13-2 Derivatives & integrals

# 13-2 Derivatives & integrals - 13.2 Derivatives...

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13.2 Derivatives & Integrals r -> (t+h) = r -> (t) + PQ -> PQ -> Since h > 0 (very small positive number) PQ -> = 1/h x (r -> (t+h)-r -> (t)/1 = r -> (t+h)-r -> (t) / h r -> (t) = Mf(t),,g(t), h(t)> r -> '(t) = <f'(t), g'(t), h'(t)> if f'(t), g'(t), h'(t) exist unit tangent vector: T -> (t) = r -> '(t)/||r -> '(t)|| r -> (t) = <cos3t, t, sin3t> Find r -> '(t) = <-3sin3t, 1, 3cos3t> Helix Application: - Coiled spring - DNA Ex: Find the parametric EQs for the tangent line to the curve r -> (t) = <t 2 -1, t+1, t+1> @ point (-1,1,1) r -> (t) = <2t,2t,1> | | | P(-1,1,1) - set x, y, and z to = 1.

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-1 = x = t 2 - 1 1 = y = t 2 +1 1 = z = t+1 x = -1 + 0t y= 1 + 0t z = 1 + 1t Properties of Derivatives - How do you take the derivative of the sum? A: Take the sum of the derivatives. (1) d/dt [M -> (t) + v -> (t)] = d/dt M -> (t) + d/dtv -> (t) = M -> '(t) + -> V -> '(t) (2) d/dt [ kx M -> (t)] = K x M -> '(t), k = constant (3) d/dt [ f(t) M -> (t)] ^ ^ function vector-val func = f'(t) M -> (t) + M -> '(t) f(t) (4) d/dt [ M -> (t) (dotproduct) V -> (t)] = M -> '(t) (dot) V -> (t) + M
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Unformatted text preview: -> (t) (dot) V-> '(t) (5) d/dt [M-> (t) x V-> (t)] = M-> '(t) x V-> (t) + M-> (t) x V-> (t) (6) d/dt [ M-> (f(t))] = M-> '(f(t)) (f'(t)) chain rule- What's a smooth curve?- A curve w/ no sharp turns and gaps. - We say a curve is smooth if r-> '(t) != 0-> for any t Integrals | ~ r-> (t) dt = R-> (t) + C ->- note the constant is a vector! Definite Integral: | t=b t=a r-> (t) dt = R-> (t) | t=b t=a = R-> (b)-R-> (a) Ex: |(cospit i -> + sinpi + j-> + tk-> ) dt = < (1/pi)sinpit, -1/pi cospit, t 2 /2 > C-> | < cospit, sinpit, t > dt = sinpit/pi i->- cospit/pi j-> + t 2 /2 k-> + C-> |-1 <0, 4/1+t 2 , 2t/(1+t 2 ) > dt | 1 (4/(1+t 2 )j-> + 2t/(1+t 2 ) k-> ) dt = 4 tan-1 t + ln(1+t 2 )k-> | 1 = [4tan-1 j-> + ln(2)k-> ] - [4tan 01 (0) j-> + ln(1) K-> ] 4(pi/4) j-> + ln(2) k -> = pi j-> + ln2 k-> = <0, pi, ln2> |du/(1+u 2 ) = tan-1 u + C |2t/(1+t 2 ) dt = |du/u = ln|u|+C ^ u = 1+t 2 , du = 2tdt...
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