Aleksandar Visekruna, Alex Shiel, Tanner Pierzina, and Sarah Lewis
Group Project 2
TA: Tony Yuan
Riemann Sums:
b
In this project, we will attempt to gain an understanding of the definite integral, f (x)dx, by computing
a
different types of Riemann sums. W
AP CALCULUS BC
Stuff you MUST Know Cold
lHopitals Rule
f (a ) 0
,
If
or =
=
g (a ) 0
f (x)
f '( x )
= lim
then lim
x a g ( x)
x a g '( x )
Properties of Log and Ln
1. ln 1 = 0
2. ln e a = a
3.eln x = x
4. ln x n = n ln x
5. ln ( ab ) = ln a + ln b
6.ln (
Week Nine Exercises
Please give complete, well written solutions to the following exercises.
1. Show that the curves
y = x3 3x + 4 and y = 3(x2 x)
are tangent to each other. That is: there is point where the curves intersect
and at this point the tangent
Week Ten Exercises
Please give complete, well written solutions to the following exercises.
1. Find the highest and lowest points on the curve
x2 + xy + y 2 = 12.
2. From the text/notes section 5.14, please do exercises 6, 13, 14, and 24.
3. Show that
| s
Week 8 Written HW
Please give complete, well written solutions to the following exercises. Such
responses are in complete sentences and should include all relevant
computations and well labeled pictures.
1. Suppose that the curve above has a horizontal ta
Week Seven Exercises
Please give complete, well written solutions to the following exercises.
1. In your own words, explain why
d2 f
dx2
=
df
dx
2
2. Section 4.12 exercise 11.
3. Prove that
d
cos x = sin x
dx
directly from the denition of the derivative.
Week Five HW and Exam review
questions
Week Five Exercises
Please give complete and well written solutions to the following exercises:
1. Use a comparison to argue that the following limit does not exist:
lim (x + sin x).
x
2. Find all values of a so that
Week Six Exercises
Please give complete, well written solutions to the following exercises.
Please give complete and well written solutions to the following exercises:
1. Section 4.5 exercise 14. You are welcome to use our dierentiation rules
here.
2. Sup
Week Four HW
Please give complete, well written solutions to the following exercises.
1. Compute
|2x 1| |2x + 1|
.
x
Hint: As x 0, will 2x 1 be positive or negative?
lim
x0
2. If
lim (f + g) = 2 and lim (f g) = 1
xa
xa
nd
lim f g.
xa
Hint: Can you use lim
Week 2 HW
Please give complete, well written solutions to the following exercises.
1. The altitude perpendicular to the hypotenuse of a right triangle is 12
centimeters. Express the length of the hypotenuse as a function of the
perimeter. What is the doma
Week 1 HW
Please give complete, well written solutions to the following exercises.
1. Sketch the region in the plane that solves the inequality
|x| + |y| 1
2. Section 1.2 exercise 6 in the class notes.
3. Section 1.2 exercise 4 in the class notes. Note th