1
136.150: Test #1
Solutions
B08.
Name:_
Student Number: _
1. Let f (x) = x 1 and let g(x) = 2x
(a) Find the composition f g(x)
(b) Find the domain of the function f g(x) .
Solution. (a) f g(x) = f (g(x) = f (2x) = 2x 1 .
(b) Because of the root we must h
1
136.150: Test #3 Solutions
B11.
Name:_
[6] 1. Find f (x) if f (x) = ( sin x )
Student Number: _
cos x
(
)
Solution. Set y = sin x cos x , so that ln y = ln sin x cos x , i.e., ln y = ( cos x ) ln ( sin x ) . Differentiate
1
1
y = ( sin x ) ln ( sin x )
1
136.150: Test #2 Solutions
B09.
Name:_
Student Number: _
x2 + 1
.
x 1
x 1+ 1x
x2 + 1
x +1
Solution. lim f (x) = lim
= lim
= lim
= 1 . So y = 1 is a
x
x x 1
x x 1
x
x 1 1x
horizontal asymptote as x .
[6] 1. Find all horizontal asymptotes of the function
1
136.150: Test #1
20 minutes
B04.
Name:_
Student Number: _
1. Find the domain of the following function. State your answer in terms of intervals.
f (x) = 1 2x
Solution. Because of the root we must have 1 2x 0 . We solve it:
1
1 2x 01 2x x where the symbo
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136.150: Test #2
B05.
Solutions
1. Find the limit or, if it does not exist, check if it is , - or neither. Show your work.
x 1
x 1
x +1
lim
(b) x
x 1
(a) lim
x1
Solution. (a)
(b)
lim
x1
x 1
= lim
x1
x 1
(
x 1
)(
x 1
)
x +1
= lim
x1
(
1
)
x +1
=
1
2
1
1
1
136.150: Test #5
20 minutes
No Calculators
B04.
Name:_
[9]
[7]
Student Number: _
1. Consider f (x) = x 3 x .
(a) Find and classify the local extrema .
(b) Sketch the graph of the function.
1
1
and x =
. Since f (x) = 6x
3
3
the second derivative test gi
1
136.150: Test #4
20 minutes
B07.
Solutions
1. Find y if y = x 2 x .
Solution. If y = x 2 x then ln y = 2x ln x . Differentiate this to get
y
= 2 ln x + 2 , from where
y
we find that y = y(2 ln x + 2) = x 2 x (2 ln x + 2) .
2. Find all local extrema of t
1
136.150: Test #3
Solutions
B06.
Name:_
Student Number: _
1. Use the definition of derivative to find the derivative f (x) of the function f (x) = x 1 .
No points will be awarded if you do not use the definition of the derivative of a function.
Solution.
1
136.150: Test #3 Solutions
B10.
Name:_
Student Number: _
[8] 1. Find f (x) . Do not simplify your answer after differentiating.
x
sin x + cos x
(b) f (x) = (1 + tan(1 x)2
(a) f (x) =
Solution. (a)
( x ) (sin x + cos x)
f (x) =
x (sin x + cos x )
(sin x