MAT170 SPR 2009 Material for 2 nd Quiz

1. Label Label the units of t and of population so you don’t forget to do it after solving: (ie: t: days, hours, pop: hundreds, millions…) 2. “The long run” is when t = the vertical asymptote if positive, or ∞ if not. Find the vertical asymptote, plug it in as t & solve for A. 3. If time is not limited by a vertical, asymptote, or or if the vertical asymptote isn’t positive, then the population will equal the horizontal asymptote. HOW TO DO THIS KIND OF PROBLEM: “The following rational function in hundreds models the population of a certain species of animal, where t is measured in days. What number does the population approach in the long run?”

1. Find the horizontal asymptote. 2. As the number of things made increases, the average cost approaches the horizontal asymptote. Graph it to see whether the price increases or decreases with quantity, and whether the difference between a small amount and a large amount is a problem for smaller businesses that can only produce small amounts. HOW TO DO THIS KIND OF PROBLEM: “The average cost of producing a thing is given by a function: when x is the number of things sold, identify the horizontal asymptote of the function and explain its meaning in this context.”

Coterminal = Given ± k(2π)Coterminal = Given ± k(2π) + if angle is negative - if angle is positive K ≈ Given /2π up downK ≈ Given /2π (round up if angle is negative, round down if angle is positive) Remember: 2π = 360° How to find Coterminal Angles:

How do you convert between radians and degrees? So by dimensional analysis: X° ( π / 180 ° ) = Θ radians And Θ radians ( 180 ° / π ) = X°

Linear speed of a point on a circle: Distance/time Where S = RΘ

Formula for length of an arc: Θ must be in radians

Formula for populations with continuous interest or growth:

Formula to figure interest compounded at a certain rate (discrete, not continuous):

x y x = a y

Y = log a xY = log a x

log b 1 = For any base b

log b b = For any base b

b log b x = For any base b

log b b x = For any base b

To change the base of a logarithm to any base a: Most calculators can’t graph y = log 3 x directly. But you can change the base to e and easily plot y = (ln x)÷(ln 3). (You could equally well use base 10.)

Rule: multiplication becomes addition:

Rule: division becomes subtraction:

Rule: exponent becomes multiplier:

The variable to the highest power in a polynomial function is the ___ of the function:

The coefficient of the variable to the highest power, is called:

When the degree of the polynomial is: Even Odd Both ends are the same > 0 Leading coefficient > 0: UP Both Ends UP x  ∞, f(x)  ∞ x  ∞, f(x)  ∞ x  -∞, f(x)  ∞ x  -∞, f(x)  ∞ <0 Leading coefficient <0 DOWN Both Ends DOWN x  ∞, f(x)  -∞ x  ∞, f(x)  -∞ x  -∞, f(x)  -∞ x  -∞, f(x)  -∞ The ends go opposite ways > 0 Leading coefficient > 0: UP RIGHT End UP x  ∞, f(x)  ∞ x  ∞, f(x)  ∞ x  -∞, f(x)  - ∞ x  -∞, f(x)  - ∞ <0 Leading coefficient <0 DOWN RIGHT End DOWN x  ∞, f(x)  -∞ x  -∞, f(x)  ∞

The graph touches the x-axis and turns around at R. R ‘s multiplicity is:

The graph crosses the x-axis at R. R ‘s multiplicity is:

11. Write equation, use 0 for y 0 = Qx n +x n-1 ….+P 2.2. Unless a zero has been given, find one with a calculator graph or: a.List all factors of P b.List all factors of Q c.Figure all possible zeros = P/Q d.Use synthetic division to test each possible zero until you find one with no remainder – it will be a factor. (X – that factor) = Write the equation again as 0 = (divisor)(quotient) 44. Use the quadratic formula with the quotient to find the other zeros Plug each factor into the original equation to be sure the answers are in the domain. How do you find the zeros of a polynomial?

11. Write equation, use 0 for y 0 = 2x 4 (x+3) 2 (x-7) For each factor in the equation, determine what makes it =0 2x 4 =0(x+3) 2 =0(x-7) 8 =0 x=0x+3=0x-7=0 x=-3x= For each zero you found: the multiplicity is equal to the number of the exponent. X=0x=-3x=7 multiplicity 4 multiplicity 2 multiplicity 8 How do you find the zeros of a single termed function?

A rational function is a function that can be written as the ratio of two polynomials where the denominator isn't zero.

To find the Vertical Asymptote:

To find the Horizontal Asymptote: Rewrite the function equation leaving out all but the products of X Like this: f(x) = 6x-2. (-2x-5)(7x-2) = 6x = 6x = 6. -2x * 7x -14x 2 -14x If you get a number, then y = that number, but if a variable remains, then consult rules.

Rule 1: After you reduce the function, if numerator (n) < denominator(m) then what is the horizontal asymptote?

Rule 2: After you reduce the function, if numerator (n) = denominator(d) then what is the horizontal asymptote? In other words: N Leading Coefficient of the Numerator D Leading Coefficient of the Denominator

Rule 3: After you reduce the function, if numerator (n) > denominator (d) then what is the horizontal asymptote? However, if n = d+1 then there is an oblique or slant asymptote