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 defi
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
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 cho
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 ) ,
(Remem
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 , th
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.
Solu
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
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
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
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 ) ,
(Remembe