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1 .   ) Find the absolute maximum value of the function f ( x ) = x 4 - 2 x 2 on the interval [ - 2 , 2 ] .
( a ) - 2
( b ) - 1
( c ) 3
( d ) 8
( e ) 63
2 .   ) Find f 0 ( 0 ) where the function is given by f ( x ) = ln r 2 x +1 x +1 .
3 .   ) Determine the interval on which the function f ( x ) = x 2 - 1 x is concave down .
4 .   ) Find a number c that satisfies the conclusion of the Mean Value Theorem for the function f ( x ) = x 2 - x on the interval [ - 5 , 1 ] .
5 .   ) Find where the function f ( x ) = x + 4 x 2 has a horizontal tangent , and determine whether f has a local minimum or maximum there .
( a ) x = - 3 √ 4 , neither local maximum nor local minimum ( b ) x = - 3 √ 4 , local minimum ( c ) x = - 3 √ 4 , local maximum ( d ) x =2 , local minimum ( e ) x =2 , local maximum
6 .   ) Find the solutions to the equation log 6 ( x ) + log 6 ( x - 5 ) = 1 .
7 .   ) Find the value of A such that the function f ( x ) = ( sin ( 4 x ) 7 x if x < 0 4 A cos ( 5 x ) if x > 0 . is continuous at x = 0 .
9 .   ) Given [ ln ( xy ) ] 2 = ln2 · l n 2 find dy dx at the point ( 1 , 2 ) .
10 .   ) Find an equation of the tangent line to the curve x 3 - x 2 y +4 y 2 = 8 at the point ( 2 , 1 ) .
( a ) y - 2 = - ( x - 1 ) ( b ) y - 1 = - 2 ( x - 2 ) ( c ) y - 1 = - 4 ( x - 2 ) ( d ) y - 1 = - ( x - 2 ) ( e ) y - 2 = - 2 ( x - 1 )
11 .   ) A function f ( x ) is defined for all positive real numbers and satisfies the equation f ( x 2 ) = x 3 for every x> 0 . Find f 0 ( 4 ) .
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Chapter 11 / Exercise 24
Differential Equations with Boundary-Value Problems
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This exam comprises the cover page and five pages of questions.
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Chapter 11 / Exercise 24
Differential Equations with Boundary-Value Problems
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MATH 1A03/1ZA3 – Winter 2015Version 1Page 11.)Find theabsolute maximum valueof the functionf(x)=x4-2x2on the interval [-2,2].
2.)Findf0(0) where the function is given byf(x)=lnr2x+1x+1.
3.)Determine the interval on which the functionf(x)=x2-1xis concave down.
4.)Find a numbercthat satisfies the conclusion of the Mean Value Theorem for the functionf(x)=x2-xon the interval [-5,1].(a)-2(b)-1/4(c)0(d)1/2(e)
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MATH 1A03/1ZA3 – Winter 2015Version 1Page 25.)Find where the functionf(x)=x+4x2has a horizontal tangent, and determine whetherfhas alocal minimum or maximum there.
6.)Find the solutions to the equation log6(x)+log6(x-5)=1.
7.)Find the value ofAsuch that the functionf(x)=(sin(4x)7xifx<04Acos(5x)ifx>0.is continuous atx=0.
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MATH 1A03/1ZA3 – Winter 2015Version 1Page 38.)Find limx!1ln(x)x-1.(a)1(b)e(c)1/e(d)0(e)+
9.)Given[ln(xy)]2=ln2·ln2finddydxat the point (1,2).
10.)Find an equation of the tangent line to the curvex3-x2y+4y2=8 at the point (2,1).
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