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definition of continuity (definition)
lim[x → c⁺] f(x) = lim[x → c⁻] f(x) = f(c)
definition of derivative (h)
definition of derivative (c)
definition of continuity (justify)
By the definition of continuity, since lim[x → c⁺] f(x) = lim[x → c⁻] f(x) = f(c) = k, f(x) is continuous.
average rate of change
(f(b) - f(a)) / (b - a)
average value of f(x)
∫ᵇₐ[f(x)dx] / (b - a)
Intermediate Value Theorem (definition)
f(x) continuous on [a, b]
k is on [a, b] interval
Intermediate Value Theorem (justify)
Since f(x) is continuous on [a, b] and k is between f(a) and f(b), then IVT Theorem applies.
Mean Value Theorem (definition)
f(x) continuous on [a, b]
f(x) differentiable on (a, b)
then, f’(c) = [f(b) - f(a)] / [b - a]
Mean Value Theorem (justify)
Since f(x) is continuous on [a, b] and is differentiable on (a, b), then MVT Theorem applies.
average rate of change
(f(b) - f(a)) / (b - a)
product rule
d/dx [f(x) * g(x)]
f’(x) × g(x) + f(x) × g’(x)
quotient rule
d/dx [f(x) / g(x)]
( f’(x) × g(x) - f(x) × g’(x) ) / (g(x)²)
chain rule
d/dx [f(g(x))]
f’(g(x)) × g’(x)
inverse derivative
(f⁻¹)’ (a)
1 / f’( f⁻¹(a) )
d/dx [sin(u)]
cos(u) × u’
d/dx [tan(u)]
sec²(u) × u’
d/dx [sec(u)]
sec(u) × tan(u) × u’
d/dx [cos(u)]
-sin(u) × u’
d/dx [cot(u)]
-csc²(u) × u’
d/dx [csc(u)]
-csc(u) × cot(u) × u’
d/dx [ln(u)]
u’ / u
d/dx [eᵘ]
eᵘ × u’
d/dx [logₐ(u)]
u’ / (u × ln(a))
d/dx [aᵘ]
aᵘ × ln(a) × u’
d/dx [sin⁻¹(u)]
u’ / √(1 - u²)
d/dx [tan⁻¹(u)]
u’ / (1 + u²)
d/dx [sec⁻¹(u)]
u’ / (|u| × √(u² - 1))
d/dx [cos⁻¹(u)]
− u’ / √(1 - u²)
d/dx [cot⁻¹(u)]
− u’ / (1 + u²)
d/dx [csc⁻¹(u)]
− u’ / (|u| × √(u² - 1))
Note
Flashcard