| Derivatives | Definite Integrals | Derivatives | Theorem/FTC | Review |
|---|---|---|---|---|
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m
(mx)' =
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Zero
integral from a to a f(x) dx =
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f'(a)
limit as h approaches zero ((f(a+h)-f(a)) / h) =
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If f(x) is continuous on [a,b] and M is between f(a) and f(b), then there must be at least one c on (a,b) where f(c)=M
Intermediate Value Theorem
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f">0 or f' is increasing
f(x) is concave up if
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nx^(n-1)
(x^n)' =
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negative integral from b to a f(x) dx
integral from a to b of f(x) dx =
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f'(a)
limit as x approaches a ((f(x)-f(a)) /(x-a)) =
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1.If l(x) ≤ f(x) ≤ u(x)
2.limit as x approaches c l(x)= limit as x approaches c u(x)=L 3.Then, limit as x approaches c f(x)=L
Squeeze Theorem
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inverse cos(x) + C
∫ (-1/ √(1-x^2)) dx=
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cos(x)
(sin(x))' =
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integral from a to c of f(x) dx + integral from c to b of f(x) dx
integral of b to a of f(x) dx =
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y-f(a)=f'(a) (x-a)
Equation of a Tangent line to f(x) at x=a is
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1.f(x) continuous on [a,b]
2.differentiable on (a,b) 3.f'(c)= ((f(b)-f(a)) / (b-a))
Mean Value Theorem
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((a^x) / (ln(x))) +C
∫ ln(x) dx
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-sin(x)
(cos(x))' =
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integral from a to b of f(x) dx plus or minus integral from a to b of f(x) dx
integral from a to b (f(x)±g(x)) dx =
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It represents the slope of the tangent.
What is the meaning of the derivative?
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1.f(x) continuous on [a,b]
2.differentiable on (a,b) 3.f(a)=f(b) 4.a < c < b 5.f'(c)=0
Rolle's Theorem
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f'(a) } slope of the tangent line
What is Instantaneous Rate of Change?
|
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Zero
(constant)' =
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Amount of change
Integral from a to b R'(t)dt=R(b)-R(a)
integral from a to b (Rate) =
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f'(x)= limit as h approaches zero ((f(x+h)-f(x)) / h) =
Limit Definition of a Derivative of f(x)
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F(b)-F(a)
FTC (fundamental theorem of Calculus) Part 1:
integral from a to b f(x) dx |
limit as x approaches a of f(g(x))= f(limit as x approaches a of g(x))
limit as x approaches a of f(g(x))=
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