Math 132 Fall 2007 Final Exam 1. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡0ππππ2( )cos x ( )sin x3x.a) 1 b) 12 c) 13 d) 14 e) 15 f) 23 g) 34 h) 32 i) 43 j) 16 Solution: d> J := Int(cos(x)*sin(x)^3, x = 0..Pi/2); := J d⌠⌡0π2( )cos x ( )sin x3x> K := student[changevar](u = sin(x), J, u); := K d⌠⌡01u3u> value(K);142. Let = = = = ( )F x d⌠⌠⌠⌠⌡⌡⌡⌡x2 + + + + 5 t4 + + + + 1 t3t. Calculate the derivative D( F )( 2 ) of F at 2. a) 4 b) 5 c) 6 d) 7 e) 8 f) −−−−4 g) −−−−5 h) −−−−6 i) −−−−7 j) −−−−8 Solution: i> F := (x) -> Int((5+t^4)/sqrt(1+t^3),t = x .. 2); := F → x d⌠⌡x2 + 5 t4 + 1 t3t> D(F)(x);− + 5 x4 + 1 x3> D(F)(2);−7 93> simplify(D(F)(2));-73. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡01x( ) + + + + x 1 ( ) + + + + x 2x. a) ln98 b) ln76 c) ln54 d) ln43 e) ln32 f) ln95 g) ln83 h) ln94 i) ln163 j) ln169 Solution: a > J := Int(x/(x+1)/(x+2),x = 0 .. 1); := J d⌠⌡01x( ) + x 1 ( ) + x 2x> R := student[integrand](J); := Rx( ) + x 1 ( ) + x 2> PFE := convert(R, parfrac, x);:= PFE − + 1 + x 12 + x 2> antiderivative := int(PFE, x); := antiderivative − + ( )ln + x 1 2 ( )ln + x 2> definiteIntegral := subs(x=1,antiderivative) - subs(x=0,antiderivative); := definiteIntegral − + + 3 ( )ln 2 2 ( )ln 3 ( )ln 1> Answer := combine(definiteIntegral, ln); := Answer −ln894. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡01 + + + + + + + + 8 x22 x 6( ) + + + + 1 x ( ) + + + + 1 x2x. a) 14 ( )ln 2 b) 12 ( )ln 2 c) ( )ln 2 d) 2 ( )ln 2 e) 3 ( )ln 2 f) 4 ( )ln 2 g) 5 ( )ln 2 h) 6 ( )ln 2 i) 7 ( )ln 2 j) 8 ( )ln 2 Solution: i > J := Int((8*x^2+2*x+6)/(1+x)/(1+x^2),x = 0 .. 1); := J d⌠⌡01 + + 8 x22 x 6( ) + x 1 ( ) + 1 x2x> R := student[integrand](J); := R + + 8 x22 x 6( ) + x 1 ( ) + 1 x2> PFE := convert(R, parfrac, x); := PFE + 2 x + 1 x26 + x 1> antiderivative := int(PFE, x); := antiderivative + ( )ln + 1 x26 ( )ln + x 1> definiteIntegral := subs(x=1,antiderivative) - subs(x=0,antiderivative); := definiteIntegral − 7 ( )ln 2 7 ( )ln 1> Answer := combine(definiteIntegral, ln); := Answer 7 ( )ln 25. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡1ex2( )ln x x.a) 13 e3 b) 13 ( − 2 e31 ) c) 13 ( − e32 ) d) 23 ( − e31 ) e) 13 ( + 2 e31 ) f) 13 ( + e32 ) g) 23 ( + e31 ) h) 19 ( + 2 e31 ) i) 19 ( + e32 ) j) 29 ( + e31 ) Solution: h> J := Int(x^2*ln(x), x = 1 .. exp(1)); := J d⌠⌡1ex2( )ln x x> K := student[intparts](J, ln(x)); #Integration by Parts with u=ln(x) := K − 13( )e3d⌠⌡1ex23x> value(K); + 29( )e3196. What is the derivative of x1x with respect to x at = = = = x12 ? a) −−−− ( )ln 2 b) −−−−12 ( )ln 2 c) − − − − 1 ( )ln 2 d) − − − − 112 ( )ln 2 e) ( )ln 2 f) 12 ( )ln 2 g) + + + + 1 ( )ln 2 h) + + + + 112 ( )ln 2 i) 14 ( )ln 2 j) 14Solution: g > restart;> eqn1 := f(x) = x^(1/x); := eqn1 = ( )f x x1x> eqn2 := map(z-> simplify(ln(z), symbolic), eqn1); := eqn2 = ( )ln ( )f x( )ln xx> eqn3 := map(z -> diff(z,x), eqn2); := eqn3 = ddx( )f x( )f x− + ( )ln xx21x2> eqn4 := D(f)(x) = solve(eqn3, diff(f(x),x)); := eqn4 = ( )( )D f x −( )f x ( ) − ( )ln x 1x2> eqn5 := subs(x = 1/2, eqn4); := eqn5 = ( )D f12−4f12 − ln121> eqn6 := subs(x = 1/2, eqn1); := eqn6 = f1214> subs(eqn6, eqn5); = ( )D f12 + ( )ln 2 17. If = = = = ( )y 0 0 and dydx = ( )cos x − − − − 1 y2, then what is ( )y x ?a) ( )sin ππππ ( )cos x b) ( )sin ( )sin x c) cosππππ ( )cos x2 d) ( )cos ( )sin x -1 e) ( )arcsin x2 f) ( )arcsin ( )arcsin x g) ( )sin ( )tan x h) ( )tan ( )sin x i) ( )arcsin ( )tan x j) ( )arcsin ( )arctan x Solution: b By the Method of Separation of Variables:> eqn1 := Int(1/sqrt(1-t^2),t = 0 .. y(x)) = Int(cos(t),t=0..x); := eqn1 = d⌠⌡0( )y x1 − 1 t2t d⌠⌡0x( )cos t t> eqn2 := map(value, eqn1); := eqn2 = ( )arcsin ( )y x ( )sin x> Answer := y(x) = solve(eqn2, y(x)); := Answer = ( )y x ( )sin ( )sin xFor those who are interested, here is how to get MAPLE to solve this differential equation without the user supplying any guidance: > ode := diff(y(x),x)=cos(x)*sqrt(1-y(x)^2); := ode = ddx( )y x ( )cos x − 1
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