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28 Cards in this Set
- Front
- Back
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L'Hopital's Rule
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if the derivative is in intederminate form then use which is 0/0 and infinity/infinity
lim x->a f(x)/g(x) = lim x->a f'(x)/g'(x) |
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optimization
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minimum or maximum problem
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critical point
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local minimum or maximum
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point of inflection
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point where concavity of the graph changes
where f"(x) = 0 |
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y=e^i(x-a)^2
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a shifts the graph left or right
divide (x-a)^2 by b and b makes the graph wider or narrower |
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y=a(1-e^-bx)
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a is the asymptote and b is the stretch or shrink of the graph
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y=a(x-h)^2 + k
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a is stretch or shrink
h shifts right and left b shifts up and down |
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global max and min
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overall maximum and minimum
highest and lowest y values substitute the cp into the f(x) set equal to zero to find y location on the graph to see which is the highest |
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Revenue
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price x quantity
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Profit
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Revenue - cost
usually written as pi |
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Marginal Cost
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C'(q)
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Marginal Revenue
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R'(q)
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maximum profit
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Marginal Cost = Marginal revenue
R'(q) = c'(q) on graph where space between is widest |
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Modeling Optimization problems
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need two equations
the quantity needed to be optimized is the thing you want the derivative of sketches find the cp and ep to evaluate global max and min |
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cosh x
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e^x + e^-x / 2
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sinh x
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e^x - e^-x / 2
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d/dx cosh x
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sinh x
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d/dx sinh x
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cosh x
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hyperbolic functions to know
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cosh(0) = 1
sinh(0) = 0 cosh(-x) = cosh x sinh (-x) = -sinh x cosh^2 x - sinh^2 x = 1 |
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Extreme Value theorem
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if f is continuous then must have a global max and global min on the closed interval
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Mean Value theorem
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f'(c) = f(b) - f(a) / b-a
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delta t
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b-a/n
n is the number of intervals and b and a is the interval |
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left and right increments
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in a chart begin with the first value but don't take the last for the right hand value
begin with the second value and end with the last value for the left hand sum average the two to find the most acurate measurement of the integral |
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sigma notation
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definite integral
number at the top tells the number of intervals number at the bottom tells where to begin |
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definite integral
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integral from a to b f(t) dt
area under the curve total change distance |
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definite integral
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F(b) - F(a) = integral from b to a F'(t) dt
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average value
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1/b-a integral from b to a f(x) dx
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antiderivative
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working backwards from the derivative
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