Aim: What is the Second Fundamental Theorem of Calculus?
Do Now: Differentiate each integral
x
ln x
a) F ( x)   3 tdt
b) F ( x ) 
1
 t dt
2
1
F '( x)  x1/3
F '( x) 
1 1

ln x x
x
Since F ( x)   f (t )dt is a function, we can find out where it is increasing, where the relative and
0
absolute extrama are, where the point of inflections are. Basically, almost any question that can be
asked about f ( x)  x3 can be asked about a function defined by an integral.
x
EX1: Let F ( x)   f (t )dt , where the graph of f is
0
shown below.
c) On what intervals is graph of F increasing?
The graph of F is increasing if F '( x)  f ( x)  0 .
It is increasing on (5,10)
d) Find the critical value(s) of F. Identify it as a
relative maximum, relative minimum or neither.
Justify your answer.
f ( x)  0
x  0,5
Relative minimum occurs at x  5 because
F '( x) changes from negative to positive at x  5 .
a) Find F (3)
3
1
F (3)   f (t )dt   (3)(6)  9
2
0
Do you know why we have different variables? If
there is the same variable, where will 2 also go?
e) On what open intervals is the graph of F
concave down?
The graph of F is concave down if
F ''( x)  f '( x)  0
(1,3)
b) Find F '(2)
f) Find the x-coordinate of the point of inflection.
Justify your answer.
F '( x)  f ( x)
F '(2)  f (2)  4
x  3 because F ''( x ) changes sign at x  3
g) Find the absolute maximum value of F on the closed interval [0,10] .
Critical value: x  5 , endpoints: x  0,10
0
F (0)   f (t )dt  0
The maximum value of F is 9.
0
5
1
F (5)   f (t )dt   (5)(6)  15
2
0
10
F (10)   f (t )dt  15  6  18  9
0
2002-AB-4: The graph of the function f shown
above consists of two line segments. Let g be the
x
function given by g ( x)   f (t )dt .
0
a) Find g (1), g (2), g '(1), g ''(1) .
b) Find what values of x in the open interval (-2, 2)
is g increasing? Explain your reasoning.
1
1
3
g (1)   f (t )dt   (1)(3) 
2
2
0
2
g (2)   f (t )dt 0
g '( x)  f ( x)
g is increasing on ( 1,1) because f ( x)  0
0
x
d
f (t )dt  f ( x)
dx 0
g '(1)  f (1)  0
g '( x) 
g ''( x) 
d
f ( x)  f '( x )
dx
g ''(1)  f '(1)  3
c) For what values of x in the open interval (-2, 2) is the graph of g concave up? Explain your reasoning.
g ''( x)  f '( x) . g is concave up on ( 2, 0) because F ''( x)  f '( x)  0 .
Aim: What is the Second Fundamental Theorem of Calculus?
Do Now: Differentiate each integral
x
ln x
a) F ( x)   3 tdt
b) F ( x ) 
1
 t dt
2
1
x
EX1: Let F ( x)   f (t )dt , where the graph of f is
c) On what intervals is graph of F increasing?
Justify your answer.
0
shown below.
d) Find the critical value(s) of F. Identify it as a
relative maximum, relative minimum or neither.
Justify your answer.
a) Find F (3)
Do you know why we have different variables? If
there is only one variable, where will 3 also go?
e) On what open intervals is the graph of F
concave down?
f) Find the x-coordinate of the point of inflection
on the graph of F. Justify your answer.
b) Find F '(2)
g) Find the absolute maximum value of F on the
closed interval [0,10] .
2002-AB-4: The graph of the function f shown
above consists of two line segments. Let g be the
a) Find g (1), g (2), g '(1), g ''(1) .
x
function given by g ( x)   f (t )dt .
0
b) Find what values of x in the open interval (-2, 2)
is g increasing? Explain your reasoning.
c) For what values of x in the open interval (-2, 2)
is the graph of g concave up? Explain your
reasoning.