Quiz 3 soln - Quiz/7 NC[3 Total In Math 1A — Quiz 3 —...

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Unformatted text preview: Quiz/7 NC [3 Total In) Math 1A —- Quiz 3 — February 15, 2008 NAME 1‘ The graph of f is given 2 Skewh the graph of f' . 1.5 I " f is shown at right, please ”5 see your quiz for the original graph off 4.1.5 05 I a 56 .I -\5 2, Find the first and second derivatives of g (t) = 29051 — Mint For what values ofn does g” (t) = g (t) 7 g ’ (t) = — 2sint— most g ” (t) = — Zemt + mint Now for the second question, we must find what derivative gives us back the original fimction, g (t) - So, g ‘” (t) =2sl'nt + most g””(t) =g’y (t) =2cost— mint =g (t) Thus we found that when n = 4 we gEt g (t) back - Additionally, this is also true if n is any multiple of 4 t 3‘ For what values ofx does the graph offlx) = 2x3 + 151:“ + 36x + 6 have a horizontal tangent? We muslfind where f (x) has horizontal tangents . . . or where the derivative iszero ~ So, f’(x)=6xZ +30x +36 So, f(x)=0 —> 6x2 +30x+36 =0 —»6(x‘+5x+6)=0 —+ (x+3)(x+2)=0 —)x=——3 orx=—2 Thusf(x) has han‘zontaltangents when x =— 3 or): =~ 2 NOTECARD BONUS QUESTION: (1 point) Find the equations of the tangent line and the normal line to the curve at the given point y=6cosx (g , 3) In order tofind the equations of the tangent and normal lines, we must find the slopes of each of these lines - So, we must first find the derivative of the function - y ’ = — 6sinx Now thederivative ofyatg isy’(g)=i 65in (:)=—6(‘L2:)='— 3 «/§ So the tangent line to thefimctton at (% , 3) has slope — 3 \/3_ Thus the equation of the tangent line is: y — 3 =-— 3 V? (x — 135) Now we know that the tangent line and normal line are perpendicular. thus their slopes must be negative reciprocals ‘ So the normal line to the flmction at (g . 3) has slope 3 \1/3. Thus the equation afthe normal line is: y — 3 = 3 3/? ( — g) ...
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