Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.165.09 Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. Friday, September 18, 2009 Main Idea: Prepare for Test Monday Key Words: Preparation Goal: Do well on the test on Monday ############################################################## # # # Practice Test # # # # Actual test is Monday, September 21, 2009 # # # # During Class 7:30 to 9:00 AM # # # ############################################################## 1. The speed of two drag racers consists of joined line segments. The vertices of the graphs are. racer A (0,0) (2,20) (4,35) (6,70) racer B (0,0) (2,15) (4,45) (6,70) (a) How far did racer A go in 6 seconds? 180 feet (b) How far did racer B go in 6 seconds? 190 feet (c) Which racer was ahead after six seconds? Racer B (d) At what times did they pass? at four seconds both had gone 75 feet (6,70) .| RACER A .' | .' | .' | .' | (4,35) | .'| | .' | | (2,20)' | | Total = 180 feet .| | | .' | | 105 | .' | 55 | | .' 20 | | | (0,0)---------|----------|----------| (6,70) .| .' | RACER B .' | .' | .' | (4,45) | .| | .' | | . ' | | .' | | Total = 190 feet (2,15) | | . | | 115 | . ' | 60 | | . '15 | | | (0,0)---------|----------|----------| ========================================================== Get["font.math"]; a[x_] := If[ x <=2, x/2 20, If[ x <= 4, 20+(x-2)7.5, 35 + (x-4)17.5]]; b[x_] := If[ x <=2, x/2 15, If[ x <= 4, 15+(x-2)15, 45 + (x-4)12.5]]; p1 = Plot[a[x],{x,0,6},PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; p2 = Plot[b[x],{x,0,6},PlotStyle->{RGBColor[0,1,0],Thickness[0.005]}]; p3 = ParametricPlot[{{2,y}},{y,0,20}, PlotStyle->{RGBColor[0,0,1],Thickness[0.003]}]; p4 = ParametricPlot[{{4,y}},{y,0,45}, PlotStyle->{RGBColor[0,0,1],Thickness[0.003]}]; p5 = ParametricPlot[{{6,y}},{y,0,70}, PlotStyle->{RGBColor[0,0,1],Thickness[0.003]}]; p6 = Show[p1,p2,p3,p4,p5,PlotLabel->"Q1 red = A, green = B, Velocity", PlotRange->All]; Export["/math/www/hentzel/class.165.09/sep18.q1x.pdf",p6]; n = 2000; aa[x_] := Sum[ x/n a[x i/n],{i,1,n}]; bb[x_] := Sum[ x/n b[x i/n],{i,1,n}]; p7 = Plot[aa[x],{x,0,6},PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; p8 = Plot[bb[x],{x,0,6},PlotStyle->{RGBColor[0,1,0],Thickness[0.005]}]; p9 = ParametricPlot[{{2,y}},{y,0,20}, PlotStyle->{RGBColor[0,0,1],Thickness[0.003]}]; p10 = ParametricPlot[{{4,y}},{y,0,75}, PlotStyle->{RGBColor[0,0,1],Thickness[0.003]}]; p11 = ParametricPlot[{{6,y}},{y,0,190}, PlotStyle->{RGBColor[0,0,1],Thickness[0.003]}]; p12 = ListPlot[{{4,75}},PlotStyle->{RGBColor[1,1,0],PointSize[0.008]}]; p13 = Show[p7,p8,p9,p10,p11,p12,PlotLabel->"Q1 red = A, green = B, Distance", PlotRange->All]; Export["/math/www/hentzel/class.165.09/sep18.q1y.pdf",p13]; p14 = Show[p6,p13,PlotLabel->"Q1 red = A, green = B, Velocity and Distance", PlotRange->All]; Export["/math/www/hentzel/class.165.09/sep18.q1z.pdf",p14]; ========================================================== 