52.}ddyx}55}121sicnostt}At t52}p4}, we have x5cos 12}p4}25 }ˇ22w},y p4}1sin 12}p4}2p4}2 }ˇ22w}, and }ddyx}5ˇ2w11.The equation of the tangent line is y5(ˇ2w11)1x2 }ˇ22w}22 }p4}2 }ˇ22w}, or y5(11ˇ2w)x2ˇ2w212 }p4}.This is approximately y52.414x23.200.53. (a)[21, 3] by 321,}53}4(b)Yes, because both of the one-sided limits as x →1 areequal to f(1)51.(c)No, because the left-hand derivative at x51 is 11 andthe right-hand derivative at x51 is 21.54. (a)The function is continuous for all values of m, becausethe right-hand limit as x→0 is equal to f(0)50 forany value of m.(b)The left-hand derivative at x50 is 2 cos (2 ?0) 52,and the right-hand derivative at x50 is m, so in order for the function to be differentiable at x50,mmust be 2.55. (a)For all x±0(b)At x50(c)Nowhere56. (a)For all x(b)Nowhere(c)Nowhere57.Note that limx→02f(x)5limx→02(2x23)523 and limx→01f(x)5limx→01(x23)3. Since these values agreewith f(0), the function is continuous at x50. On the otherhand,f9(x)55, so the derivative is undefined atx50.(a)[21, 0) <(0, 4](b)At x50(c)Nowhere in its domain58.Note that the function is undefined at x50.(a)[22, 0) <(0, 2](b)Nowhere(c)Nowhere in its domain
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Fall '08 term at University of Florida.