Math 109: Midterm 1 Fall 2011 Julia Chen

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About me Julia Chen , 2A Biotech/CA Email: julia.qy.chen@gmail.com Note: This course was differently structured last year Wise words of advice: Do not donate blood 3 hours before your midterm. If you have a graphing calculator, use it to check your answers!

Agenda Chapter 0 (30 min) Exponents, algebraic expressions, factoring, equations, Chapter 1 (30 min) Word problems, inequalities, absolute value, summation, sequences Chapter 2 (30 min) functions, graphing, 3D

0.3 Exponents and Radicals In an, a is the base and n is the exponent. Evaluate it by multiplying a by itself n number of times Example: 34 = 3 x 3 x 3 x 3 = 81 A radical is the opposite of an exponent. The expressionn√ais a radical, also written as a1/n. Therefore, the nth root of a real number = the number to the power of the inverse of n Exponent is a fraction - Think: the numerator is how many times you’re putting it together, the denominator is how many times you’re breaking it down.

Exponent and Radical Rules Simplify Simplify

Solutions Example: Example:

0.4 Operations with Algebraic Expressions Algebraic expressions have variables and coefficients and can contain one or many terms Terms are separated by addition or subtraction Like terms differ only in their coefficients. They can be added or subtracted from each other. Special algebraic expressions Quadratic Difference of squares Perfect square Algebraic expressions can be divided. Sometimes, long division is used.

0.5 Factoring Factors are expressions that are multiplied to make a larger expression In c=ab, a and b are both factors Always factor as completely as possible Look for perfect squares, difference of squares, perfect cubes Decomposition method for quadratics If (ax2+ bx + c) factors into (x+q) and (x+p), then qp = c and q+p = b

0.6 Equations in Fractions We can simplify algebraic expressions by multiplying top and bottom by the same number Factor the polynomials, cancel out like terms When adding and subtracting fractions, make one large fraction by finding the lowest common denominator (LCD). Then simplify numerator When dividing expressions in fractions, invert and multiply Remember your limits! Denominators can’t be 0

0.7 Equations An equation is composed of two expressions that are equal To solve an equation means to find all values of each variable These values are called the solutions of the equation When only one variable solutions is called a root The set of all solution is called the solution set Equivalent equations have the same solution sets. To obtain an equivalent equation, perform the same function(+,-,x,/) to both sides of the equation NOTE: raising both sides to a power changes the solution set

0.8 Quadratic Equations These are written in the form ax2 + bx+ c = 0 These can be solved by factoring Quadratic formula is used when factoring is too difficult

1.1 Applications of Equations Mathematical equations are used to model practical problems Always define your variables (let x be…) Example problems could include: Total costs (= fixed costs and variable costs) Total revenue (price x quantity) Profit (revenue – total costs) Always finish word problems with a concluding sentence

Word problem example A total of $10 000 was invested in two business ventures, A and B. At the end of the first year, A and B yielded returns of 6% and 534%., respectively, on the original investment. How was the original amount allocated if the total amount earned was $588.75. let x represent the original amount allocated in A let 10000-x represent the original amount allocated in B 0.06x+10000-x0.0575=588.75 0.06x+575-0.0575x=588.75 0.0025x=13.75 x=5500 ∴$5500 was the original amount of A, and $ 4500 was the original amount of B

1.2 Linear Inequalities Inequality: a statement of one quantity in relation(>, ≥, =, <, ≤) to another quantity A closed interval: a set of all real numbers for x Example: b ≥ x ≥ a where endpoints are a and b If a and b are in the interval, denoted as [a,b] If a and b are not in the interval, denoted as (a,b)

Rules for inequalities 1. We can (+,-,x,/) the same number from both sides If multiply/divide by negative number, inequality switches to opposite sign 2. We can put 1 over both sides (reciprocal of each) inequality switches to opposite sign 3. If both sides are bigger than 0, we can raise both sides to the same exponent 4. We can replay any side of an inequality by an equivalent expression by simplifying

Linear Inequality example Solve Solve 2x-3<4 x-3<2 x<5 ∴x ∈(-∞,5) Always remember to state all possible values

1.3 Applications of Inequalities Word problem profit example: 1. The combined cost for labour and materials is $21 per heater. Fixed costs are $70000. If the selling price of a heater is $35, how many must be sold for the company to earn a profit? Let q be the number of heaters that must be sold. Then their cost is 21q and the total cost for the company is 21q + 70000. The total revenue from the sale of q heaters will be 35q. 21q+70000 < 35q 70000/14 < 14q/14 5000 < q ∴the company must sell more than 5000 heaters

1.4 Absolute Value Equation The absolute value of a number is the distance that number is away from 0 (always positive) Is denoted |x|. So if x = -5, |x| = 5 because -5 is 5 numbers away from 0 Example: Solve |3 – 2x| ≤ 5 It’s like solving 2 inequalities simultaneously Get rid of abs sign: -5 ≤ 3 - 2x ≤ 5 Isolate x: -5-3 ≤ -2x ≤ 5 – 3 -8 ≤ -2x ≤ 2Simplify (÷by -2) 4 ≥ x ≥ -1 Rewrite -1 ≤ x ≤ 4

