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Unformatted text preview: 2007-2008 FALL MATH 119 WEEK 3 RECITATION QUESTIONS 1)Blind application of implicit diﬀerentiation to the equation x4 + y 4 = x2 y 2 leads to the formula
dy dx = 2x3 −xy 2 x2 y −2y 3 Why is this result meaningless ? 2)Write an equation of the tangent line to the curve which is the graph of the equation sin(x + 2y ) = 2xcosy
1 at the origin.(Ans:y = 2 x) 3)A rocket that is launched vertically is tracked by a radar station located on the ground 3 mi from the launch site.What is the vertical speed of the rocket at the instant that its distance from the radar is 5 mi and this distance is increasing at the rate of 5000 mi/h ? (Ans:6250 mi/h) 4)Sand is being emptied from a hopper at the rate of 10 f t3 /s.The sand forms a conical pile whose height is always twice its radius.At what rate is the radius of the pile increasing when its height is 5f t ?(Ans: 54 f t/s) π 5)Show that if A,B ,and k are constants and y = Acos(kt) + Bsin(kt) then y satisﬁes the second order diﬀerential equation of simple harmonic motion
d2 y dt2 + k2 y = 0 6)Find y = d2 y dx2 if y = y (x) is deﬁned implicitly by the equation x2 − xy + y 2 = 9 d (Ans: dxy = − (2y54x)3 ) 2 − 2 7)Use a linear approximation L(x) to an appropriate function f (x),with an appropriate value of a,to estimate the number cos43◦ 1 8)(Acceleration of grawity)The acceleration a of grawity at an altitude of h miles above the surface of the earth is given by a = g ( RR h )2 + ∼ where g ∼ 32f t/s2 is the acceleration at the surface of the earth,and R = 3960 = miles is the radius of the earth.By about what percentage will a decrease if h increases from 0 to 10 miles ? (Ans:−0.5 ◦/◦ approximately ) 2 ...
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