Mathematical Prerequisites

## Level of Mathematical Background Required

This course assumes mathematical facility in the following topics usually associated with "precalculus" and introductory statistics

1. Ordered Pairs, n-tuples
2. Linear_equation
3. Sets
4. Solving inequalities and equations
5. Properties of functions
6. Composite_functions
7. Matrices
8. Frequency_distribution
9. axioms Basic probability through simple conditional probability
10. Basics of Boolean algebra

You have some familiarity with these concepts:

1. Polynomial_functions
2. Rational_functions
3. Exponential_functions
4. Logarithmic_functions
5. Sequence]s and [[series_(mathematics) series]
6. Mathematical_induction
7. Limits

## Self Test

[[table]]
[[row]]
[[cell style="vertical-align: top;"]]
On the grid to the right …

1.

Q124. On the grid below …

…draw the line $y=2x + 1$
…shade the area for $x>5$ and the area for $y < 3$
…shade the area for y >= 2x + 1

Q125. Write an equation for the line passing through the points (3,2) and (0,5).

#### Sets and Logic

4. Write an equation for the line passing through the points (3,2) and (0,5).

If A = {red, blue, yellow, green} and B = {blue, green, violet, orange},

5. what is A B?

6. what is A intersection B?

If A and B are sets that are neither identical nor disjoint, sketch them in a VENN Diagram, and label the areas representing…

7. …A and B

8. …A but not B

#### Inequalities

9. If a > b and b > c what do we know about a and c?
 $a > c$ $a < c$ $a = c$ $a \leq c$ $a \geq c$ depends what c is

If a < b then

10.

(1)
\begin{align} a + c \bigcirc b + c \end{align}

11.

(2)
\begin{align} a − c \bigcirc b − c \end{align}

12. If $3 x + 2 y = 7$ and $x + y = 10$, what are $x$ and $y$?

13. If $a > 0$ and $b > 0$ and $a < b$ then $\frac{1} {a} \bigcirc \frac{1}{b}$?

#### Algebra

14. Simplify $(ab + ac)=$

15. Simplify $\frac{(a + b) (3a + 4c)}{(a + b) (a - b)}=$

16. Simplify $\frac{(a + b) (3a + 4c)}{(a^2 - b^2)}=$

17. Simplify $\frac{(a - b) (c - d)}{(d - c) (b - a)}=$

#### Functions

18. If $f(x) = x^2$, what is the range of $f(x)$?

19. If $1000 = 10^x$ what is $x$?

20. If $f(x) = 3x + 2$ and $g(x) = x^2$, what is f(g(4))?

#### Notation and such

21. Consider the matrix below. What is element a2,3?
$A = \left( \begin{tabular} {c c c} 3 & 4 & 2 \\ 1 & 1 & 7 \\ 1 & 8 & 8 \\ 4 & 0 & 5 \end{tabular} \right)$
22. If $S_{n+1} = 2 S_n + 3$, and $S_4 = 33$, what is $S_5$?

#### Matrices

23. $\left( \begin{tabular} {c c c} 3 & 4 & 2 \\ 1 & 1 & 7 \\ \end{tabular} \right) \times \left( \begin{tabular} {c c} 3 & 4 \\ 1 & 1 \\ 1 & 8 \end{tabular} \right) =$

#### Probability and Statistics

24. Suppose we have age, years of education, and income data on four individuals:

Unsupported math environment "tabular"

Express the third individual's data as an ordered 3-tuple.

25. What is the median income?

26. Sketch a plot of income as a function of years of education.

27. Sketch a normal distribution

28. Sketch a uniform distribution

29. If you throw a single six-sided fair die 300 times, what will the frequency distribution of the results look like? What do we call this distribution?

30. If you throw two six sided fair dice 300 times, what will the frequency distribution of the results look like? What do we call this distribution?

31. If you measure the height of all the students at Mills College what will the frequency distribution of the results look like? What do we call this distribution?

32. What are the possible outcomes from throwing a single die?

33. If we write P(event) to represent the probability of event, what is P(outcome is odd)?

34. What is P(outcome > 4)?

35. What is the probability that the outcome is both odd and greater than four?

36. What is the probability that the outcome is either odd or greater than four?

#### Logs and exponents

37. An investment of $100 earns 10% annual simple interest. What is it worth in three years? 38. If$y = 10^x$, what is$\log_{10} y$? 39.$10^3 \times 10^6 =$40.$a^b + a^c =$41.$5 a^5 + 3 a^3 =\$

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