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19 Cards in this Set
- Front
- Back
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Define Maximum Value |
Suppose that c is a critical number of a continuous function f defined on an interval. - If f'(x) > 0 for all x < c and f'(x) < 0 for all x > c, then f(c) is the absolute maximum value of f |
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Define Minimum Value |
Suppose c is a critical number of a continuous function f defined on an interval - If f'(x) < 0 for all x < c and f'(x) > 0 for all x > c then f(c) is the absolute minimum value of f |
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What is the distance formula? |
d^2 = (x2 - x1)^2 + (y2 - y1)^2 |
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What is the equation for the linearization of f at a? |
L(x) = f(a) + f'(a)(x - a) |
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What is the linearization approximation or the tangent line approximation of f at a? |
f(x) ~ f(a) + f'(a)(x - a) |
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Define a differential |
If y = f(x), where f is a differentiable function, then the differential dx is an independent variable; that is dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation: dy = f'(x)dx |
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Equation of Sinh(x) |
Sinh(x) = e^x - e^-x / 2 |
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Equation of Cosh(x) |
Cosh = e^x + e^-x / 2 |
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What is the equation of a tangent line? |
lim x -> a [ f(x) - f(a) / (x - a) ] |
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What is the derivative of a function at a number? |
f(a) = lim h -> 0 [ f(a + h) - f(a) / h ] |
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The number e |
lim n -> infinity ( 1 + 1/n) ^ n |
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Inflection Point |
Where a function changes concavity |
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State the Intermediate Value Theorem |
Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b) where f(a) does not equal f(b), then there exists a number c in (a,b) such that f(c) = N |
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State Rolle's Theorem |
Let f be a function that satisfies the following: 1) f is continuous on [a,b] 2) f is differentiable on (a,b) 3) f(a) = f(b) Then there is a number c in (a,b) such that f'(c) = 0 |
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The Squeeze Theorem |
If f(x) < g(x) < h(x) when x is near a and lim x -> a [f(x)] = lim x -> a [h(x)] = L, then lim x-> a [g(x) = L |
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Fermat's Theorem |
If f has a local max at c and is differentiable there, then f'(c) = 0 |
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State the limit laws |
1. The limit of a sum is the sum of the limits 2. The limit of a difference is the difference of the limits 3. The limit of a constant times a function is the constant times the limit of the function 4. The limit of a product is the product of the limits 5. The limit of a quotient is the quotient of the limits (provided the denominator is not 0) |
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Formal definition of the chain rule |
F'(x) = f'[g(x)] * g'(x) We differentiate the outer function f (at the inner function g(x) ) and then we multiply by the derivative of the inner function. |
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Continuity requires 3 things |
1) f(a) is defined (a is in the domain) 2) Lim x-> a f(x) exists 3) Lim x->a f(x) = f(a) A function f is continuous on an interval if it is continuous at every number in the interval. |
