Riemann Sums, FTC, Definite Integrals
Big Ideas
Rectangular Approximation (left, right, midpoint)
n
Riemann Sum:
∑ f ( x )∆x
k =1
k
Definite Integral:
k
limit of a Riemann Sum
n
b
k =1
a
lim ∑ f ( xk )∆xk =
∫ f ( x) dx
n →∞
Net accumulation of a rate of change
Recognize Riemann sum as a definite integral
Integral defined functions
Fundamental Theorem of Calculus (FTC)
Integral as area:
Integrals by geometry
Net area (positive area, negative area)
Verbal description and interpretation of definite integrals
Units
Trapezoidal approximation
Integrating "right to left" vs. "left to right"
Mike Koehler
4-1
Riemann Sums, FTC, Definite Integrals
Mike Koehler
4-2
Riemann Sums, FTC, Definite Integrals
Approximations with Rectangles
(Finding the Area by Approximating with Rectangles)
Area
The area under a curve y = f ( x) can be approximated through the use of Riemann sums.=
n
∑ f ( x )∆x
k =1
k
k
.
LRAM n = Sum of n rectangles using the left-hand x-coordinate of each interval to find the height of the rectangle.
RRAM n = Sum of n rectangles using the right-hand x-coordinate of each interval to find the height of the rectangle.
MRAM n = Sum of n rectangles using the midpoint x-coordinate of each interval to find the height of the rectangle.
Let f ( x=
) x 2 + 1 on [ 0, 3] .
If there are 3 intervals, what is the value of ∆xk ?
If there are 6 intervals, what is the value of ∆xk ?
If there are n intervals and all n intervals are the same width, what is the value of ∆xk ?
For each of the following, draw the indicated rectangles and find the area of the rectangles.
On the first figure, draw three rectangles using the left-hand rule. (LRAM 3 )
Find the area of each rectangle, and find the sum of the areas.
Mike Koehler
4-3
Riemann Sums, FTC, Definite Integrals
Draw six rectangles using the lefthand rule. (LRAM 6 )
Find the area of each rectangle, and
find the sum of the areas.
Draw and compute RRAM 6
Draw three rectangles using the
mid-point rule. (MRAM 3 )
Find the area of each rectangle, and
find the sum of the areas.
Mike Koehler
4-4
Riemann Sums, FTC, Definite Integrals
Use the DRAWRECT program on the TI-84+ to complete the following table for the function f ( x=
) x + 1 . Use
this program up to n = 50 , and then use the RAM program (Rectangular Approximation Method).
Discuss whether these will under estimate or overestimate the actual area.
2
n
6
10
50
100
500
LRAM
MRAM
RRAM
Let f ( x) = 3x on [ −1,3] .
Use the DRAWRECT and RAM programs to complete the following table. Work in groups of three, with each
person taking one of the three types of approximation. Discuss whether these will under estimate or overestimate
the actual area.
n
LRAM
MRAM
RRAM
8
10
50
100
500
The definite integral is the limit of a Riemann sum as the number of intervals approach infinity (width of the
interval approaches zero) as indicated in the following formula:
n
b
k =1
a
f ( xk )∆xk =
lim
∫ f (x) dx .
n→∞ ∑
The fnInt ( m 9:fnInt( ) option on the calculator produces a numerical integral of the function. It computes an
approximation of the total area from
=
x a=
to x b . The TI-84+ with current operating system will show an integral
unless the calculator is in Classic Mode. The syntax on a TI-83+ is fnInt( function, variable of integration, left
endpoint, right endpoint ).
Sketch a graph and shade the indicated region of each of the following functions. Use fnInt to approximate the
value of each definite integral.
1.
∫ (x
3.
∫ (x
2
2
0
−2
+ 1)dx
2
0
+ 1)dx
Why is the answer to problem 3 a negative number?
5.
∫
2
−2
2.
∫ (x
4.
∫ (x
0
−2
1
0
2
2
+ 1)dx
− 1)dx
Why is the answer to problem 4 a negative number?
sin( x)dx Explain the answer to problem 5?
Draw a sketch for each problem. Use fnInt to determine the area between the curve and the x-axis.
1.
4.
7.
∫
∫
2
0
x 3 dx
2
−1
x 3 dx
2.
5.
∫
∫
0
−2
1
−2
x 3 dx
3.
x 3 dx
6.
∫
∫
2
−2
x 3 dx
−2
0
x 3 dx
Give a geometric explanation why some of the answers are negative and some are positive. If any of the
answers equal zero, give a reason why.
Mike Koehler
4-5
Riemann Sums, FTC, Definite Integrals
Integral Defined Functions
What is the relationship between the area function A( x) and the original function f (t ) ?
Problem 1. Given f (t ) = 2 on [0, x] . Find an area function, A( x) , which represents the area under the graph of f
from
=
t 0=
to t x for different values of x.
x
Fill in the chart below by evaluating A( x) = ∫ 2 dt .
0
x
A( x)
0
1
2
3
-1
To the right there is a sketch of the function
=
y f=
(t ) 2 on [0, x] and the region from 0 to
x shaded.
-2
y
8
Plot the values of A( x) on the axis.
Sketch the graph of A( x) .
4
Find a function A( x) that fits the data points.
.
A( x) =
−4
−3
−2
−1
1
2
x 3
4
x
−4
−8
Complete the steps above for the following functions:
f (t ) = 2 on [1, x]
x
A( x)
=
f (t ) 2 on [−1, x]
x
A( x)
Mike Koehler
∫
x
1
2 dt
0
∫
x
−1
0
1
2
3
-1
-2
1
2
3
-1
-2
2 dt
4-6
Riemann Sums, FTC, Definite Integrals
Problem 2.
Let f (t ) be defined by the graph shown on the right.
x
Let g ( x) = ∫ f (t )dt .
0
Rewrite the following integrals in terms of g .
∫
1
∫
−2
∫
4
0
0
1
f (t )dt =
f (t )dt =
f (t )dt =
x
x
Let h( x) = ∫ f (t )dt and k ( x) = ∫ f (t )dt . Fill in the chart below.
−3
x
g ( x)
1
-3
-1
0
1
3
4
h( x )
k ( x)
Plot the data from the table on the graph to the right.
8
y
What do you notice about the three graphs?
6
4
Express the functions h( x) and k ( x) in terms of g ( x) .
2
−3
Earlier in the course, we found the antiderivatives of
functions. We also learned that a function has infinitely
many antiderivatives, each differing by a constant.
Is this fact confirmed by the graph? Explain.
−2
1
−1
2
3
4
x
−2
−4
−6
−8
Mike Koehler
4-7
Riemann Sums, FTC, Definite Integrals
Problem 3. Let f (t=
) 2t − 2 . Sketch the graph of f on [−2, 4] on the graph below.
Let =
A( x)
∫ ( 2t − 2 ) dt .
x
0
Remember, the function A represents the area between the graph of f and the x-axis.
Use the graph of f to answer the following questions.
A(0) _______
=
A(1) _______
=
A(2) ______
A(3) _______
=
A(4) _______
A(−1) _______=
A(−2) _______
Plot the coordinates of the points found above on the same axis below as the sketch of the graph of f (t ) .
Sketch what you think A( x) might look like.
Verify your sketch by graphing fnInt(2T-2,T,0,X) on the calculator.
Find a function A( x) that fits the data points. A( x) =
The function A( x) has a minimum at x =?
The function f ( x) has at zero at x =?
The minimum of the function A( x) occurs at a
of f ( x) .
Enter the functions shown on the screen
to the right. Note the graph type for Y3.
Press WINDOW and change Xres to 4.
Graph in a decimal window (ZOOM 4).
What can you conclude about the
function A( x) and the function f ( x) ?
Mike Koehler
4-8
Riemann Sums, FTC, Definite Integrals
Problem 4: How do other functions behave?
x
Investigate the function g ( x) = ∫ cos(2t )dt .
0
Enter the functions shown on the screen to the right. Note the graph type for
Y3.
Press WINDOW and change to [ 0, 2π ] by [ −1.5,1.5] and change Xres to 4.
What can you conclude about the derivative of the function g ( x) ?
x
Add the function h( x) = ∫ cos(2t )dt by graphing in Y4 as shown on the
π 4
right.
This changes the lower limit of the integral.
How is this graph of Y4 different from Y2?
How does this relate to the graph of Y1/Y3?
Discuss any relationships between the function cos(2t ) and the graphs of g ( x) and h( x) ?
Where do the graphs of g ( x) and h( x) appear to have maximums and minimums?
How does this relate to the function cos(2t ) ?
Problem 5:
Investigate the function =
k ( x)
∫ (t
x
0
2
− 1) dt .
Enter the functions shown on the screen to the right. Note the graph type for
Y3.
Press WINDOW and change Xres to 4.
Graph in a ZOOM 4 window.
What can you conclude about the derivative of the function k ( x) ?
Find a new function with a new lower limit of integration such that the graph
of Y4 will be at least one unit higher than the graph of Y2.
Mike Koehler
4-9
Riemann Sums, FTC, Definite Integrals
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus (Part 1)
x
Define A( x) = ∫ f (t ) dt
a
 x

