x) = 2*x + 3 f 
       
var('x y') solve(x == 2*y+3, y) 
       
g(x) = 1/2*x - 3/2 g 
       
f(g(x)) 
       
g(f(x)) 
       
[f(x) for x in [0..10]] 
       
[g(x) for x in _] 
       

    • Extra topics we've covered:

      • Factorials
      • Permutations
      • Combinations
      • Binomial Theorem
      • Binomial Probability
 
       

Answer the following with

  • A - I know it's true.
  • B - I believe it's true.
  • C - I really don't know.
  • D - I believe it's false.
  • E - I know it's false.
 
  1. \sum_{x=1}^{n}x = \frac{n(n+1)}{2}
  2. \sum_{i=1}^{\infty}\frac{1}{2^i} = 1
  3. (p + q)^3 = \sum _{r=0}^{3} (_{3}C_{r})(p^{3-r})(q^{r})
  4. (p + q)^n = \sum _{r=0}^{n} (_{n}C_{r})(p^{n-r})(q^{r})
  5. The third term in the expansion of (p + q)^n is _{n}C_{2}p^{n-2}q^{2}.
  6. \sum_{r=0} ^{n} (_{n} C _{r}) = 1
  7. If you toss 10 coins into the air, the probability that 5 of them will land heads and 5 will land tails is 50\%.
  8. If you guess on 10 questions on a True/False test, the probability that you will guess 5 right and 5 wrong is 50\%.
  9. If you guess on 10 questions on a Multiple Choice test with 4 choices per question, the probability that you will guess all of them wrong is about 6\%.
  10. If you guess on 10 questions on a Multiple Choice test with 4 choices per question, the probability that you will guess 5 right and 5 wrong is about 6\%.
  11. If you toss a die 10 times, the probability that you will get 7 fours is 35%.
  12. \sum_{k=1}^{\infty} \frac{1}{3^k} = \frac{1}{2}
  13. \sum_{k=1}^{\infty} \frac{1}{n^k} = \frac{1}{n-1}