Download Major Quiz 3 Solutions for MA140-01 - Mathematics - Prof. Joe A. Stickles and more Quizzes Calculus in PDF only on Docsity!
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Major Quiz 3 Solutions
They call me________________________
1.) Find each of the following. Simplify as much as possible!
9 points (3 each)
(a) ( )
2
cos 4
d x
e x
dx
(b) ( tan 1 (co s ))
d x dx
โ
#10 from section 2.7 homework #31 from section 2.8 homework
2 2
(2)[cos 4 ] sin 4
x x
e x + e โ x
2
( sin )
1 (cos )
x
x
2 2
2 cos 4 4 sin 4
x x
e x โ e x
2
sin
1 cos
x
x
(c) ( )
2
sin(cos3 ) 3sec 2
d
x x
dx
Just like section 2.6 homework
cos(cos3 )[( x โ sin 3 )(3)] x โ 6sec 2 sec 2 x x tan 2 x
2
โ 3sin 3 cos(cos3 ) x x โ12sec 2 x tan 2 x
2.) Derive the derivative of csc x.
4 points section 2.
( ) csc
sin
f x x
x
2
(0)(sin ) (1)(cos )
(sin )
x x
f x
x
2
cos cos 1
sin sin sin
x x
f x
x x x
f '( ) x = โ cot x โ
csc x
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3.) Use the position function
2 s t ( ) = t + 8 to find the velocity at time t = 2. (Assume units
of meters and seconds.) 5 points
#25 from section 2.5 homework
1 (^2 2 ) s t ( ) = t + 8 = ( t +8) 1 (^12 ) '( ) ( 8) (2 ) 2
s t t t
โ = +
1 (^12 ) '(2 ) ( 2 8) (2 2 ) 2
s
โ = + โ
2 2 1 '(2 ) / 12 2 3 3
s = = = m s
4.) Use logarithmic differentiation to find the derivative of
ln
x
f x = x.^ 6 points
#43 from section 2.7 homework
ln
x
f x = x
ln
ln[ ( )] ln( )
x
f x = x
ln[ f ( x )] = ln x [ln( x )]
2
ln[ f ( x )] = ln x
'( ) 2(ln )
f x x
f x x
f '( x ) 2 (ln x ) f ( x )
x
1 ln
'( ) 2 (ln )
x
f x x x
x
ln
2 ln
x
x x
f x
x
5.) For f ( x ) = cos x, find
( 77 ) f ( x )and
( 123 ) f ( x ). 6 points
Just like #37 from section 2.6 homework except changed sin x to cos x
f '( x ) = โ sin x
( 77 ) f ( x ) = f '( x ) = โsin x
f ''( x ) = โcos x
f '''( x ) = sin x
( 123 ) f ( x ) = f '''( x ) = sin x ( 4 ) f ( x ) = cos x
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8.) Use the following table of values to answer the questions below. 8 points
Similar to #29, 31, and 33 from section 2.5 homework and previous sections
(a) Let (^) h x ( ) = f ( x ). Find (^) h (4) and (^) h '(4).
h (4) = f ( 4) = f (2) = 4
1 (^12) '( ) '( ) 2
h x f x x
โ = โ
1 1 2 1 1 1 3 '(4) '( 4) (4) '(2) ( 3) 2 2 2 4 4
h f f
โ (^) = โ
= โ
(^) = โ = โ
(b) Let (^) j x ( ) = f ( g x ( )). Find (^) j (4)and (^) j '(4).
j (4) = f ( g (4)) = f (2) = 4
j '( ) x = f '( g x ( )) โ
g '( ) x
j '(4) = f '( g (4)) โ
g '(4) = f '(2) (3)โ
= โ( 3)(3) = โ 9
x f ( ) x g x ( ) f '( ) x g '( ) x
2 4 -8 -3 -
4 -7 2 5 3