d  A( x)  d=
=

f (t ) dt  f ( x)


∫

 dx  a
dx 

Fundamental Theorem of Calculus (Part 2)
x
G ( x) = ∫ f (t ) dt
An antiderivative of f exists.
F=
( x) G ( x) + C
F is also an antiderivative of f . Any other antiderivative will differ by a constant.
a
[G (b) + C ] − [G (a) + C ]
F (b) − F (a=
)
= G (b) − G (a )
∫
= ∫
=∫
=
b
a
b
a
b
a
a
f (t ) dt − ∫ f (t ) dt
a
f (t ) dt − 0
f (t ) dt
b
F (b) − F (a) where F '( x) =
f ( x)
∫a f (t) dt =
The value of a definite integral of any continuous function f can be calculated without taking limits or calculating
Riemann sums as long as an antiderivative of f can be found.
Example:
∫
2
0
2
x 2 dx =
1 3
1
1
8
x = 23 − 03 =
3 0 3
3
3
Evaluate the following definite integrals.
∫ ( 2 x − 2 ) dx
3
0
∫ ( 3x
3
1
∫
π
0
2
+ 2 x )dx
sin( x)dx
Mike Koehler
4 - 10
Riemann Sums, FTC, Definite Integrals
Fundamental Theorem of Calculus Part 1 Revisited
What is
d
dx
( ∫ t dt ) ?
x
d
dx
2
1
( ∫ t dt ) = dxd ( t )
x
x
1 3
3
1
2
1
d 1 3 1
( x − 3)
dx 3
= 13 (3 x 2 ) − 0
=
= x2
What is
d
dx
Example:
(∫
x
a
d
dx
)
f (t ) dt ?
b. Write
( ∫ (sin t ) dt )
x
a. Let F be the antiderivative of f (t ) .
∫
x
a
f (t ) dt in symbolic form using FTC P2.
F ( x) − F (a)
c. Take the derivative of the answer to (b) with respect to x.
2
2
Let F be the antiderivative of f (t ) (i.e. F '(t ) = sin 2 (t ) ) and write
d
dx
( ∫ (sin t ) dt ) in symbolic form using FTC
x
2
2
P2: F (=
t)
F ( x) − F (2)
x
2
Take the derivative with respect to x:
d
0 sin 2 (=
x) f ( x)
( F ( x) − F (2)=) F '( x) −=
dx
Find the following derivatives:
d
dx
(∫
x
−2
1 + e5t dt
)
d
dx
d  x2
5 + t 3 dt 

dx  ∫2

d
dx
(∫
sin x
x2 + x
( ∫ ln(1 + t ) dt )
6
2
x
d  x3

 2 cos(2t )dt 
dx  ∫ x

( t + t )dt )
Mike Koehler
4
4 - 11
Riemann Sums, FTC, Definite Integrals
Mike Koehler
4 - 12
Riemann Sums, FTC, Definite Integrals
AP Multiple Choice Questions
2008 AB Multiple Choice
10 17 79 85
2008 BC Multiple Choice
8 81 91
2003 AB Multiple Choice
2.
∫
1
0
e −4 x dx
−e −4
4
A)
23.
d  x2

