FST First Semester

182 days ago by sarahbaekk

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
# Def of function: A variable so related to another that for each value assumed by one there is a value determined for the other. # Range: the "range" is the set of all possible values of a function for the values of the variable. # Domain: The set of all possible values of an independent variable of a function. # Composition: Set of numbers that are combined # Inverse: Of or relating to an inverse or an inverse function. 
       

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]] 
        
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
f(x) = x^2 + 2*x - 5 [f(x) for x in [0..10]] 
        
[-5, -2, 3, 10, 19, 30, 43, 58, 75, 94, 115]
[-5, -2, 3, 10, 19, 30, 43, 58, 75, 94, 115]
domain = [-10..10] [f(x) for x in domain] 
        
[75, 58, 43, 30, 19, 10, 3, -2, -5, -6, -5, -2, 3, 10, 19, 30, 43, 58,
75, 94, 115]
[75, 58, 43, 30, 19, 10, 3, -2, -5, -6, -5, -2, 3, 10, 19, 30, 43, 58, 75, 94, 115]
[(x,f(x)) for x in domain] 
        
[(-10, 75), (-9, 58), (-8, 43), (-7, 30), (-6, 19), (-5, 10), (-4, 3),
(-3, -2), (-2, -5), (-1, -6), (0, -5), (1, -2), (2, 3), (3, 10), (4,
19), (5, 30), (6, 43), (7, 58), (8, 75), (9, 94), (10, 115)]
[(-10, 75), (-9, 58), (-8, 43), (-7, 30), (-6, 19), (-5, 10), (-4, 3), (-3, -2), (-2, -5), (-1, -6), (0, -5), (1, -2), (2, 3), (3, 10), (4, 19), (5, 30), (6, 43), (7, 58), (8, 75), (9, 94), (10, 115)]
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] 
        
[(75, -10), (58, -9), (43, -8), (30, -7), (19, -6), (10, -5), (3, -4),
(-2, -3), (-5, -2), (-6, -1), (-5, 0), (-2, 1), (3, 2), (10, 3), (19,
4), (30, 5), (43, 6), (58, 7), (75, 8), (94, 9), (115, 10)]
[(75, -10), (58, -9), (43, -8), (30, -7), (19, -6), (10, -5), (3, -4), (-2, -3), (-5, -2), (-6, -1), (-5, 0), (-2, 1), (3, 2), (10, 3), (19, 4), (30, 5), (43, 6), (58, 7), (75, 8), (94, 9), (115, 10)]
points([(y,x) for (x,y) in F]) 
       
 
       
f(x) = 3*x + 19 f 
        
x |--> 3*x + 19
x |--> 3*x + 19
var('x y') solve(x == 10*y+7, y) 
        
[y == 1/10*x - 7/10]
[y == 1/10*x - 7/10]
g(x) = 2/3*x - 5/4 g 
        
x |--> 2/3*x - 5/4
x |--> 2/3*x - 5/4
f(g(x)) 
        
2*x + 61/4
2*x + 61/4
g(f(x)) 
        
2*x + 137/12
2*x + 137/12
[f(x) for x in [0..10]] 
        
[3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]
[3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]
[g(x) for x in _] 
        
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

    • 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}