Emily Duval
Calc III
Mini Quiz:
Question 1 (10). Find the derivative of the following functions:
a.
b.
c.
d.
e.
Question 2 (15). a. State the definition of the derivative of a function
b. Use the definition of the derivative to compute
at a point
.
for
Qu
Emily Duval
Calc III
One Sided Limits
SOLUTION 14 : Consider the function
i.) The graph of f is given below.
ii.) Determine the following limits.
a.)
.
b.)
.
c.) We have that
does not exist since
does not equal
.
d.)
.
e.)
.
f.) We have that
since
.
g.) W
Emily Duval
Calc III
Limit Proofs by Limit Definition
SOLUTION 1 :
Prove that
. Begin by letting
be given. Find
, then
, i.e.,
so that if
, i.e.,
inequality is always true, no matter what value is chosen for . For example,
, then it follows that
. But thi
Emily Duval
Calc III
Implicit Differentiation
SOLUTION 1 : Begin with x3 + y3 = 4 . Differentiate both sides of the equation, getting
D ( x3 + y 3 ) = D ( 4 ) ,
D ( x3 ) + D ( y 3 ) = D ( 4 ) ,
(Remember to use the chain rule on D ( y3 ) .)
3x2 + 3y2 y' =
Emily Duval
Calc III
Solving the LImit of a Function
SOLUTION 1 :
.
SOLUTION 2 :
(Circumvent the indeterminate form by factoring both the numerator and denominator.)
(Divide out the factors x - 2 , the factors which are causing the indeterminate form
limi
Emily Duval
Calc III
Mini Quiz 1 Solutions
Question 1 (10). Find the derivative of the following functions:
a.
Solution:
by the product rule and the chain rule.
b.
Solution:
by the chain rule.
c.
Solution:
by the quotient rule.
d.
Solution:
First simplify
Emily Duval
Calc III
Limit Proofs by Limit Definition
SOLUTION 4 : Prove that
. Begin by letting
(which depends on ) so that if
be given. Find
, then
. Begin with
and `solve for" |x-1| . Then,
iff
iff
iff
iff
.
We will now `replace" the term |x+1| with an
Emily Duval
Calc III
Limit Proofs by Limit Definition
SOLUTION 9 : Prove that
. Begin by letting
(which depends on ) so that if
be given. Find
, then
. Begin with
and `solve for" | x - 3 | . Then,
iff
iff
iff
iff
iff
iff
iff
.
We will now `replace" the te
Emily Duval
Calc III
Logarithmic Differentiation
SOLUTION 1 : Because a variable is raised to a variable power in this function, the ordinary rules of
differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begi
Emily Duval
Calc III
Implicit Differentiation
SOLUTION 1 : Begin with x3 + y3 = 4 . Differentiate both sides of the equation, getting
D ( x3 + y3 ) = D ( 4 ) ,
D ( x3 ) + D ( y3 ) = D ( 4 ) ,
(Remember to use the chain rule on D ( y3 ) .)
3x2 + 3y2 y' = 0