3
 sin ( t ) dt  =
dx  ∫ 0

− cos ( x 6 )
A)
77.
B)
B)
1 e −4
−
4 4
−4e −4
C)
e −4 − 1
D)
sin ( x 3 )
C)
sin ( x 6 )
3
D) 2 x sin ( x )
E)
4 − 4e −4
E)
2 x sin ( x 6 )
The regions A, B, and C in the figure on the right are
bounded by the graph of the function f and the xaxis. If the area of each region is 2, what is the value
of
A)
Mike Koehler
∫ ( f ( x) + 1) dx ?
3
−3
-2
B)
-1
C)
4
D)
4 - 13
7
E)
12
Riemann Sums, FTC, Definite Integrals
85.
If a trapezoidal sum over approximates
∫
4
0
∫
4
0
f ( x) dx and a right Riemann sum under approximates
f ( x) dx , which of the following could be the graph of y = f ( x) ?
A)
B)
C)
D)
E)
92.
Let g be the function =
given by g ( x)
g decreasing?
A) −1 ≤ x ≤ 0
D) 1.772 ≤ x ≤ 2.507
Mike Koehler
∫
x
0
sin ( t 2 ) dt for − 1 ≤ t ≤ 3 . On which of the following intervals is
B) 0 ≤ x ≤ 1.772
E) 2.802 ≤ x ≤ 3
4 - 14
C) 1.253 ≤ x ≤ 2.171
Riemann Sums, FTC, Definite Integrals
2003 BC Multiple Choice
18.
The graph of the function f shown in the figure
on the right has horizontal tangents at
=
x 3=
and x 6 .
If g ( x) = ∫
A)
27.
0
2x
0
f (t ) dt , what is the value of g ′(3) ?
B)
-1
C)
-2
D)
-3
E)
-6
d  x3

2
 ln ( t + 1) dt  =
dx  ∫ 0

A)
D)
Mike Koehler
2 x3
x6 + 1
2 x 3 ln ( x 6 + 1)
B)
E)
3x 2
x6 + 1
3 x 2 ln ( x 6 + 1)
4 - 15
6
C) ln ( x + 1)
Riemann Sums, FTC, Definite Integrals
1998 AB Multiple Choice
2.
The graph of a piecewise-linear function f , for
−1 ≤ x ≤ 4 , is shown on the right. What is the
4
value of ∫ f ( x) dx ?
−1
A)
7.
B)
2.5
x2 − 1
∫1 x dx =
1
A) e −
e
C)
4
D)
5.5
E)
8
e
D)
11
1
B) e 2 − e
e2 − 2
C)
e2
1
−e+
2
2
e2 3
−
2 2
E)
b
If f is a linear function and 0< a < b, then ∫ f ′′( x) dx =
a
A)
15.
0
B)
1
C)
b2 − a 2
2
ab
2
D) b − a
E)
2
D) 3
E) 18
D) -3 and 3
E) -3, 0, and 3
x
3
If F ( x) =
∫ t + 1 dt , then F ′(2) =
0
A)
20.
-3
B)
-2
What are all values of k for which
A)
Mike Koehler
-3
B)
0
C)
∫
k
−3
x 2 dx = 0 ?
C)
3
4 - 16
Riemann Sums, FTC, Definite Integrals
1997 AB Multiple Choice
3.
∫
If
b
a
A)
f ( x) dx =
a + 2b, then
a + 2b + 5
24.
The expression
A)
D)
∫
B)
0
b
a
5b − 5a
C)
7b − 4a
D)
7b − 5a
E)
7b − 6a
1  1
2
3
50 
+
+
+ +

 is a Riemann sum approximation for
50  50
50
50
50 
x
dx
50
1
∫ ( f ( x) + 5) dx =
1 1
x dx
50 ∫ 0
1
B)
∫
E)
1 50
x dx
50 ∫ 0
0
x dx
C)
1 1 x
dx
50 ∫ 0 50
1997 BC Multiple Choice
22.
The graph of f is shown in the figure to the right.
x
If g ( x) = ∫ f (t ) dt , for what value of x does g ( x) have a
a
maximum?
A)
B)
C)
D)
E)
Mike Koehler
a
b
c
d
It cannot be determined from the information given.
4 - 17
Riemann Sums, FTC, Definite Integrals
1985 BC Multiple Choice
2
2
2
1  1   2 
 3n  
If n is a positive integer, then lim   +   +  +    can be expressed as
n →∞ n
 n  
 n   n 
45.
A)
D)
∫
∫
1
0
3
0
2
1 1 
B) 3∫   dx
0 x
 
