unit 2 Flashcards
(50 cards)
1
Q
the constant rule
A
- if f(x) = c, where c is constant, then f’(x) = 0
- f(x) = 3, f’(x) = 0
2
Q
what does f’(x) mean?
A
slope
3
Q
the power rule
A
- 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
4
Q
the constant multiple rule
A
- 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)
5
Q
the sum rule
A
- if f(x) and g(x) differentiable, and h(x) = f(x) + g(x)
then…
h’(x) = f’(x) + g’(x)
6
Q
the difference rule
A
- if f(x) and g(x) differentiable, and h(x) = f(x) - g(x)
then…
h’(x) = f’(x) - g’(x)
7
Q
what must you do if ur answer has a fraction exponent?
A
turn it into a radicand
8
Q
what must u do if ur answer has a term w/ negative exponent?
A
move the exponent down and make it positive
9
Q
do we use chain rule or product rule first?
A
- product rule first
- use chain rule to derive terms
10
Q
how do u find revenue in terms of price increases?
A
- make eqn for price
- make eqn for items/ppl
- multiply both eqns by each other
11
Q
when finding revenue in terms of price increases, how do u make a simplified form expression?
A
derive it using product rule
12
Q
when finding revenue in terms of price increases, how do u find ROC and items at particular price?
A
- 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
13
Q
product rule
A
h’(x) = h’(x)g(x) + h(x)g’(x)
14
Q
steps for using product rule
A
- write statement
- find limits on side and plug in
- distribute brackets
- combine like terms
*factoring not needed
15
Q
what happens if u have #sqrtvariable^n?
A
- n/#
- the variable alr there is numerator, number from radicand becomes denom
16
Q
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
A
REDUCE IT
17
Q
when using the constant multiple rule/deriving long eqns w/ adding/subtracting, wht must u rmbr to do
A
- 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
18
Q
if u have two terms connected by + , one is fraction where numerator is negative, wht do u do?
A
- get rid of the + sign
- move the “-“ to the middle instead
19
Q
if u have more than 2 brackets multiplied by each other, wht do u do?
A
add more letters to product rule
- h(x)g(x)j(x) …etc
20
Q
x(x-1)(6x+3)
A
- x counts as a term
- use product law but with three terms
21
Q
two tangent line equations at x-point given, need eqn of tangent to curve of y=f(x)g(x) at x-point given
A
- 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.
22
Q
displacement
A
- distance an object has moved from the origin over a period of time
- s(t)
- units: m
23
Q
velocity
A
- rate of change of displacement (s) with respect to time
- s’(t) = v(t)
units: m/s
24
Q
acceleration
A
- rate of change of velocity (v) with respect to time
- s’‘(t) = v’(t) = a(t)
- units: m/s^2
25
what is a(t) in relation to displacement?
the second derivative
26
describe the relationship of displacement, velocity, and acceleration
- derivative of displacement = velocity
- derivative of velocity = acceleration
27
how to find velocity at a particular time given displacement equation
1. make velocity eqn by finding derivative of given eqn. write v(t)=f'(x) and ADD UNITS to eqn!
2. plug in the given time to get the velocity at the time, add units
3. final ans. if negative, put direction down in brackets. if pos, goes upwards.
28
how do u calculate when a tossed object is momentarily at rest?
1. rest means no ROC in displacement, so use velocity eqn previously derived
2. set velocity equation to 0 and isolate for time
3. at ___ seconds, the object is at rest
**check that time is in the domain.
29
when is arrow moving upward/downward?
1. draw sketch
2. recognize that axis of symm occurs at the momentary rest area BC that is the vertex
3. when time is 0 to vertex, it's going up, when time is greater than vertex, it's going downwards
30
max height of arrow/height of arrow at time of momentary rest
1. plug the time (at axis of symm) into the eqn for displacement/height
2. answer with proper units. "at ___ sec, the max height is ___."
31
find when the arrow hits the ground
1. set displacement equation to 0
2. use quadratic equation: (-b+-sqrt(b^2-4ac))/2a
3. use positive time
**write "inadmissable" on -ve
32
what must u do when using quadratic eqn
write INADMISSABLE under the negative time
33
how to find velocity when arrow hits the ground
1. plug in the time where arrow hits ground (prev. found) into the velocity eqn
2. solve, add units and direction
34
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
35
when i object moving in pos/neg direction
- pos: v(t) greater than 0
- neg: v(t) less than 0
36
when is velocity increasing/decreasing
- increase: a(t) greater than 0
- decrease: a(t) less than 0
37
when do we use chain rule
(many terms)^n OR sqrt(many terms)
38
what is chain rule
- (n)(orig eqn)^n-1 (derivative of orig eqn)
39
steps of chain rule
1. plug everything in
2. expand derivative UNLESSS using product/quotient rule after
40
quotient rule
- use when fraction
f'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2
- subtract instead
- denom is denom
41
find point on graph where tangent is horizontal, given eqn
1. derive original eqn
2. set derivation to 0 (bc horizontal tangent means m=0), cross multiply AND FACTOR
3. set each term to 0 and isolate for x
4. plug isolated x-value into original eqn to get pt. f(x) = y is the point
42
revenue function
R(x) = xp(x)
- x: number of units of product/service sold
- p(x): price per unit
43
cost function
C(x) is the total cost of producing x units of prodt/service
- cost of labour
44
profit function
P(x) = R(x) - C(x)
- total profit made, diff between revenue and cost
45
what does marginal mean in business?
find the derivative
46
demand function
p(x)
- p: # of units of prodt/services tht can be sold at price of x
- x: price
47
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
48
marginal revenue
1. R'(x) = derive
2. R'(#) = plug in
49
what does it mean when marginal revenue is 0
- revenue is at max
- selling more will decrease revenue
50
what must u include once you've found marginal profit?
/PER ITEM
**i.e. $16.20 per big mac
