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)