1
dx
x2
2
E)
x dx
3
C)
2
1
∫ 0  x  dx
3
2
3∫ x dx
0
1973 AB Multiple Choice
30.
x−4
dx =
x2
1
A)
−
2
∫
2
1
Mike Koehler
B)
ln 2 − 2
C)
ln 2
4 - 18
D)
2
E)
ln 2 + 2
Riemann Sums, FTC, Definite Integrals
Special Focus: The Fundamental
Theorem of Calculus
Multiple-Choice Questions on the Fundamental Theorem of Calculus
1. 1969 BC12
f
If F(x) =
(A) 2xe_X
e_t
d
2
t, then F’(x) =
2
(B) 2xe_X
e_+l
2
(C)
—
e
(D) e_X
2
7
1
(E) e_
2. 1969 BC22
If f(x) =
$
0 3
dt, which ofthe following is FALSE?
2
(A) f(O) = 0
(B)
f is continuous at x for all
x
0
(C)f(l)>0
(D)f’(l)=—
(E) f(—l)>0
3. 1973 AB2O
If F andf are continuous functions such that F’(x) = f(x) for all x, then
f bf(x)dx is
(A) F’(a) F’(b)
(B) F’(b) F’(a)
(C) F(a)—F(b)
(D) F(b)— F(a)
(E) none of the above
—
—
4. 1973 BC45
Suppose g’(x) <0 for all x 0 and F(x) =
following statements is FALSE?
j tg’(t)dt for all x
0. Which of the
(A) F takes on negative values.
(B) F is continuous for all x > 0.
g(t)dt
(C) F(x) = xg(x)-
f
(D) F’(x) exists for all x > 0.
(E) F is an increasing function.
AP
MikeCalculus:
Koehler 2006—2007 Workshop Materials
4 - 19
Riemann Sums, FTC, Definite Integrals
83
Special Focus: The Fundamental
Theorem of Calculus
5. 1985 AB42
drxl
I V1+t 2 dt=
J
dx 2
—
(A)
(B) 1 + x
2 —5
1+x2
(D)
—
±
1+x2
1
1
2Jl+x2
2q5
(E)
(C) 1 + x
2
6. 1988 AB13
If the functionf has a continuous derivative on [O,c], then
(A) f(c)
f(O)
-
(E) f”(c)
—
(B) f(c)
f(O)J
(C) f(c)
=
(D) f(x) + c
f”(O)
7. 1988 AB25
=
For all x> 1, if f(x)
(A) 1
-
f’ f’(x)dx
(B)
fx
1
-dt, then f’(x) =
(C) mx —1
(D) mx
(E) eX
8. 1988 BC14
If F(x)
=
J
X
+t
3
(A) 2xl + x
6
x2
(E)f
2
3t
r
dt, then F’(x) =
(B) 2xl + x
3
(C) l +
(D)
+
dt
211+t
9. 1993 AB41
drx
I cos(2itu)du is
dx” 0
—
(A) 0
84
Mike Koehler
(B) *sinx
2t
(C) c
1
os(2)
2t
4 - 20
(D) cos(2)
(E) 2cos(2)
Riemann Sums, FTC, Definite Integrals
AP Calculus: 2006—2007 Workshop Materials
Special Focus: The Fundamental
Theorem of Calculus
10. 1993 BC41
=
Let f(x)
jx3x
2 dt. At what value of x is f(x) a minim
et
um?
(A) For no value of x
(B)
(C)
(D) 2
(E) 3
11. 1997 AB78
I
3
I
_•_•
I
I
The graph off is shown in the figure above. If f
f(x)dx= 2.3 and F’(x)
3
1
then F(3) F(O)
—
(A) 0.3
=
f(x),
=
(B) 1.3
(C) 3.3
(D) 4.3
(E) 5.3
12. 1997 BC22
The graph off is shown in the figure above. If g(x)
does g(x) have a maximum?
(A) a
(B) b
(C) c
=
f
fQ) dt, for what value of x
(D) d
(E) It cannot be determined from the information given.
AP
Calculus:
Mike
Koehler 2006—2007 Workshop Materials
4 - 21
Riemann Sums, FTC, Definite Integrals
85
Special Focus: The Fundamental
Theorem of Calculus
13. 1997 BC88
Let f(x) =
$
:2
sin t dt At how many points in the closed interval
.
[o,
]
does the
instantaneous rate of change offequal the average rate of change offon that interval?
(A) Zero
(B) One
(C)Two
(D) Three
(E) Four
14.1997BC89
Iffis the antiderivative of
X
2
such that f(l) =
5
l+x
(A) —0.012
(B) 0
(C) 0.016
(D) 0.376
,
then f(4) =
(E) 0.629
15. 1998 AB9
6
12
24
Hours
The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph
shown above. Of the following, which best approximates the total number of barrels
of oil that passed through the pipeline that day?
(A) 500
86
Mike Koehler
(B) 600
(C) 2,400
(D) 3,000
4 - 22
(E) 4,800
Riemann Sums, FTC, Definite Integrals
AP Calculus:
2006—2007 Workshop Materials
Special Focus: The Fundamental
Theorem of Calculus
16. 1998 AB11
1ff is a linear function and 0< a
(A) 0
(B) 1
(C)
<
(D) b
b, then
-
a
f”(x) dx
(E)
2
b
-
a
17. 1998 AB15
If F(x) =
fJt3 +1
(A) —3
(B) —2
dt, then F’(2) =
(C) 2
(D) 3
(E) 18
18. 1998 AB88
(lnx)
3
Let F(x) be an antiderivative of
If F(1)
x
.
(A) 0.048
(B) 0.144
APMike
Calculus:
Koehler 2006—2007 Workshop Materials
(C) 5.827
=
0 then F(9)
(D) 23.308
4 - 23
=
(E) 1,640.250
Riemann Sums, FTC, Definite Integrals
87
_______________
Special Focus: The Fundamental
Theorem of Calculus
19. 1998 BC88
—
0
Let g(x)
=
f
X
h
a
f(t) dt, where a
x
b. The figure above shows the graph of g on
[a, b]. Which of the following could be the graph off on [a, b]?
(A)
y
0
(D)
Mike Koehler
V
(E)
y
(C)
y
/
V
0
88
a
(B)
I)
4 - 24
Workshop
2006—2007
AP’ Calculus:
Riemann
Sums, FTC,
DefiniteMaterials
Integrals
Special Focus: The Fundamental
Theorem of Calculus
20. 2003 AB22
6
C)
The graph off’, the derivative off, is the line shown in the figure above. Iff(O)
then f(1) =
(A)O
(B)3
(C)6
(D)8
=
5,
(E) 11
21. 2003 AB82/BC82
The rate of change of the altitude of a hot-air balloon is given by r (t) =
2 +6
4t
for 0 t 8. Which of the following expressions gives the change in altitude of the
balloon during the time the altitude is decreasing?
—
(A)
(B)
3.5 14
11.572
f(t)dt
2.667
(C)
(D)
(E)
r(t)dt
r(t)dt
3.5 14
I r’(t)dt
.‘1.572
f
2.667
r’(t)dt
Mike
Koehler 2006—2007 Workshop Materials
AP
Calculus:
4 - 25
Riemann Sums, FTC, Definite Integrals
89
Special Focus: The Fundamental
Theorem of Calculus
22. 2003 AB84
A pizza, heated to a temperature of 350 degrees Fahrenheit (° F) is taken out of an
oven and placed in a 75° F room at time t = 0 minutes. The temperature of the pizza
t degrees Fahrenheit per minute. To the nearest
4
is changing at a rate of —1 lOe_
degree, what is the temperature of the pizza at time t = 5 minutes?
(A) 112°F
(B) 119°F
(C) 147°F
(D) 238°F
(E) 335°F
23. 2003 AB91
A particle moves along the x-axis so that at any time t> 0, its acceleration is given by
a(t)
=
t).
ln(l + 2
If the velocity of the particle is 2 at time t
=
1 then the velocity of
the particle at time t = 2 is
(A) 0.462
(B) 1.609
(C) 2.555
24. 2003 AB92
Let g be the function given by g(x)
following intervals is g decreasing?
=
(D) 2.886
fsin(t2)dt for —l
(E) 3.346
x
3. On which of the
(A)—lx0
(B) 0xl.772
(C) l.253x2.17l
(D) 1.772 x 2.507
(E) 2.802x3
90
Mike Koehler
4 - 26
Riemann
Sums, FTC,
Definite
Integrals
AP Calculus:
2006—2007
Workshop
Materials
Special Focus: The Fundamental
Theorem of Calculus
25. 2003 BC18
7
6
4
I
0
—,
—j
Graph oI/
The graph of the functionf shown in the figure above has horizontal tangents at x
and x =6. If g(x)
(C)—2
(B)—l
(A)O
fQ) dt,
=
=
3
what is the value of g’(3)?
(E) —6
(D) —3
26. 2003 BC27
d
—if
dx 0
x
+1)dt
2
ln(t
3
2x
(A)
6
x +1
(E)
(B)
=
2
3x
6
x+l
(C) in(x6 + i)
(D) 2x31n(x6 +
3x21n(x6+l)
MikeCalculus:
Koehler 2006—2007 Workshop Materials
AP
4 - 27
Riemann Sums, FTC, Definite Integrals
91
Special Focus: The Fundamental
Theorem of Calculus
27. 2003 BC8O
lOOe_Olt
tons per day, where time t is
Insects destroyed a crop at the rate of
—3t
2
measured in days. To the nearest ton, how rany tons did the insects destroy during
the time interval 7
(A) 125
t
14?
(C) 88
(B) 100
(D) 50
(E) 12
