UIZ # 3 Problems 3 1. Show that the function f ( x )=e x +2 x+1 is one-to-one. Graphing does not count as proof. (Hint: Use the derivative of the function.) 2. Find a formula for the inverse function f 1 7 ( f (e )) =e 1 f ( x) of f ( x )=3ln ( x) . Verify that 7 ' 3. Fill in the values of f 1 ( x ) and ( f 1) (x ) for support your answers. x f (x) 1 1 1 2 2 1 f ' (x ) 0 2 4 f 1 (x ) x=1,1, 2 . Provide detail to ' ( f 1) (x ) 4. In the triangle below, angle is (a) the arcsine of what number? (b) the arctangent of what number? (c) the arcsecant of what number? (d) the arccosine of what number? 3 4 5. Calculate the derivatives. (a) y (arctan( x) 5) 6 (b) (c) y tan( x 2 1) arcsin( x ) y ln[ arc sec( x )] 6. Evaluate each integral based on inverse trigonometric functions. An answer with no sufficient detail will not receive full credits. (a) dx 1 16 x (b) x (c) 2 1 16 x 2 25 e 2t e 4t 9 dt dx QUIZ # 3 Problems 3 1. Show that the function f ( x )=e x +2 x+1 is one-to-one. Graphing does not count as proof. (Hint: Use the derivative of the function.) 3 +2 x+ 1 y =( 3 x + 2 ) ( e ' y=e x 2 x 3+2 x+ 1 ) 3+ 2x+ 1 ' We see that 3 x 2 +1is always positivee x is also always positive . So y >0 Since f ' ( x ) >0, hence this is an increasing function hence one one function 2. Find a formula for the inverse function 1 f ( x) of f ( x )=3ln ( x) . Verify that f 1 ( f (e 7) ) =e 7 y=3ln ( x ) switch x y Hence , f 1 ( x )=e3 x e 3 ( 4 ) e 3+4 e x=3lny lny=3x f 1 ( f ( e7 ) )=f 1 (3ln e7 ) y=e 3 x f 1 ( 37 ) f 1 (4 ) 7 ' 3. Fill in the values of f 1 ( x ) and ( f 1) (x ) for support your answers. x f (x) 1 1 1 2 2 1 ' f (x ) 0 2 4 1 f (x ) -1 2 1 x=1,1, 2 . Provide detail to ' ( f 1) (x ) 0 4 -2 for f 1 (1 ) , we need find value of x when f ( x )=1, so x=1 1 for f (1 ) , we ned find value of x when f ( x )=1, so x=2 for f 1 ( 2 ) , we need find value o f x when f ( x )=2, so x =1 ' for ( f 1 ) (1 ) , we need find f ' ( x ) for which f 1 ( x )=1, so f ' ( x )=0 1 ' ' 1 ' for ( f ) (1 ) , we need find f ( x ) for which f ( x )=1, so f ( x )=4 1 ' ' 1 ' for ( f ) ( 2 ) , we need find f ( x ) for which f ( x )=2 , so f ( x )=2 4. In the triangle below, angle is (a) the arcsine of what number? (b) the arctangent of what number? (c) the arcsecant of what number? (d) the arccosine of what number? 32+4 2=5 3 3 a sin = 5 =tan 1 ( 34 ) c sec= 5 4 =se c1 ( 54 ) =sin1 d cos= 3 5 () 4 5 4 3 b tan= 4 =cos1 ( 45 ) 5. Calculate the derivatives. (a) y (arctan( x) 5) 6 ( x ) +5 1 tan 6 y ' = (b) y tan( x 2 1) arcsin( x ) y=tan ( x2 +1 ) sin 1 ( x ) (c) y ' =2 xsec 2 ( x2 +1 ) sin1 ( x ) + tan ( x 2 +1) 1 ( 1x ) 2 y ln[ arc sec( x)] (x ) sec 1 y=ln (x) sec 1 ' 1 y ' = 1 sec ( x ) 1 1 ' y = 1 sec ( x ) x x 21 6. Evaluate each integral based on inverse trigonometric functions. An answer with no sufficient detail will not receive full credits. (a) dx 1 16 x 2 dx dx = 2 116 x 1 ( 4 x )2 u=4 x du=4 dx dx= du 4 1 1 sin u+ C 4 1 1 sin 4 x +C 4 (b) x 1 16 x 2 25 du 4 1 (u ) 2 1 du 4 1( u )2 dx 1 dx 1 dx x 16 x 225 25 2 4x