2. Find the following limit. Show all of your work. 2-Sqrt[11-x] Limit -------------- x->7 2 x - 49 2-Sqrt[11-x] (2 + Sqrt[11-x]) Limit -------------- ---------------- x->7 2 x - 49 (2 + Sqrt[11-x]) 4-(11-x)] Limit ----------------------------- x->7 2 ( x - 49 ) (2 + Sqrt[11-x]) x-7 Limit ----------------------------- x->7 ( x+7) (x-7) (2 + Sqrt[11-x]) 1 1 1 Limit ----------------------------- = ---- = ------ <=== Answer x->7 14*4 56 ( x+7) (2 + Sqrt[11-x]) ==================================== Get["font.math"]; f[x_] := (2-Sqrt[11-x])/(x^2-49); p1 = Plot[f[x],{x,0,11},PlotStyle->{Thickness[0.01]}]; p2 = ListPlot[{{7,1/56}},PlotStyle->{RGBColor[0,1,1],PointSize[0.010]}]; p3 = Plot[1/56,{x,0,7},PlotStyle->{RGBColor[0,1,1],Thickness[0.005]}]; p4 = ParametricPlot[{7,y},{y,0,1/56}, PlotStyle->{RGBColor[0,1,1],Thickness[0.005]}]; p5 = Show[p1,p2,p3,p4,PlotLabel->"Q2 f[x_] := (2-Sqrt[11-x])/(x^2-49)", PlotRange->All]; Export["/math/www/hentzel/class.165.09/sep18.q2.pdf",p5]; ======================================== 3. Find the equation of the tangent line to 3 y = x + 3 x + 3 at the point (1,7). / 2 y = 3x + 3; / y (1) = 6; y-7 = 6(x-1) <===== Answer; ======================== Get["font.math"]; f[x_] := x^3 + 3x + 3; p1 = Plot[f[x],{x,-2,2},PlotStyle->{Thickness[0.005]}]; p2 = ListPlot[{{1,7}},PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p3 = Plot[7+6(x-1),{x,-1,2},PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; p4 = ListPlot[{{1,7}},PlotStyle->{RGBColor[1,0,1],PointSize[0.015]}]; p5 = Show[p1,p2,p3,p4,PlotLabel->"Q3 y = x^3 + 3 x + 3"]; Export["/math/www/hentzel/class.165.09/sep18.q3.pdf",p5]; ======================== 4. Find the points (x,y) where the tangent line 3 to y = x + x + 1 is horizontal. / 2 y = 3x + 1 / 2 y = 0 when 3x + 1 = 0 2 x = -1/3 There are no such points <======== Answer ====================== Get["font.math"]; f[x_] := x^3 + x + 1; p1 = Plot[f[x],{x,-1.5,1.5},PlotStyle->{Thickness[0.02]}]; p2 = Show[p1,PlotLabel->"Q4 y = x^3 + x + 1", PlotRange->All,AspectRatio->Automatic]; Export["/math/www/hentzel/class.165.09/sep18.q4.pdf",p2]; ====================== 5. Explain why calculus uses radians instead of degrees. Sin[x] Limit ------- = 1 only when x is in Radians. x->0 x / / The rules for derivatives Sin[x] = Cos[x] and Cos[x] = -Sin[x] are only true when x is in radians. If we used degrees, each of the derivatives would have complicated constants. 6. Give the definition of the derivative of a function y = f(x). / f(x+h) - f(x) f (x) = Limit -------------- h->0 h 7. A spherical balloon is filling at the rate of 6 cubic feet per minute. At what rate is the radius changing when the balloon holds 40 cubic feet. 3 V = 4/3 Pi r When V = 40 then 3 40 = 4/3 Pi r 1/3 r = ( 30/Pi ) 2 dV/dt = 4 Pi r dr/dt 2/3 6 = 4 Pi (30/Pi) dr/dt 6 ------------------ = dr/dt 2/3 4 Pi (30/Pi) 6 ------------------ = dr/dt <==== Answer 1/3 2/3 4 Pi 30 1/3 6 30 ----------------- = dr/dt 