28. 2003 BC87
A particle moves along the x-axis so that at any time
v(t) = cos(2
—
t2).
t
0, its velocity is given by
The position of the particle is 3 at time t
=
0. What is the position
of the particle when its velocity is first equal to 0?
(A) 0.411
Mike Koehler
(B) 1.310
(C) 2.816
4 - 28
(D) 3.091
(E) 3.411
Riemann Sums, FTC, Definite Integrals
AP Free Response Questions
2012 AB3
Let f be the continuous function defined on [ −4,3]
whose graph, consisting of three line segments and a
semicircle centered at the origin, is given on the
x
right. Let g be the function given by g ( x) = ∫ f (t ) dt .
1
a)
b)
c)
d)
Find the values of g (2) and g (−2) .
For each of g ′(−3) and g ′′(−3) , find the value or state that it does not exist.
Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each
of these points, determine whether g has a relative minimum, relative maximum, or neither a
minimum nor a maximum at the point. Justify your answers.
For − 4 < x < 3 , find all values of x for which the graph of g has a point of inflection. Explain your
reasoning.
2009 AB5
Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected
points in the closed interval 2 ≤ x ≤ 13 .
a) Estimate f ′(4) . Show the work that leads to your answer.
13
b)
Evaluate ∫ ( 3 − 5 f ′( x) ) dx . Show the work that leads to your answer.
2
c)
d)
Use a left Riemann sum with subintervals indicated by the data in the table to approximate
∫
13
2
f ( x) dx .
Show the work that leads to your answer.
Suppose
=
f ′(5) 3 and f ′′( x) < 0 for all x in the closed interval 5 ≤ x ≤ 8 . Use the line tangent to the graph of
f at x = 5 to show that f (7) ≤ 4 . Use the secant line for the graph of f on 5 ≤ x ≤ 8 to show that f (7) ≥
Mike Koehler
4 - 29
4
.
3
Riemann Sums, FTC, Definite Integrals
2009 AB6
The derivative of a function f is defined by
 g ( x)
f ′( x) =  − x 3
5e − 3
for − 4 ≤ x ≤ 0
for 0 < x ≤ 4
.
The graph of the continuous function f ′ , shown in the
figure to the right, has x-intercepts at x = −2 and
5
x = 3ln   . The graph of g on − 4 ≤ x ≤ 0 is a
3
semicircle, and f (0) = 5 .
a) For − 4 ≤ x ≤ 4 , find all values of x at which the graph of f has a point of inflection. Justify your answer.
b) Find f (−4) and f (4)
c) For − 4 ≤ x ≤ 4 , find the value of x at which has an absolute maximum. Justify your answer.
2003 AB4 BC4
Let f be a function defined on the closed interval
−3 ≤ x ≤ 4 with f (0) = 3 . The graph of f ′ , the
derivative of f , consists of one line segment and a
semicircle, as shown on the right.
Graph of f '
a)
c)
On what intervals, if any, is f increasing? Justify your answer.
Find the x-coordinate of each point of inflection of the graph of f on the open interval − 3 < x < 4 . Justify
your answer.
Find an equation tangent to the graph of f at the point ( 0, 3) .
d)
Find f (−3) and f (4) . Show the work that leads to your answers.
b)
Mike Koehler
4 - 30
Riemann Sums, FTC, Definite Integrals
1999 AB5
The graph of the function f , consisting of three line
x
segments, is give above. Let g ( x) = ∫ f (t )dt
1
a)
b)
c)
d)
Compute g (4) and g (−2) .
Find the instantaneous rate of change of g , with respect to x, at x = 1 .
Find the absolute minimum value of g on the closed interval [ −2, 4] .
The second derivative of g is not defined
at x 1=
=
and x 2 . How many of these values are x-coordinates of
points of inflection of the graph of g ? Justify your answer.
1997 AB5
The graph of a function f consists of a semicircle and two
line segments as shown above. Let g be the function
x
given by g ( x) = ∫ f (t )dt .
0
a)
Find g (3)
b)
Find all values of x on the open interval ( −2,5 ) at which g has a relative maximum. Justify you answers.
c)
d)
Write an equation for the line tangent to the graph of g at x = 3.
Find the x-coordinate of each point of inflection of the graph of g on the open interval (-2,5). Justify your
answer.
Mike Koehler
4 - 31
Riemann Sums, FTC, Definite Integrals
1995 AB6
The graph of a differentiable function f on the closed interval [1,7 ] is shown above.
=
Let h( x)
a)
b)
c)
d)
∫
x
1
f (t )dt for 1 ≤ x ≤ 7 .
Find h(1) .
Find h′(4)
On what interval or intervals is the graph of h concave upward? Justify your answer.
Find the value of x at which h has its minimum on the closed interval [1,7]. Justify your answer.
1995 BC6
Let f be a function whose domain is the closed interval
[0,5] .
The graph of f is shown on the right.
x
Let h( x) = ∫ 2
0
+3
f (t )dt .
Graph of f
a)
b)
Find the domain of h .
Find h′(2)
c)
At what x is h( x) a minimum? Show the analysis that leads to your conclusion.
Mike Koehler
4 - 32
Riemann Sums, FTC, Definite Integrals
1976 AB6
x
Given 5 x 3 + 40 =
∫ f (t )dt :
c
a)
Find f ( x)
Find the value of c .
b)
( x)
If F=
Mike Koehler
∫
3
x
1 + t16 dt , find F ′( x) .
4 - 33
Riemann Sums, FTC, Definite Integrals
Textbook Problems
Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, l2012
Section
6.1
6.2
6.3
QQ p 297
6.4
6.5
QQ p 319
6.R
Questions
18 28
37 - 40
2 4 7 15
1234
57 58 59 75
10 (use trapezoidal rule)
1234
46 51 52 54 58('99AB3)59 60('99AB5)
P 283 Exploration 1
Handouts
Mike Koehler
4 - 34
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Section 1
1.
You wish to measure the quantity of water that is
piped into a factory during a typical morning. A
gauge on the water line gives the flow rate (in cubic
meters per hour) at any instant.
The flow rate is 80 m3 hr at 6 am and increases
steadily to 280 m3 hr at 10 am.
Give your best estimate of the total volume of water
used by the factory between 6 am and 10 am.
Use the graph at the right to show your work.
2.
At time t , in seconds, your velocity, in feet/second, is
given by v(t ) = 1 + t 2 for 0 ≤ t ≤ 6 .
Use ∆t =2 to estimate the distance traveled during
the time.
Find both LRAM and RRAM and average the two.
Use the graph at the right to show your work. Draw
both the LRAM and RRAM rectangles.
3.
A man is training for a race. His friend follows him on a bicycle and clocks his speed every 15 minutes.
The runner starts out strong, but after an hour and a half he stops. The runner's data is in the following table.
Time (min.)
Speed (mph)
0
12
15
11
30
10
45
10
60
8
75
7
90
0
a.
Assuming the runner's speed is never increasing, give upper and lower estimates for the distance ran during
the first half hour.
b.
Assuming the runner's speed is never increasing, give upper and lower estimates for the distance ran during
the entire hour and a half.
Mike Koehler
4 - 35
Riemann Sums, FTC, Definite Integrals
4.
A car comes to a stop six seconds after the driver applies
the brakes. While the brakes are on, the following
