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CALCULUS ONE
MAKEUP NOV. 6, 2006 NAME:(in ink)_____________________________
SHOW ALL CALCULATIONS. SIMPLIFY ANSWERS.
Page 1 of 1.
BEST EIGHT PROBLEMS COUNT FOR A SCORE OUT OF 100. Answers may be rounded to four decimals.
1. Evaluate the limits if possible. Use ∞ or − ∞ wherever it is appropriate.
9x 2 + x + 2
lim
(a)
x d −∞
5 − 2x
−
lim
x d −∞
lim
xd0
lim
d0
9x 2 + x + 2
1
x2
1
x (5
=
=
lim
xd3
9x 2 + x + 2
1
x (5
− 2x )
=
− 9 + 1/x + 2/x 2
5/x − 2
(2x − 1 )(x − 3 )
lim
=
xd3
(x − 8 )(x − 3 )
(Assume x < 0.)
=
−3
−2
=
3
2
2x − 1
= −1
x − 8
x $ csc 10x
x
10x
lim
lim
=
=
=
xd0
xd0
2
2 $ sin 10x
20 $ sin 10x
1
1
1
lim
$1 =
=
=
20 d 0 sin 20
20
20 $ sin dy
2. (a) If y = x 3 sin 4 (5x ) find dx
dy
dx
1
x
lim
x d −∞
=
− 2x )
2x 2 − 7x + 3
x 2 − 11x + 24
lim
(b)
xd3
(c)
lim
=
x d −∞
d
( 3 ) $ sin 4 (5x )
dx x
+ x3 $
(Expression in factored form.)
d
( 4 ( ))
dx sin 5x
= 3x 2 sin 4 (5x ) + 4x 3 sin 3 (5x ) cos(5x ) $ 5
= x 2 sin 3 (5x ) $ (3 sin(5x ) + 20x cos(5x ))
(b) If x 4 y 3 − x 3 y 5 = 3x − 7y
dy
Find dx by implicit differentiation.
d
d
3 5)
( 4 3
= dx (3x − 7y ),
dx x y − x y
4x 3 y 3 + 3x 4 y 2 y ∏ − 3x 2 y 5 − 5x 3 y 4 y ∏ = 3 − 7y ∏,
3x 4 y 2 y ∏ − 5x 3 y 4 y ∏ + 7y ∏ = 3x 2 y 5 − 4x 3 y 3 + 3,
y∏ =
GO TO PAGE 2.
3x 2 y 5 − 4x 3 y 3 + 3
3x 4 y 2 − 5x 3 y 4 + 7
CALCULUS ONE
MAKEUP NOV. 6, 2006 NAME:(in ink)_____________________________
SHOW ALL CALCULATIONS. SIMPLIFY ANSWERS.
Page 2 of 2.
BEST EIGHT PROBLEMS COUNT FOR A SCORE OUT OF 100. Answers may be rounded to four decimals.
3.
⎧
⎧ x 2/3 ,
⎫
1
if x [ 1
⎪ x 3 sin
⎪ 2
⎪
,
g(x) = ⎨ 3 x + 1, if 1 < x < 6 ⎬, f(x) = ⎨
x2
⎪ x − 1,
⎪
⎪
if x m 6
0,
⎩
⎭
⎩
⎫
x ! 0 ⎪
⎬
⎪
x = 0 ⎭
(a) For what value(s) of x is g discontinuous? Sketch a graph of g. (Below.)
lim
x 2/3 = 1,
x d 1−
1 !
5
3,
lim
( 2 x + 1) =
x d 1+ 3
5
3,
lim
(x − 1 ) = 5,
x d 6+
lim
( 2 x + 1 ) = 5,
x d 6− 3
so g is discontinuous at x = 1.
(b) For what value(s) of x is g continuous, but not differentiable?
2
= 3x 1/3 ,
d
( 2/3 )
dx x
x = 6 is
2
3,
2
3x 1/3
d ∞
as x d 0. The left hand derivative at
but the right hand derivative there is 1.
The function g is continuous at 0 and 6, but is not differentiable at those numbers.
∏
(c) Find f (x ) for x ! 0, using rules of differentiation.
(x
3 )∏
1
sin 2
x
3x 2 sin
∏
1
x2
1
+ x 3 sin 2
x
− 2 cos
= 3x 2 sin
1
x2
+ x 3 cos
1
x2
$
1
x2
(d) State f (0 ) . (No work Required.)
GO TO PAGE 3.
∏
lim
hd0
1
h
h 3 sin
1
h2
− 0 = 0
−2
=
x3