Math 250 Chapter 1 Sections 1.1 to 1.9

exponential decay

0 < a < 1

properties of the natural logarithm

1. ln(AB)=lnA+lnB 2. ln(A/B)=lnA-lnB 3. ln(A^P)=plnA 4. lne^x=x 5. e^lnx=x ln1=0 bc e^0=1 lne=1 bc e^2=e

continuous rate of change

P = (P0)(e^kt) ; k is the rate of increase or decrease

exponential function

P = (Po)(a)^t ; output is always greater than 0; P0 is the initial quantity; a is the factor by which P changes when t increases by 1; a = 1 + r (r is the decimal representation of the percent rate of change; may be positive for growth or negative for decay)

constant % rate of change

P = Po a^t

profit function

P(q) = R(q) - C(q) ; profit = revenue - cost

exponential growth

a > 1

exponential function

a constant percent, or relative, rate of change

Function

a rule that takes certain numbers as inputs and assigns each a definite output number

average velocity

change in distance / change in time

Output

dependent variable

decreasing exponential function

elimination of a drug from the body approx. 40% eliminated every hour Q = f(x) = 250 (0.6)^t 0.6 bc 60% is left in your body each hour how much is left after 4 hours? f(4) = 250(0.6)^4 = 32.4 mg

average rate of change

f(b) - f(a) / b - a

proportionality

f(x) = kx^p we say that f(x) is directly proportional to x if there is a nonzero constant k

f(x) = x+1 =x + h + l - (x + 1) =x + h + 1 - x -1 f(x+h) - f(x) =h

f(x) = x+1 What is f(x+h) - f(x)?

directly proportional

f(x) = x^2 k=1 p=2 f(x) = 13x^5 k=13 p=5 f(x) = square root of x =x^1/2 f(x) = square root of x cubed =x^1/3 (cubed = 1/3)

f(x) = x^2 =x^2 + 2xh + h^2 - x^2 = 2xh + h^2 f(x+h) = (x +h)^2 = x^2 + 2xh + h^2

f(x) = x^2 What is f(x+h) - f(x)?

fog = f(g(x)) = f(x+3) =(x+3)^2 =x^2+6x+9 gof = g(f(x)) = g(x^2) = x^2+3

f(x) = x^2 g(x) = x + 3

compose f with another function g

fog = f(g(x)) f is the outside function (g(x)) is the inside function

supply curve

for a given item, relates the quantity, q, of the item that manufacturers are willing to make per unit time to the price, p, for which the item can be sold

inversely proportional

g(x) = 1/x = x^-1 g(x) = 1/ square root of x = x^-1/2 (square root = 1/2)

exponential functions

g(x) = 2^x g(0) = 2^0 = 1 g(1) = 2^1 = 2 g(10) = 2^10 = 1024 g(-1) = 2^-1 = 1/2 g(-3) = 2^-3 = 1/8 g(-10) = 2^-10 = 1/1024

cost function

gives the total cost of producing a quantity, q, of some good; C(q) = C0 (fixed cost) + Mq (marginal cost; slope)

revenue function

gives the total revenue received by a firm from selling a quantity, q, of some good; R(q) = Mq (marginal revenue; price multiplied by quantity)

concave down

graph of a function bends downward as we move left to right

concave up

graph of a function bends upward as we move left to right

linear function

graphs are straight lines; constant rate of change

budget constraint

how much company or person can spend to have a certain amount of one thing and a certain amount of another thing

Input

independent variable

increasing exponential function

invested $1,000 at 5% interest Q = Ab^t Q = 1,000 (1.05)^t

constant of proportionality

k

natural logarithm

lnx = c means e^c = x

slope

m = y2 - y1 / x2 - x1

half-life

of an exponentially decaying quantity is the time required for the quantity to be reduced by a factor of one half

doubling time

of an exponentially increasing quantity is the time required for the quantity to double

power function

one quantity is proportional to the power of another quantity; f(x) is proportional to a constant power of x; if k is the constant of proportionality, and if p is the power, then --> f(x) = kx^p

inversely proportional

p < 0

directly proportional

p > 0

equilibrium price and equilibrium quantity

point where supply and demand cross; the market naturally settles to the equilibrium point

break-even point

point where the profit is zero and revenue equals cost; R(q) = C(q)

slope

rate; m

demand curve

relates the quantity, q, of an item demanded by consumers per unit time to the price, p, of the item

velocity

the average rate of change of height with respect to time

Domain

the set of all input numbers

Range

the set of resulting output numbers

vertical intercept

value of y when x is zero; b

Decreasing function

values of f(x) decrease as x increases

Increasing function

values of f(x) increase as x increases

relative change

without units; change in P / P0 (initial) = P1 - P0 / P0

linear function

y = f(x) = b + mx

power function

y = x/5 --> 1/5x y = 3/x^2 --> 3(1/x^2) --> 3x^-2 y = 3.5 ^x --> exponential function, so it can't be made to look like a power function y = (5x)^3 --> (5^3)(x^3) --> 125x^3 y = (3x^5)^2 --> 3^2(x^5)^2 --> 9x^10

point-slope form

y2 - y1 = m (x2 - x1)