1/3 4 Pi 30 1/3 30 ------------------ = dr/dt 1/3 20 Pi 6/(4 Pi^(1/3) 30^(2/3) ) = 0.106078 ft/min <==== Answer ============================= Get["font.math"]; r[v_] = ( 3 v/(4 Pi))^(1/3) p1 = Plot[ r[6 t] ,{t,0,10}]; A = Table[{ t, r[6 t]},{t,1,10}]; p2 = ListPlot[A,PlotStyle->{RGBColor[1,0,0],PointSize[0.010]}]; p3 = Plot[r[36],{x,0,6},PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; p4 = Plot[r[42],{x,0,7},PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; p5 = ListPlot[ {{40/6,r[40]}},PlotStyle->{RGBColor[0,1,0],PointSize[0.01]}]; p6 = Show[p1,p2,p3,p4,p5, PlotLabel->"Q7 r[v_]=( 3 v/(4 Pi))^(1/3) dots = 1 minute", PlotRange->All, Axes->True, AxesOrigin->{0,0}, AxesLabel->{Time,Radius}, TicksStyle->Directive[Red,12], AxesStyle->Directive[Orange,15] ]; Export["/math/www/hentzel/class.165.09/sep18.q7x.pdf",p6]; ========================================================= Get["font.math"]; r[v_] = ( 3 v/(4 Pi))^(1/3) H = Table[ ParametricPlot[ r[6 t]{Cos[w],Sin[w]},{w,0,2 Pi}, PlotStyle->{Thickness[0.005]}],{t, 1,10}]; p7 = ParametricPlot[ r[40]{Cos[w],Sin[w]},{w,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.002]}]; K = Table[ ParametricPlot[ {v,y},{y,-0.1,0.1}, PlotStyle->{RGBColor[1,0,1],Thickness[0.002]}],{v,0,r[60],0.1}]; p8 = Show[K,H,p7,AspectRatio->Automatic,PlotRange->All, PlotLabel->"q7 Size each minute, red=40 cu ft"]; Export["/math/www/hentzel/class.165.09/sep18.q7y.pdf",p8]; ========================================= 8. (a) State the chain rule. / / / (fog) (x) = f (g(x)) g (x) (b) Prove the chain rule. fog(x+h) - fog(x) Limit ---------------------- h->0 h f(g(x+h)) - f(g(x)) g(x+h) - g(x) Limit ---------------------- * ----------------- h->0 g(x+h) - g(x) h f(g(x+h)) - f(g(x)) g(x+h) - g(x) Limit ---------------------- * Limit ----------------- h->0 g(x+h) - g(x) h->0 h / / f (g(x)) * g (x) In problems 9 through 20 Find the derivatives. Do NOT simplify 3 9. y = 19 x + 7 x + 5 / 2 y = 57 x + 7 2 14 x + 3 x 10. y = -------------- 3 x + 1 3 2 2 / (x +1)(28x + 3) - (14 x + 3 x)(3x ) y = --------------------------------------- 3 2 (x +1) 3 11. y = x Sin[x] / 2 3 y = 3x Sin[x] + x Cos[x] 12. y = Sin[ Sin[x] ] / y = Cos[Sin[x]] Cos[x] 10 13. y = ( Sin[x] + Cos[x] ) / 9 y = 10 (Sin[x] + Cos[x]) ( Cos[x] - Sin[x] ) 10 14. y = 12 ( Sin[3 x] ) / 9 y = 120 ( Sin[3x] ) Cos[3x] 3 2 3 5 15. y = ((x + 5) + 7) / 2 3 4 2 2 y = 5 ((x + 5) + 7) 3 (x + 5) 2 x Sin[x] 16. y = -------- Cos[x] / Cos[x] Cos[x] - Sin[x](-Sin[x]) y = ---------------------------------- 2 Cos [x] 4 3 17. y = Pi + 3 Pi + 2 Pi + 6 / y = 0 3 3/2 18. y = (x + 3 x + 1) / 3 1/2 2 y = 3/2 (x + 3 x + 1) (3x + 3) 5 -13/2 19. y = ( 2 x + 7 x -16) / 5 -15/2 4 y = -13/2 (2 x + 7 x - 16) ( 10 x + 7) 3 4 20. y = 3 ( Sin[x ] ) / 3 3 3 2 y = 12 ( Sin[x ] ) Cos[x ] 3x 21. y = Sin[x] + Cos[x] + Sec[x] / y = Cos[x] - Sin[x] + Sec[x] Tan[x] 22. y = Cos[x] + Ctn[x] + Csc{x] / 2 y = -Sin[x] - Csc [x] - Csc[x] Ctn[x]