velocities are recorded.
Time since brakes applies (sec.)
Velocity (ft. / sec.)
0
88
2
45
4
16
6
0
a.
Give lower and upper estimates for the distance the car
traveled after the brakes were applied.
b.
On a sketch of velocity versus time, show the lower and
upper estimates from part (a).
5.
Two cars travel in the same direction along a straight
road. The figure on the right show the velocity v of each
car at time t .Car B starts two hours after car A and car B
reaches a maximum velocity of 30 mph .
a.
For approximately how long does each car travel?
b.
Estimate car A's maximum velocity.
c.
Approximately how far does each car travel?
6.
Two cars start from rest at a traffic light and accelerate
for several minutes. The graph at the right shows their
velocities as a function of time.
Which car is ahead after one minute?
Which car is ahead after two minutes?
Give reasons for your answers.
Mike Koehler
4 - 36
Riemann Sums, FTC, Definite Integrals
7.
A truck driver starts a trip by driving at 40 mph for 2 hours and the cruises at 60 mph for 1.5 hours. The
driver stops for lunch which takes a half hour and finishes the trip by driving at 30 mph for another hour.
(Assume the changes in speed take place instantly.)
a.
b.
c.
On the axis below left, graph the speed of the truck as a function of the time t in hours.
On the axis below right, graph the distance traveled D as a function of time.
Find a formula for the distance from the starting point as a function of time for the last hour of the trip.
8.
As you come over a hill while driving your car, your spot an obstacle in the road 200 feet ahead. You
immediately apply your brakes, begin slowing down, and discover that the object is a dead skunk in the
middle of the road. The speed decreases through the 8 seconds it takes to stop, although not necessarily at a
uniform rate. Your velocity every two seconds is recorded in the following table.
Time since brakes applies (sec.)
Velocity (ft. / sec.)
0
80
2
50
4
25
6
10
8
0
a.
What is your best estimate of the total distance that your car traveled before coming to rest? Show the work
that leads to your answer.
b.
Which of the following statements can you justify from the information given?
(i)
(ii)
(iii)
Mike Koehler
The car stopped before getting to the skunk.
The data is inconclusive. The skunk may or may not have been hit.
The skunk was hit by the car.
4 - 37
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Section 1
Answers
1.
720 cubic meters
2.
LRAM = 46 feet, RRAM = 108 feet
Average = 77 feet
3.
a. Upper = 5.75 miles Lower = 5.25 miles
b. Upper = 14.5 miles Lower = 11.5 miles
4.
a. Upper = 294 feet Lower = 122 feet
5.
a. Car A: 8 hours Car B: 4 hours
b. 60 mph
c. Car A: 240 miles Car B: 60 miles
Car 1 after one minute.
Car 2 after two minutes.
6.
7.
a)
b)
c) f (t )= 170 + 30(t − 4), 4 ≤ t ≤ 5
8.
a) 250 feet
b) Statement ii.
Mike Koehler
4 - 38
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Section 2
The graph of a function f shown below consists of two straight lines and two one-quarter circles.
Evaluate each of the following integrals.
1.
∫
2
3.
∫
5
5.
∫
4
7.
∫
9
9.
∫
12
0
0
4
5
9
f ( x)dx
2.
∫
5
f ( x)dx
4.
∫
9
f ( x)dx
6.
∫
15
f ( x) dx
8.
∫
15
f ( x)dx
10.
∫
9
Mike Koehler
4 - 39
2
5
0
12
15
f ( x)dx
f ( x)dx
f ( x)dx
f ( x)dx
f ( x)dx
Riemann Sums, FTC, Definite Integrals
Answers
1
3
2
9π
4
3
9π
+3
4
4
−
5
0
6
−9 −
7
4π
8
-4
9
−9
10
12
Mike Koehler
4 - 40
16π
=
−4π
4
7π
4
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Section 3.1
The graph of a function f is sketched below.
Find each of the following.
3
1. ∫ f (t )dt
0
−2
4. ∫
0
7. ∫
( 3 − f (t ) ) dt
0
f (t )dt
4
2. ∫
0
−3
f (t )dt
0
5. ∫ (2 f (t ) + 1)dt
−3
8. ∫
4
0
f (t ) dt
3
3. 4 ∫ f (t )dt
0
3
6. ∫ ( f (t ) + t )dt
0
9.
∫
4
0
f (t )dt
10. The average value of f on [0, 4] .
Mike Koehler
4 - 41
Riemann Sums, FTC, Definite Integrals
Answers
1.
5
2.
3
2
3.
20
4.
-2
5.
6
6.
9.5
7.
8
8.
6
9.
4
10.
1
Mike Koehler
4 - 42
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Section 3.2
Suppose that: ∫
5
5
5.
8
8
−2
−2
5
0
f ( x)dx =
18 ∫ g ( x)dx =
5
−11 ∫ f ( x)dx =
−3 ∫ f ( x)dx =
15
∫ h( x)dx =
−2
Evaluate the following integrals.
−2
1.
∫ ( f ( x) + g ( x) )dx
5.
∫
2.
∫ ( f ( x) − h( x) )dx
6.
∫ ( h( x) + 1)dx
3.
∫
7.
∫
4.
∫ ( −4 g ( x) )dx
8.
∫ ( f ( x) )dx
5
−2
5
−2
5
−2
8
f ( x) dx + ∫ 2 f ( x) dx
5
5
−2
5
f ( x)dx
5
−2
7
0
g ( x − 2)dx
5
0
9 - 12. Suppose that a car travels on an east-west road with eastward velocity v(t=
) 60 − 20t mph at t hours.
4
9.
Evaluate ∫ v(t )dt . Interpret the answer in car talk. (Hint: draw graph of v(t ) .)
10.
Find the average value of v(t ) on the interval [0, 4] .
11.
Let s (t ) be the cars speed at time t. Evaluate
0
∫
4
0
s (t )dt and interpret the answer in car talk.
(Hint: draw graph of s (t ) .)
12.
What is the car's average speed between t = 0 and t = 4?
Mike Koehler
4 - 43
Riemann Sums, FTC, Definite Integrals
Answers
1.
23
2.
29
3.
12
4.
-20
5.
-18
6.
-4
7.
5
8.
18
9.
80 miles – displacement
10.
20 mph
11.
100 miles – total distance traveled
12.
25 mph
Mike Koehler
4 - 44
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Sections 1 to 3
1.
Suppose that C (t ) represents the cost of heating your house, measured in dollars per day, where t is time
90
measured in days and t = 0 corresponds to January 1. Interpret ∫ C (t ) dt and
0
2.
90
1
C (t ) dt .
∫
0
90 − 0
Two cars start from rest at a traffic light and
accelerate for several minutes. The graph at the
right shows their velocities as a function of time.
Which car is ahead after one minute?
Which car is ahead after two minutes?
3. For the even function f shown on the right,
consider the average value of f over the
following intervals:
I.
0 ≤ x ≤1
II.
0≤ x≤2
III.
0≤ x≤5
IV.
−2 ≤ x ≤ 0
For which interval is the average value of f
least?
For which interval is the average value of f
greatest?
For which pair of intervals are the average
values equal?
Mike Koehler
4 - 45
Riemann Sums, FTC, Definite Integrals
4.
For the even function shown in number 3. Answer the following. Answers will be definite integrals.
2
2
0
−2
Suppose you know ∫ f ( x) dx , what is ∫
f ( x) dx ?
5
5
2
0
2
0
Suppose you know ∫ f ( x) dx and ∫ f ( x) dx , what is ∫ f ( x) dx ?
Suppose you know ∫
5
−2
f ( x) dx and ∫
2
−2
5
f ( x) dx , what is ∫ f ( x) dx ?
0
5.
A piece of steel at 1500oF is removed from the oven and placed in a room at 70oF. The temperature T of the
steel, t minutes after it starts cooling is given by T= 70 + 1430e −0.2t . Find, to the nearest degree, the average
temperature of the steel over the first hour. (Calculator should be used.)
