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MAT 3749 2.3 Part II Team Homework
Names: ______________________________________________________________
1. (5 points) Prove that Power Rule.
Let n   . If y  f ( x)  x n , then f ( x)  nx n1 .
n
n
n
(Hint: You may use the formula  x  h     x k h n k for n 
k 0  k 
use summation notations instead of “   ” type of notations.)
Last Edit: 7/28/2017 10:51 PM

. You are expected to
2. (4 points) Prove the Product Rule.
Let f and g be differentiable functions. If F  x   f ( x)  g ( x) , then
F   x   f ( x) g ( x)  f ( x) g ( x) .
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3. (5 points) Prove the Quotient Rule.
f ( x)
, then
g ( x)
f ( x) g ( x)  f ( x) g ( x)
F x 
.
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 g ( x) 
Let f and g be differentiable functions. If F  x  
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4. (12 points) The following is another proof of the Chain Rule.
(a) Let g be differentiable at a . Let a function 1  h  be defined by
1  h  
Show that
g  a  h  g a 
h
 g  a 
(1)
lim 1  h   0 .
h 0
(b) The statement in part (a) can be rephrased as:
If g be differentiable at a , then g  a  h   g  a   hg   a   h1  h  where lim 1  h   0 .
h 0
Similarly, we have the following statement:
If f be differentiable at b , then f  b  k   f  b   kf  b   k 2  k  where lim  2  k   0 .
k 0
Show that lim  2  g  a  h   g  a    0 .
h0
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(c) (The Chain Rule) Suppose g is differentiable at a , and f is differentiable at
b  g  a  . Let F  x   f  g  x   . Then F is differentiable at a and
F   a   f   g  a   g  a  .
(Hint: Let k  g  a  h   g  a  i.e. g  a  h   g  a   k  b  k .)
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5. (4 points) Use the Lemma proved in class to prove the following statement.
If f is differentiable at d , then f is continuous at d .
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