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how does the number of primes affect differentation?
number of primes = number of times we differentiate
f(x) =x^5-3x^3+8x^2-5x+10
find f'(x)
5x^4-9x^2+16x-5
f''(x) -> f(x)=x^5-3x^3+8x^2-5x+10
5(4x^3)-9(2x)+16 -> 20x^3-18x+16
f'''(x) ->x^5-3x^3+8x^2-5x+10
60x^2-18
F''''(x) -> x^5-3x^3+8x^2-5x+10
60(2x) -> 120x
4 Step Method for solving derivative
1. F(x+h) -> replace all x in function w/(x+h)
2. f(x+h)-f(x) -> subtract the original function
3. f(h+h)-f(x)/h -> divide by h
4. take limit -> plug in 0 for all h
in terms of demand & supply, when is the equilibrium graphically?
where the demand curve and supply curve intersect
how do you find equilibrium?
add the equations of the supply & demand & solve when = to 0
Break even point
Break-Even Point = FixedCosts/(1-VariableCosts/Sales
how do you find a limit?
plug in numbers approaching the limit defined by the function.
domain:
possible numbers that can make sense for the solution
informal definition of limit
plugging a series of numbers in close to the limit as trial/error.
how do you graphically tell where the limit is?
where the 'hole' in the graph is
how to find limit
plug in the limit into the function
what does lim/x->0+ =1 mean?
as the right hand limit approaches 0, 1 is the closest it can get
lim/x->a=l only if?
lim/x->a+ AND limt/x->a-
a function is not graphically continuous when?
you are not able to complete the graph without lifting the pencil
a function is continuous if?
1. f(a) is defined
2. lim/x->a exists
3. lim/x->a=f(a)
formal definition fo continuity
A function f is continuous at x = x0 if
exists and is f(x0).
formula for average rate of change
f(b)-f(a)/b-a
what relation does the slop of the secant line have to the rate of change?
quantity increases at the rate of the value of m
limit definition of derivative:
f(x+h)-f(x)/h
f'(a) representa what of the tangent line?
the slope
what is the difference between avg rate of change & instantaneous rate of change?
instant = specific point of change, avg = general change
1. what is the rate of change of f(x) when x=a?
2. at what rate is f(x) growing/increasing/decreasing/changing when x=a
3. how is f(x) fast/quickly grow/increas/decreas/chang when x=a