6.
Oil is leaking from a ruptured tanker at the rate of 500e −0.4t gallons hour . To the nearest gallon, what is the
average rate of leakage of oil in the first five hours, starting at time t = 0 ? (Calculator should be used.)
7.
x) 4 x 3 − 2 x over the interval 2 ≤ x ≤ 3 .
Find the average value of the function f (=
8.
=
and ∫ f ( x) dx 2, =
then ∫ f ( x) dx
If f ( x) is an odd function,
∫ f ( x) dx 8,=
9.
and ∫ f ( x) dx 2,=
then ∫ f ( x) dx
=
If f ( x) is an even function,
∫ f ( x) dx 10,=
10.
If
11.
If=
∫ f ( x) dx k , then
12.
The table gives the values of a continuous function.
x
20 25 30
f ( x) 42 38 31
4
f ( x) dx
∫=
−2
5
3
3, and
6
3
3
0
0
2
−2
3
5
5
−3
3
0
f ( x) dx
∫=
−2
∫
5
3
1, then
x) + 5] dx
∫ [ f (=
6
4
1
f ( x) dx − ∫ =
f ( x + 2) dx
3
35
29
40
34
45
48
50
60
Use a right Riemann sum to estimate the average value of f on [ 20,50] .
Mike Koehler
4 - 46
Riemann Sums, FTC, Definite Integrals
13.
If f ( x) is the function shown on the right, then when
will
14.
∫
a
f ( x) dx, a ≥ 0 equal 0?
0
The graph of a function h is shown at the right.
List from smallest to largest:
(i)
The average value of h over the
interval [ 0,10].
(ii)
The average rate of change of h over the
interval [ 0,10].
15.
(iii)
h′(5)
(iv)
∫
(v)
∫
(vi)
∫
10
0
5
0
h( x) dx
10
6
h( x) dx
h( x) dx
A truck driver starts a trip by driving at 40 mph for 2 hours and the cruises at 60 mph for 1.5 hours. The
driver stops for lunch which takes a half hour and finishes the trip by driving at 35 mph for another hour.
(Assume the changes in speed take place instantly.)
a. Graph the speed of the truck as a function of the time t in hours.
b. Graph the distance traveled D as a function of time.
c. Find a formula for the distance from the starting point as a function of time for the last hour of the trip.
d. What is the average speed for the entire trip (including lunch)?
e. It appears that the driver was never going this speed. What small change in the description of the journey
would allow us to conclude that the drive did hit this average speed?
Mike Koehler
4 - 47
Riemann Sums, FTC, Definite Integrals
16.
17.
1
and the x-axis from
=
x 10
=
to x 20 is the same as the area
x
between the curve and the x-axis from
=
x 1=
to x 2 . Find another integral along the x-axis where the area
between the curve and the interval is the same. (Do not use a calculator on this problem.)
Show that the area between the curve y =
The graph of g ( x) is shown at the right.
For which value of c in [ −1,5] will
∫
c
−1
g ( x) dx be the largest.
Justify your answer.
Write an expression involving a definite
integral that gives the average value of
g for − 1 ≤ x ≤ 5 .
Use the graph to estimate the average value
and draw the value on the graph.
18.
The plot of the function g (t ) is shown on the
right.
Explain how the graph of g (t ) signals that the
x
plot of f ( x) = ∫ g (t ) dt
0
Goes down as t advances from 0 to 1.
Goes up as t advances from 1 to 3.
Goes down as t advances from 3 to 4.
Mike Koehler
4 - 48
Riemann Sums, FTC, Definite Integrals
19.
The function f is continuous for all real numbers. The average value of f ( x) on the closed interval
[ −1, 4] is 6.
20.
4
−1
[ f ( x) + 3] dx,
a≥0?
For the function f ( x)= 5 − 2 x , use a graph (not a calculator) to evaluate the following definite integrals.
∫
21.
What is ∫
2
0
f ( x) dx
∫
4
1
f ( x) dx
∫
−1
−5
∫
f ( x) dx
Sketch the graph of a function f with the property that
3
f ( x) dx
−2
∫
5
1
5
f ( x) dx < ∫ f ( x) dx .
1
Explain why the function f you sketched has this property.
22.
A car accelerates from rest at a rate of 2 meters second 2 . Then the brake is applied and the car decelerates at
a rate of 4 meters second 2 until it stops. The total travel time is 12 seconds. What is the total travel
distance?
Hint: find the two velocities with the initial conditions given in the problem then graph the velocities.
Mike Koehler
4 - 49
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5
Answers
1.
Total cost and average cost per day to heat your
house for 3 months.
2.
Car 1 after one minute. Car 2 after two
minutes.
3.
III, I, II and IV
4.
2 ∫ f ( x) dx
2
0
∫
∫
5
0
5
−2
5
f ( x) dx − ∫ f ( x) dx
2
1 2
f ( x) dx − ∫ f ( x) dx
2 −2
5.
189oF
6.
216 gallons per hour
7.
60
8.
-6
9.
7
10.
8
11.
2k
1200
= 40
30
13.=
a 0=
a 2
12.
14.
15.
15. b)
c) f (t )= 170 + 35 ( t − 4 ) , 4 ≤ t ≤ 5
d) 41 mph
e) speed change does not take place instantaneously.
16.
Area is ln(2)= ln(20) - ln(10) and ln(2) - ln(1) Any
interval [ a, 2a ] , a > 0 .
17. Largest value when c = 3 .
5
1
g ( x) dx
∫
−
5 − ( −1) 1
Average value approximately 1/3.
18.
Accumulating negative area, accumulating positive
area, accumulation negative area.
19.
45
20.
6
21.
The function f will have more area above the xaxis than below the x-axis.
22.
96 meters ( v1 (t ) =2t
0
44
20
vi, iii, ii, i, iv, v
v2 (t ) =−4t + 48 )
a)
Mike Koehler
4 - 50
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Section 4
The graph of f is defined
over the closed interval [-5, 5]
and is made up of straight
lines and a semicircle as
shown in the graph on the
right.
Let h( x) = ∫
x
−2
f (t )dt .
Answer the following
questions.
Graph of f (t )
Give reasons for your
answers.
1. h(2)
=
2. h(−5)
3. h(4)
=
4. h(5)
5. h '(0)
=
6. h '(4)
7.
Where in the interval [−5,5] does h( x) have a relative minimum?
8.
Where in the interval [−5,5] does h( x) have a relative maximum?
9.
On which subinterval(s) of [−5,5] is h( x) increasing?
10. On which subinterval(s) of [−5,5] is h( x) decreasing?
11. On which subinterval(s) of [−5,5] is h( x) concave down?
12. On which subinterval(s) of [−5,5] is h( x) concave up?
13. At which x-coordinates in [−5,5] does h( x) have point(s) of inflection?
Mike Koehler
4 - 51
Riemann Sums, FTC, Definite Integrals
Answers
1
2π
2
4.5
3
2π − 2
4
2π − 3
5
′(0) f=
h=
(0) 2
6
h′(4) = f (4) = −2
7
8
−2
5
2
−5
because h′( x) =
f ( x) changes sign from negative to positive
′
because h ( x) = f ( x) is negative to the left of 5
because h′( x) = f ( x) changes sign from positive to negative
because
=
h′( x) f ( x) is negative to the right of − 5
9
[ −2, 2]
10
[ −5, −2] [ 2,5]
11
( 0, 4 )
12
( −5, 0 ) ( 4,5)
13
0, 4 because h′( x) = f ( x) changes from increasing to decreasing or decreasing to increasing
Mike Koehler
4 - 52
because h′( x) =
f ( x) > 0
because h′( x=
) f ( x) < 0
because h′( x) = f ( x) is decreasing
because h′( x) =
f ( x) is increasing
Riemann Sums, FTC, Definite Integrals
AP Calculus
Chapter 5 Section 4
Fundamental Theorem of Calculus Part 1
x
Define A( x) = ∫ f (t ) dt
a
d
d  x

