Integration and Slopes

Calculus Level 3

If f ( a ) = ∫ a b g ( b ) d x \displaystyle f(a)=\int_a^bg(b)\ dx , then what is the slope of f ( a ) f(a) ?

− g ( b ) -g(b) − G ( b ) -G(b) g ( b ) g(b) g ′ ( b ) g'(b) − g ′ ( b ) -g'(b)

Nicholas Walters by Nicholas Walters , 21, USA

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1 solution

Tom Engelsman
Mar 20, 2019

Essentially, f ( a ) = g ( b ) ⋅ ( b − a ) f(a) = g(b) \cdot (b-a) as g ( b ) g(b) is a constant within the given definite integral. The slope of f ( a ) f(a) is just the following derivative:

d d a f ( a ) = d d a [ b ⋅ g ( b ) − a ⋅ g ( b ) ] = − g ( b ) . \frac{d}{da} f(a) = \frac{d}{da}[b \cdot g(b) - a \cdot g(b)] = \boxed{-g(b)}.

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