MATA37H3 Lecture 6: Fundamental Theorem of Calculus Part II
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Mata37 - lecture 6 - fundamental theorem of calculus part ii. De nition: see page 390, let a, b r, a < b. If f is continuous on [a, b], de ne f (x) = continuous on [a, b]. F is differentiable on (a, b), and f (cid:48)(x) = f (x) x [a, b: f (cid:48)(x) = f (x) = (cid:90) x f (t)dt) a d dx a (cid:90) x f (t)dt, x [a, b]. Examples: example 1: compute d dx (cid:90) x. 1 + t4 is continuous on r because 1 + t4 0 t r. so f (t) is continuous on. 1 + t4 dt) = f (x) = 1 + x4: example 2: let h(x) = 1 + t4 is continuous on r because 1 + t4 0 t r. so f is continuous on (cid:90) x (cid:112) (cid:90) x (cid:112) 1 + t4dt) : example 3: nd g(cid:48)(x) where g(x) = arctan(t)dt.