=
A( x) ) =
(
 f (t ) dt  f ( x)
dx
dx  ∫a

What is
( ∫ t dt ) ?
d
dx
x
d
dx
2
1
( ∫ t dt ) = dxd ( t )
x
x
1 3
3
1
2
1
d 1 3 1
( x − 3)
dx 3
= 13 (3 x 2 ) − 0
=
What is
(∫
d
dx
x
a
= x2
a. Let F be the antiderivative of f (t ) .
)
f (t ) dt ?
b. Write
∫
x
a
f (t ) dt in symbolic form using FTC P2.
F ( x) − F (a)
c. Take the derivative of the answer to (b) with respect to x.
d
) ) F ′( x) −=
0 f ( x)
( F ( x) − F (a=
dx
)
(
d x
sin 2 t ) dt
(
∫
2
dx
Let F be the antiderivative of f (t ) (i.e. F '(t ) = sin 2 (t ) ) and write
Example:
d
dx
F (=
t)
F ( x) − F (2)
x
( ∫ (sin t ) dt ) in symbolic form using FTC P2.
x
2
2
2
d
0 sin 2 (=
x) f ( x)
( F ( x) − F (2)=) F '( x) −=
dx
Take the derivative with respect to x.
Find the following derivatives:
x
d
1 + e5t dt
∫
−
2
dx
d
dx
d  x2
5 + t 3 dt 

dx  ∫2

d  x3

 2 cos(2t )dt 
dx  ∫ x

)
(
d
dx
(∫
sin x
x2 + x
( ∫ ln(1 + t dt )
6
2
x
( t + t )dt )
Mike Koehler
4
4 - 53
Riemann Sums, FTC, Definite Integrals