1.5 Summation Notation A way to express the sum of a sequence of numbers from m to n inclusively: Rules: Factor constants Commutative Change boundaries Break down

Summation Equations & Examples 1) 2) 3) Examples: Solve: Solve

1.6 Sequences A string of numbers that have a pattern Notation: n is the last number and the length of the sequence We write ak for the kth number of the sequence An infinite sequence is written As k approaches infinity (a very, very large number), it will approach a limit, L A recursively defined sequence uses previous numbers in the sequence to generate numbers Ex. Fibonnaci Sequence

Arithmetic and Geometric Sequences Arithmetic sequence: starting with a value and always adding another fixed value kth term of an arithmetic sequence (bk) is given by bk= (k − 1)d + a Where a is the original value and d is the difference Geometric sequence: starting with a value and always multiplying by another fixed value recursive! The kth term of a geometric sequence (ck) is given by ck= ark−1

Sums of Sequences Sequences can be added together (summation notation) aka. series The sum of the first n terms of an arithmetic sequence (bk) with first term a and common difference d is given by The sum of the first n terms of a geometric sequence (ck) with first term a and common ratio r is given by where r ≠1

More Sequence properties Sum of an infinite geometric sequence If |r|>1 and rbecomes a very large number, Sn ∞ If |r|<1 and r becomes a very large number, the values in the sequence will become approach 0 This means Sn  (a/1-r) This is taken directly from the equation

2.1 Functions Functions are denoted f(x) For every x-value, there is one y-value All x-values are in the domain, y-values in range In a rational function (function in fractions), consider if there is any value of x that would make the denominator 0. The domain can’t include that value. Two functions are equal if they have the same domain, and for every value of x, f(x) = g(x)

Demand and Supply Functions The demand equationp = f(q) describes the relationship between the price per unit p of a certain product and the number of units q of the product that consumers will buy (demand) per week at the stated price. The supply equationp = g(q) describes the price per unit p of a certain product that producers are willing to supply q units of per week at the stated price. The domain of these functions is q≥ 0 since q is a number of units which cannot be negative.

2.2 Special Functions Constant Functions: f(x)=c looks like a straight, horizontal line Polynomial Functions In the form f(x) = cnxn + cn−1 xn−1 + . . . + c1x + c0 Rational Functions A quotient of polynomials *watch out for restrictions on x* Case-defined function Kind of like piecewise, no specific pattern

Factorial Notation Denotated r! and represents the product of all positive integers from 1 to r Used in probability where there is a total amount (n) and a selected group (r) where n>r

2.3 Combinations of Functions Sometimes, functions can be combined. Can be added, subtracted, multiplied, divided (f – g)(x) = f(x) – g(x) , same for products, quotients and sums Composition Notation: (f o g) means putting g(x) as x in f(x) f(x) = x2and g(x) = x+1, (f o g)(x) = (x+1)2 Associative: (f o g)o h = f o (g o h)

2.4 Inverse Functions An inverse function is notated by f -1 Satisfies f o g = identity function = g o f Think: two functions cancelling each other out When inverse exists, function must be one-to-one One-to-one means that no two inputs (x) can produce the same output (y) Finding the inverse: make an equation with y in terms of x, exchange all y and x and re-arrange to for y in terms of x

2.5 Graph in Rectangular Coordinates Consist of y and x axis, Intercepts are where the function touches an axis ordered pairs are labeled (x,y) Find domain and range Line tests are performed Vertical and Horizontal – be on the lookout for functions that intersect line more than once

2.6 Symmetry If there is symmetry about the y-axis, the values x and –x will give the same output. Therefore, the two sides are mirror images Similarly, if there is symmetry about the x-axis, the values yand –ywill give the same output. A graph is symmetrical about the origin if f(-x) = -f(x) (example: cubic) A function and its inverse will be symmetrical about the line y = x

2.7 Translations and Reflections y = f(x) + c shift c units upward y = f(x) − c shift c units downward y = f(x − c) shift c units to the right y = f(x + c) shift c units to the left y = −f(x) reflect about x-axis y = f(−x) reflect about y − axis y = cf(x), c > vertically stretch away from x-axis by a factor of c y = cf(x), c < 1 vertically shrink toward x- axis by a factor of c

2.8 Functions of Several Variables If f is a function of two variables, we can write z = f(x, y) where the domain of f is the set of all ordered pairs (x, y) for which the expression for z is defined In other words, it’s a function that has two variables To find z, plug in the values of (x, y) 3-dimensional planes (use your hand!) Sketching a plane in 3 steps Draw a corner, take turns finding x-axis (set y and z to 0) , y-axis, and z-axis intercepts

2.8 Traces Example: z = x2 y-axis can be anything Shape: parabola Example: x2 + y2 + z2 = 25 Find radius first (square root of 25) no value will ever be bigger than the radius Example: z = 4 - x2 Look at the shape (open down parabola) Look at the maximum

Best of luck on your midterm!

Math 109: Midterm 1 Fall 2011 Julia Chen