Download What is the Second Fundamental Theorem of Calculus

Aim: What is the Second Fundamental Theorem of Calculus?
Objectives: to see that a function can be defined by an integral and that it can be differentiated to find
the maximum or minimum.
Grouping: students are given opportunity to work in cooperative setting during this class. Time is
purposely set aside for students to work collaboratively on at least one set of exercise problems. During
this time, students are encouraged to explain concepts and materials and solutions to each other.
Furthermore, students get to present their solutions on the board.
Differentiated instruction: all students are held to the highest standard. However the students that are
not performing well are grouped with students that are excelling to serve as study partners. Teacher will
informally assess their understanding by asking them questions to ensure that they are keeping up.
Assessment: Class is given multiple occasions to work individually and in groups. Teacher circulates
the room to assist and also assess student understanding. Furthermore, students are encouraged to
explain work to each other. This is another opportunity for teacher to assess student understanding.
Lastly, when lesson is finished, students are grouped for class work, questions that are designed as exit
slip problems.
Homework Review (10 minutes): Students are assigned problems to present on board. Students will
also answer questions. Teacher can go over a problem if no one in class could explain it
Lesson Development: [10 min] The Fundamental Theorem of Calculus deals with a definite integral
b
which represents the total change so  f '( x)dx  f (b)  f (a)  numerical value . What if the upper
a
t
t
bound is replaced by a variable or a function?

f '( x)dx  f (t )  f (a)  function . We say
a
 f '( x)dx
is
a
a function defined by an integral. Just like any function, we can differentiate, evaluate and find max and
min.
x
x
1
1
More examples of functions defined by an integral: g ( x)   sin tdt or h( x)   sin t  1dt and
sin x
f ( x) 
 t dt .
2
1
2
To evaluate, for example, g (2) , we need to evaluate the definite integral  sin tdt Just note that the
1
upper bound has a different variable as the integrand function to avoid confusion, so you don’t put 2
2
inside the integrand so you have something like  sin 2d 2 which is total nonsense.
1
Rules for differentiating a function defined by an integral is based on The Second Fundamental
Theorem of Calculus: If f is continuous on an open interval containing a, then
g ( x)
x


d 
d 
  f (t )dt   f ( g ( x)) g '( x)
  f (t )dt   f ( x) or
dx  a
dx  a


x2
x
d
Example:
sin tdt  sin x
dx 1
d
sin tdt  (sin x 2 )2 x
dx 1
[4 min]
EX1: Differentiate each function [5 min]
x
x
ln x
sin t
b) f ( x)  
dt
t
0
a) g ( x)   t  1dt
2
0
g '( x)  x 2  1
f '( x) 
c) h( x) 
sin x
x
x
h '( x)  cos(ln x)
x
EX2: Suppose g ( x)   500e0.02t dt [4 min]
0
F '(2)
a) Find g '( x ) ?
F '( x )  | cos x |
0.02 x
F '(2)  cos 2
b) Evaluate g '(1) .
g '(1)  500e0.02
EX4: Evaluate [8 min]
d
a)
cos tdt
dx 
d 
b)

dx 
ANS: 2 x cos x 2
ANS:
x2
d)
d
dx
ANS:
d
g)
dx
ln x

e
t
dt
2
ln x 1
ln x
( )
2 x
2x
tan( x 2 )

f (t )dt
0
ANS: f (tan x 2 )sec( x 4 )2 x
d
e)
dx
1
x
EX3: Suppose F ( x)   | cos t |dt Evaluate
0
g'( x)  500e
 cos tdt

sin x

0

t  1dt 

sin x  1(cos x)
e

d 
c)
  f (t )dt 
dx  0

x
ANS: f (e x )e x
4
x
 t dt
2
e
1
1
x
ANS: x x 1/2 
2
2
f)
d 2
t dt
dx x
ANS:  x 2
EX5: 2007AB-3: [6 – 8 min]
x
f ( x)
f '( x)
g ( x)
g '( x )
1
6
4
2
5
2
9
2
3
1
3
10
-4
4
2
4
-1
3
6
7
The function f and g are differentiable for all real numbers. The table above gives table for the functions
g ( x)
and their derivatives at selected values of x. Let w be the function given by w( x) 

f (t )dt . Find the
1
value of w '(3) .
d
w '( x) 
dx
g ( x)

f (t )dt  f ( g ( x))  g '( x)
1
w '(3)  f ( g (3))  g '(3)  f (4)(2)  2
Now that we learn how to differentiate functions defined by an integral, we may proceed further to find
the intervals of increasing or decreasing, relative maximum and minimum. [8 min]
x
EX6: Let F ( x)   f (t )dt , where the graph of f is shown below.
0
a) On what interval is F ( x) increasing?
F '( x)  f ( x)  0
x 1
(1,  )
(- ,1)
+
F ( x) is increasing on (1,  ) because F '( x)  0
b) On what interval is F ( x) decreasing?
F ( x) is increasing on ( ,1) because
F '( x)  0
c) Does the graph of F ( x) have a relative
extremum?
F ( x) has a relative minimum at x  1 because
F '( x) changes from negative to positive there.
x
EX7: Let F ( x)   f (t )dt , where the graph of f is shown below. Does F ( x) have a relative extremum?
0
F '( x)  f ( x )  0
x2
(, 2)
(2, )
+
-
F ( x) have a relative maximum at x =2
because F '( x) changes from positive to
negative there.
HW#59: P329: 95 – 97, 103 – 105, 107
HW#59 Solutions:
95)
96)
97)
103)
104)
105)
107)
Aim: What is the Second Fundamental Theorem of Calculus?
EX1: Differentiate each function
x
x
ln x
sin t
b) f ( x)  
dt
t
0
a) g ( x)   t  1dt
2
0
x
c) h( x) 
 cos tdt

x
EX2: Suppose g ( x)   500e0.02t dt [4 min]
EX3: Suppose F ( x)   | cos t |dt Evaluate
0
0
F '(2)
Find g '( x ) and evaluate g '(1) .
EX4: Evaluate
x2
d
a)
cos tdt
dx 
d)
d
dx
ln x

e
t
dt
2
b)
e)
d 

dx 
d
dx
sin x

0

t  1dt 

e

d 
c)
  f (t )dt 
dx  0

x
4
x
2
 t dt
e
f)
d 2
t dt
dx x
d
g)
dx
tan( x 2 )

f (t )dt
0
EX5: 2007AB-3:
x
f ( x)
f '( x)
g ( x)
g '( x )
1
6
4
2
5
2
9
2
3
1
3
10
-4
4
2
4
-1
3
6
7
The function f and g are differentiable for all real numbers. The table above gives table for the functions
g ( x)
and their derivatives at selected values of x. Let w be the function given by w( x) 

f (t )dt . Find the
1
value of w '(3) .
x
EX6: Let F ( x)   f (t )dt , where the graph of f is shown below.
0
a) On what interval is F ( x) increasing?
b) On what interval is F ( x) decreasing?
c) Does the graph of F ( x) have a relative
extremum?
x
EX7: Let F ( x)   f (t )dt , where the graph of f is shown below. Does F ( x) have a relative extremum?
0