Fundamental Theorems of Calculus
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In a Nut Shell: The
Fundamental Theorems of Calculus show that differentiation and integration are inverse
processes. It establishes the
connection between differential and integral
calculus. The Fundamental Theorems
of Calculus have two parts. |
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The Fundamental Theorem of Calculus (Part 1) If f is
continuous on [ a, b ], then the function
g defined by x g(x) = ∫ f(t) dt a
≤ x ≤
b a is continuous on [ a, b ] and differentiable on (a, b),
and g’(x) =
f(x) |
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The Fundamental Theorem of Calculus (Part 2) If f is
continuous on [a , b] , then b ∫ f(x) dx =
F(b) - F(a)
a where F
is any antiderivative of f,
that is, a function such that
F’ = dF/dx = f. |
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- Finding Derivatives of Integrals h2(x) where h2(x) is the upper limit of
integration G(x) =
∫ f(t) dt h1(x) where h1(x) is the lower limit of
integration Strategy: Take derivative using the chain rule: dG(x)/dx = f(h2(x)) dh2(x))/dx - f(h1(x)) dh1(x))/dx |
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