x 16 1 dx 5 2 2 4x x 4 5 4x 5 let x = secu= u=sec 1 dx= tanu secu du 4 5 4 1 5 tanu secu du 2 2 4 5 5 5 secu sec 2 u 4 4 4 tanu du 2 4 5 sec u1 1 tanu du 5 tanu 1 du 5 u 1 4x sec 1 +c +c 5 5 5 (c) e 2t e 4t 9 dt () ( ) ( ) () ( ) 2t 2t e 4et +9 dt e 3 ( ) 2t let e =3 tanu= u=tan1 2t 2 e dt=3 u 3 sec 2 u dt= du 2 e2 t e2t 3 sec 2 u 2t 2 9 du e 4et +9 dt= 9 tan u+ 2e 2 t sec u 2 1 du 6 1 6 1 u+C 6 e2t 3 ( )+C 1 tan1 6 3 1 2 sec u du 2 18 tan u+ 1 \f QUIZ # 3 Problems 3 1. Show that the function f ( x )=e x +2 x+1 is one-to-one. Graphing does not count as proof. (Hint: Use the derivative of the function.) 3 +2 x+ 1 y =( 3 x + 2 ) ( e ' y=e x 2 x 3+2 x+ 1 ) 3+ 2x+ 1 ' We see that 3 x 2 +1is always positivee x is also always positive . So y >0 Since f ' ( x ) >0, hence this is an increasing function hence one one function 2. Find a formula for the inverse function 1 f ( x) of f ( x )=3ln ( x) . Verify that f 1 ( f (e 7) ) =e 7 y=3ln ( x ) switch x y Hence , f 1 ( x )=e3 x e 3 ( 4 ) e 3+4 e x=3lny lny=3x f 1 ( f ( e7 ) )=f 1 (3ln e7 ) y=e 3 x f 1 ( 37 ) f 1 (4 ) 7 ' 3. Fill in the values of f 1 ( x ) and ( f 1) (x ) for support your answers. x f (x) 1 1 1 2 2 1 ' f (x ) 0 2 4 1 f (x ) -1 2 1 x=1,1, 2 . Provide detail to ' ( f 1) (x ) 0 4 -2 for f 1 (1 ) , we need find value of x when f ( x )=1, so x=1 1 for f (1 ) , we ned find value of x when f ( x )=1, so x=2 for f 1 ( 2 ) , we need find value o f x when f ( x )=2, so x =1 ' for ( f 1 ) (1 ) , we need find f ' ( x ) for which f 1 ( x )=1, so f ' ( x )=0 1 ' ' 1 ' for ( f ) (1 ) , we need find f ( x ) for which f ( x )=1, so f ( x )=4 1 ' ' 1 ' for ( f ) ( 2 ) , we need find f ( x ) for which f ( x )=2 , so f ( x )=2 4. In the triangle below, angle is (a) the arcsine of what number? (b) the arctangent of what number? (c) the arcsecant of what number? (d) the arccosine of what number? 32+4 2=5 3 3 a sin = 5 =tan 1 ( 34 ) c sec= 5 4 =se c1 ( 54 ) =sin1 d cos= 3 5 () 4 5 4 3 b tan= 4 =cos1 ( 45 ) 5. Calculate the derivatives. (a) y (arctan( x) 5) 6 ( x ) +5 1 tan 6 y ' = (b) y tan( x 2 1) arcsin( x ) y=tan ( x2 +1 ) sin 1 ( x ) (c) y ' =2 xsec 2 ( x2 +1 ) sin1 ( x ) + tan ( x 2 +1) 1 ( 1x ) 2 y ln[ arc sec( x)] (x ) sec 1 y=ln (x) sec 1 ' 1 y ' = 1 sec ( x ) 1 1 ' y = 1 sec ( x ) x x 21 6. Evaluate each integral based on inverse trigonometric functions. An answer with no sufficient detail will not receive full credits. (a) dx 1 16 x 2 dx dx = 2 116 x 1 ( 4 x )2 u=4 x du=4 dx dx= du 4 1 1 sin u+ C 4 1 1 sin 4 x +C 4 (b) x 1 16 x 2 25 du 4 1 (u ) 2 1 du 4 1( u )2 dx 1 dx 1 dx x 16 x 225 25 2 4x x 16 1 dx 5 2 2 4x x 4 5 4x 5 let x = secu= u=sec 1 dx= tanu secu du 4 5 4 1 5 tanu secu du 2 2 4 5 5 5 secu sec 2 u 4 4 4 tanu du 2 4 5 sec u1 1 tanu du 5 tanu 1 du 5 u 1 4x sec 1 +c +c 5 5 5 (c) e 2t e 4t 9 dt () ( ) ( ) () ( ) 2t 2t e 4et +9 dt e 3 ( ) 2t let e =3 tanu= u=tan1 2t 2 e dt=3 u 3 sec 2 u dt= du 2 e2 t e2t 3 sec 2 u 2t 2 9 du e 4et +9 dt= 9 tan u+ 2e 2 t sec u 2 1 du 6 1 6 1 u+C 6 e2t 3 ( )+C 1 tan1 6 3 1 2 sec u du 2 18 tan u+ 1 \f