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3 Pages

Hprobs4_sol

Course: MATH 101, Fall 2009
School: East Los Angeles College
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Word Count: 342

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Homework MATH101 Solution Sheet 4 1. For f to be dierentiable it must also be continuous. So, for continuity at x = 0 we require that the left and right limits of f should be the same and equal to f (0). For dierentiability, the left and right derivatives must also be equal at x = 0. Thus 1 lim f (x) = = lim f (x) = 1 , x0 b x0+ so we require b = 1. For the left derivative at x = 0 f (x) = and on the right f...

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Homework MATH101 Solution Sheet 4 1. For f to be dierentiable it must also be continuous. So, for continuity at x = 0 we require that the left and right limits of f should be the same and equal to f (0). For dierentiability, the left and right derivatives must also be equal at x = 0. Thus 1 lim f (x) = = lim f (x) = 1 , x0 b x0+ so we require b = 1. For the left derivative at x = 0 f (x) = and on the right f (x) = 1. 2 1 2 a a = 2, 2 (ax + b) b 1 . 2 (1 + x) 1 2 = Hence we require a = In summary, dierentiability of f (x) at x = 0 requires a= 1 2 and b = 1. 2. Dierentiating both sides of (x + y)2 = gives 2(x + y)(1 + y ) = and dierentiating a second time gives 2(1+y )(1+y )+2(x+y)y = From (1), at x = 0, y = 1, 2(1 + y ) = y 1 From (2) 8 + 2y = y (3 1)2 = y + 8 1 so y = 0 . so y = 3 . (1 + x)2 ((1 + x)y + y y ) ((1 + x)y y)2(1 + x) . (1 + x)4 (2) (1 + x)y y (1 + x)2 (1) y 1+x 3. Dierentiating both sides 2x2 of + 1 = sin(xy) + y gives 4x = (xy + y) cos(xy) + y y (1 + x cos(xy)) = 4x y cos(xy) so: 4x y cos(xy) . y = 1 + x cos(xy) At (x, y) = (0, 1) 1 = 1 . 1 The equation of the tangent is given by y = y = x + c = 1 when x = 0. and rearranging this gives: Thus the equation of the tangent at (x, y) = (0, 1) is y = 1 x. 4. (i) As t 0 0 10(sin t t) so can apply LHpitals rule (LHR): o t3 0 10(cos t 1) 10(sin t t) = lim lim 3 t0 t0 t 3t2 Again the ratio tends to 0/0 so we can apply LHR a second time: lim t0 5 10( sin t) 10(cos t 1) = lim = . 2 t0 3...

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