unit 2 Flashcards
the constant rule
- if f(x) = c, where c is constant, then f’(x) = 0
- f(x) = 3, f’(x) = 0
what does f’(x) mean?
slope
the power rule
- if f(x) = x^n, where n is a natural number, then f’(x)=nx^n-1
- f(x)=x^12, f’(x) = 12n^11
the constant multiple rule
- if f(x)=cf(x), where c is a constant, then f’(x) = cf(x)
- f(x)=5x^12, f’(x)=5(x^12) = 5(12x^11)
the sum rule
- if f(x) and g(x) differentiable, and h(x) = f(x) + g(x)
then…
h’(x) = f’(x) + g’(x)
the difference rule
- if f(x) and g(x) differentiable, and h(x) = f(x) - g(x)
then…
h’(x) = f’(x) - g’(x)
what must you do if ur answer has a fraction exponent?
turn it into a radicand
what must u do if ur answer has a term w/ negative exponent?
move the exponent down and make it positive
do we use chain rule or product rule first?
- product rule first
- use chain rule to derive terms
how do u find revenue in terms of price increases?
- make eqn for price
- make eqn for items/ppl
- multiply both eqns by each other
when finding revenue in terms of price increases, how do u make a simplified form expression?
derive it using product rule
when finding revenue in terms of price increases, how do u find ROC and items at particular price?
- set price eqn = given price, isolate for variable
- plug the variale into items eqn to get total items
- plug variable into derived simplified expression to get price per increase
***rmbr to put 3. into units as $/price increase
product rule
h’(x) = h’(x)g(x) + h(x)g’(x)
steps for using product rule
- write statement
- find limits on side and plug in
- distribute brackets
- combine like terms
*factoring not needed
what happens if u have #sqrtvariable^n?
- n/#
- the variable alr there is numerator, number from radicand becomes denom
if u derive something nd the final ans has a constant tht can be divided by the constant at the bottom, wht do u do
REDUCE IT
when using the constant multiple rule/deriving long eqns w/ adding/subtracting, wht must u rmbr to do
- actually derive everything
- turn everything into fractions to make it easy to multiply (denom of 1 on all)
- *write f’(x) WITH apostrophe
- DON’T multiply both brackets, its nawt multiplication
if u have two terms connected by + , one is fraction where numerator is negative, wht do u do?
- get rid of the + sign
- move the “-“ to the middle instead
if u have more than 2 brackets multiplied by each other, wht do u do?
add more letters to product rule
- h(x)g(x)j(x) …etc
x(x-1)(6x+3)
- x counts as a term
- use product law but with three terms
two tangent line equations at x-point given, need eqn of tangent to curve of y=f(x)g(x) at x-point given
- make sketches, based on pos/neg m-value
- find f(x) and g(x) by plugging x-value into both tangent eqns: write y = ### = # = f(x). find f’(x) and g’(x) by deriving both tangent eqns.
- find m-value by plugging in vals from step 2 into y=f(x)g(x). write y’ = # = m
- make tangent eqn. plug step 2 vals into y=f(x)g(x) to get y-value. plug m, y, and x into y=mx+b and isolate for b.
*final equation w/o y and x inputted.
displacement
- distance an object has moved from the origin over a period of time
- s(t)
- units: m
velocity
- rate of change of displacement (s) with respect to time
- s’(t) = v(t)
units: m/s
acceleration
- rate of change of velocity (v) with respect to time
- s’‘(t) = v’(t) = a(t)
- units: m/s^2
what is a(t) in relation to displacement?
the second derivative
describe the relationship of displacement, velocity, and acceleration
- derivative of displacement = velocity
- derivative of velocity = acceleration
how to find velocity at a particular time given displacement equation
- make velocity eqn by finding derivative of given eqn. write v(t)=f’(x) and ADD UNITS to eqn!
- plug in the given time to get the velocity at the time, add units
- final ans. if negative, put direction down in brackets. if pos, goes upwards.
how do u calculate when a tossed object is momentarily at rest?
- rest means no ROC in displacement, so use velocity eqn previously derived
- set velocity equation to 0 and isolate for time
- at ___ seconds, the object is at rest
**check that time is in the domain.
when is arrow moving upward/downward?
- draw sketch
- recognize that axis of symm occurs at the momentary rest area BC that is the vertex
- when time is 0 to vertex, it’s going up, when time is greater than vertex, it’s going downwards
max height of arrow/height of arrow at time of momentary rest
- plug the time (at axis of symm) into the eqn for displacement/height
- answer with proper units. “at ___ sec, the max height is ___.”
find when the arrow hits the ground
- set displacement equation to 0
- use quadratic equation: (-b+-sqrt(b^2-4ac))/2a
- use positive time
**write “inadmissable” on -ve
what must u do when using quadratic eqn
write INADMISSABLE under the negative time
how to find velocity when arrow hits the ground
- plug in the time where arrow hits ground (prev. found) into the velocity eqn
- solve, add units and direction
when is object speeding up/slowing down
- speeding up: v(t)a(t) greater than 0
- slowing down: v(t)a(t) less than 0
when i object moving in pos/neg direction
- pos: v(t) greater than 0
- neg: v(t) less than 0
when is velocity increasing/decreasing
- increase: a(t) greater than 0
- decrease: a(t) less than 0
when do we use chain rule
(many terms)^n OR sqrt(many terms)
what is chain rule
- (n)(orig eqn)^n-1 (derivative of orig eqn)
steps of chain rule
- plug everything in
- expand derivative UNLESSS using product/quotient rule after
quotient rule
- use when fraction
f’(x) = [f’(x)g(x) - f(x)g’(x)] / [g(x)]^2 - subtract instead
- denom is denom
find point on graph where tangent is horizontal, given eqn
- derive original eqn
- set derivation to 0 (bc horizontal tangent means m=0), cross multiply AND FACTOR
- set each term to 0 and isolate for x
- plug isolated x-value into original eqn to get pt. f(x) = y is the point
revenue function
R(x) = xp(x)
- x: number of units of product/service sold
- p(x): price per unit
cost function
C(x) is the total cost of producing x units of prodt/service
- cost of labour
profit function
P(x) = R(x) - C(x)
- total profit made, diff between revenue and cost
what does marginal mean in business?
find the derivative
demand function
p(x)
- p: # of units of prodt/services tht can be sold at price of x
- x: price
making demand and revenue eqn
DEMAND
1. isolate n in units eqn
2. plug eqn into p(x) price eqn
REVENUE
1. xp(x)
2. expand x to rest of eqn
marginal revenue
- R’(x) = derive
- R’(#) = plug in
what does it mean when marginal revenue is 0
- revenue is at max
- selling more will decrease revenue
what must u include once you’ve found marginal profit?
/PER ITEM
**i.e. $16.20 per big mac
