Welcome to the
Jeopardy of Calculus
AB. The questions are
based on Chapter 4
Materials.
Let’s go to the
Game Board
Categories
Area
Integration Mental Funds U-Substitution Potpourri
$ 100
$ 100
$ 100
$ 100
$ 100
$ 200
$ 200
$ 200
$ 200
$ 200
$ 300
$ 300
$ 300
$ 300
$ 300
$ 400
$ 400
$ 400
$ 400
$ 400
$ 500
$ 500
$ 500
$ 500
$ 500
Final Jeopardy
Area
100
Find the sum.
20
2
i

i 1
Answer
Answer
The distributive property of summation says that
we can place any constant outside of the sigma
notation.
20
2 i
i 1
 (20)(20  1) 
= 2
  2 210   420
2


Categories
Area
200
Find the sum.
4
1

2
k 0 k  1
Answer
Answer
We evaluate this sum by replacing k
with 0, then 1,2,3, and 4 respectively.
1
1
1
1
1
158
 2
 2
 2
 2

0  1 1  1 2  1 3  1 4  1 85
Categories
Daily Double!!!
• Choose the amount you
would like to wager.
(Up to half your score)
Question
Area
300
Find a formula for the sum of n terms. Use the
formula to find the limit as n infinity.
n
16i
lim x  2
i 1 n
Answer
Answer
Evaluate the summation:
After simplifications:
And the answer:
Categories
 16   n  n  1 
lim x  2  

 n  2 
1

8  lim x 1  lim x 
n

81  0  8 1  8
Area
400
Determine a value that best approximates the area of
the x-axis and the graph of the function over the
indicated interval.
f(x)= 4 – x2, [ 0,2]
Answer
Answer
First sketch a graph of the function:
Then approximate counting the squares:
A=1+1+1+1+0.5+0.5+0.5=5.5
Categories
Area
500
Find the sum:
 2i  1 

 2 

i 1  n
1000
Answer
Answer
1000
1
Solve the summation:
 2i  1

1000 i 1
1000  2 1002

 1.002
1000
1000
Categories
Integration
100
x

4
x

2
xdx

3
Answer
Answer
Integrate each part separately.
Don’t forget the C! Then rewrite
to:
3x  4  C
2
Categories
Integration
200
3dx

Answer
Answer
Integrate
3x  C
Categories
Integration
300
(
t

sin
t
)
dt

2
Answer
Answer
First, integrate t2 separately from
–sint. Then integrate –sint. (Chain
Rule) Then rewrite to:
2t  cos t  C
Categories
Integration
400
Solve the differential equation:
f (s)  6s  8s , f (2)  3
'
Answer
3
Answer
First, integrate the equation. Then plug in
2 for x, and find C. Then rewrite to:
f (2)  3s  4s  23
2
Categories
4
Integration
500
Calculate the indefinite integral:
5x  4 x  2 x
dx

x
4
Answer
2
Answer
First divide the entire equation by the x in the
denominator. Then integrate each piece
independently of each other.
5 4
2
x  2x  2x  C
4
Categories
Mental Funds
100
Find the area of the region
bounded by the equation:
y  2 x  3x  2
2
On the interval [0,2].
Answer
Answer
2 3 3 2
2
x  x  2x 0
3
2
2 3 3 2
10
(2)  (2)  2(2) 
3
2
3
Categories
Mental Funds
200
Find the average value of:
f ( x )  3x  2 x
2
On the interval [1,4].
Answer
Answer
1 4 2
1 3
2 4
3x  2 xdx =  x  x 



1
3 1
3
1
 43  42   13  12  

3
48
 16
3
Categories
Mental Funds
300
Evaluate the definite integral:

1
0
Answer
(2t  1) dt
2
Answer
1
First Expand: 0 (4t 2  4t  1)dt
1
Then Integrate:
4 3

2
t

2
t

t
 3

0
Then plug in the
4 3
Numbers: (1)  2(1)2  (1) 
3

Answer:
Categories
1
3
Mental Funds
400
Using the second fundamental
theorem of Calculus find F’(x):
F  x  
t  2 dt

2
Answer
x
Answer
Simply plug in x for t.
F '( x)  x  2x
2
Categories
Mental Funds
500
Using the second fundamental
theorem of Calculus find F’(X)
x3
F  x    sin t 2dt
0
Answer
Answer
Using chain rule derive the
inside and then plug in the
3
x.
F '( x)  sin( x ) (3x )  3x sin x
3 2
Categories
2
2
6
U-Substitution
100
2sin
x
cos
xdx

Answer
Answer
sin
2xdx

Thus: u = 2x; du = 2dx; ½du =
dx
 ½(cosu)  ½cos2x
½cos2x
Categories
U-Substitution
200
x
dx
 x2  1
Answer
Answer
U = x2+1; du = 2x dx; ½du = x
dx du
½ u =½

Categories
U-Substitution
300
cos x
dx
 sin x
Answer
Answer
x  1, x  3
Categories
U-Substitution
400
sec x tan x
dx
 sec x  1
Answer
Answer
Categories
y  2x 1
y  3
x2
U-Substitution
500
2
csc x
dx
 cot 3 x
Answer
Answer
y  ( x  2)  3
2
Categories
Potpourri
100
Derive:
(2x  3)  (5x  3x  5)
2
Answer
2
Answer
(4 x  0)  (10 x  3  0)  6 x  3
3(2 x  1)
Categories
Potpourri
200
Find the limit:
sin x
lim
x
x 0
Answer
Answer
You should know that as x approaches 0 in the
limit:
sin x
lim
x
x 0
The limit equals one.
Categories
Potpourri
300
Derive this equation:
y  2xy  2x  4
2
Answer
2
Answer
2 yy  2( xy  y(1))  4 x  0
'
'
2 yy  2xy  2 y  4x  0
'
'
4 x  2 y 2 x  y
y 

2 y  2x
yx
'
Categories
Potpourri
400
Integrate:
3
x

Answer
2
x  2dx
3
Answer
( x  2)
 ( x  2) (3x )dx  3/ 2
3
3
1/ 2
Categories
2
3/ 2
2 3
 C  ( x  2)3/ 2  C
3
Potpourri
500
Integrate the indefinite Integral:
2
t
(
t

)
dt

t
2
Answer
Answer
Expand:
 (t 3  2t )dt
2
Integrate: 1 t 4  2t  C  1 t 4  t 2  C
4
2
4
1
Answer: t 4  t 2  C
4
Categories
Final Jeopardy
•Think about your
wager for final
Jeopardy!
• The category is:
The Angle between two
Vectors!
•Just Kidding!
• Your Category is:
Area Approximations!
Decide wager
amount
Wager some, all, or
nothing…
Use Trapezoid Rule to
Approximate:
n=4
1
0 1  x3
1
Solution:
Solution:
We now use Trapezoid Rule to
approximate the definite
integral.
2 1
 f (1)  2 f (1.25)  2 f (1.5)  2 f (1.75)  f (2)
8
Answer: [2.0599]
Area!
• Using integration, and arithmetic sums, to
determine the area produced between the
graph and the x-axis.
Categories
Integration!
• Using basic integration to determine both
definite and indefinite integrals. Don’t
forget the + C!
Categories
Mental Funds!
• Understand the fundamental theorems of
calculus for this part…you’ll need them.
TRUST ME!
Categories
U-Sub!
• Using U-Substitution, find the integrals of
some problems.
• Warning…some of these problems are
rather difficult!
Categories
Potpourri!
• Anything and Everything From the Three
Previous Chapters! HAHAHAHA!
• Solvers Beware!
Categories