University of Oregon: Math 246
Final Exam, March 14, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 2 hours
1. (6 points) State carefully the following denitions
WORKSHEET FOR 1.11 (WITH SOLUTIONS)
MATH 246, FALL 2012
Summary of last time: Last week we discussed a model of the heart, which went as follows:
There are three things to keep in mind: (1) the pacemaker, (2) the AV node, and (3)
the heart.
The pacemake
Math 246
Homework #1
(1.1) #22. (a) Find the equation of the circle with center (2, 5) and radius 3.
(b) Where does the circle intersect the y -axis?
(c) Does the circle intersect the x-axis? Explain.
Solution: (a) A circle with center (h, k ) and radius
Math 246
Homework #2
(2.1) #3. Evaluate the limit using a table or graph: lim
x1 x2
x
+1
Solution: With a calculator, we can come up with a table:
x
x
2+1
x
0.9 0.4972
0.99 0.49997
0.999 0.4999997
1.1 0.4977
1.01 0.49998
1.001 0.4999998
x
It appears that
Math 246
Homework #3
(2.2) #1. Let f (x) = x2 cos
1
x
, x=0
(a) Use a graphing calculator to sketch the graph of y = f (x).
(b) Show that for x = 0, x2 x2 cos
1
x
x2 .
(c) Use your result in (b) and the Sandwich Theorem to show that
1
lim x2 cos
= 0.
x0
Math 246
Homework #4
x2 + 1 is continuous at c = 1.
Solution: Well, f (x) = (a b)(x), where a(x) = x and b(x) = x2 + 1. Since b is a
polynomial, its continuous at every number; in particular, its continuous at 1. Since a
is a power function (a(x) = x1/2 )
Math 246
Homework#5
(3.1) #18. Use the formal denition to nd the derivative of f (x) =
for x = 1.
1
x+1
Solution: Well,
f (x + h) f (x)
= lim
f (x) = lim
h0
h0
h
1
h0 h
= lim
1
1
(x + h) + 1 x + 1
1
(x+h)+1
1
h0 h
= lim
1
x+1
h
(x + 1) (x + h + 1)
(x + h
Math 246
Homework #6
(3.4) #1. Dierentiate with respect to the indicated variable:
f (x) = (2x 1)3
Solution: By the Chain Rule,
f (x) = (2x 1)3
= 3(2x 1)2 (2x 1)
= 3(2x 1)2 (2)
= 6(2x 1)2
(3.4) #4. Dierentiate with respect to the indicated variable:
f (s)
Math 246
Homework #7
(3.3) #3. Dierentiate with respect to the indicated variable:
2
f (x) = e2x +3x1
Solution: By the Chain Rule,
2
f (x) = e2x
+3x1
2
+3x1
2
+3x1
= e2x
= e2x
2x2 + 3x 1
(4x + 3)
(3.6) #7. Dierentiate with respect to the indicated variabl
Math 246
Homework #8
(4.1) #4. Use a graphing calculator to determine all local and global
extrema of the f (x) = x2 4x + 2 on [0, 3].
Solution: First, we compute f (x) = 2x 4. Then f (x) = 0 when x = 2. Now
f (0) = 2, f (2) = 2, f (3) = 1. From the pictu
Math 246 Fall 2001
Solutions
Quiz 1
1. Find an equation of the line perpendicular to the line 4x 7y = 3 and passing through
the point (1, 3).
Solution: Lets put the equation 4x 7y = 3 into slope-intercept form so we can see its
slope.
4x 7y = 3
7y = 4x +
Math 246 Fall 2001
Solutions
Quiz 3
1
compute the derivative f (x).
x2
Solution: We compute:
1
f (x + h) f (x)
(x+h)2
f (x) = lim
= lim
h0
h0
h
h
1. For f (x) =
1
h0 h
= lim
x2
(x + h)2
2
x2 (x + h)2
x (x + h)2
1
x2
1
h0 h
= lim
1
1
2
2
(x + h)
x
x2 (x2
Solutions to Extra Practice for Exam 1
Math 246, Fall 2012
1. A function L models total number of desert locusts in a swarm (in millions) as a function of
the size of the swarm n (in square kilometers). Researchers determine that L(3) = 170, while
L (3) =
University of Oregon: Math 246
Quiz 1, Jan. 7, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 15 minutes
1. Let f be dened by
1
x
f (x) = sin x
x
(a) Graph f bel
University of Oregon: Math 246
Quiz 2, Jan. 26, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 15 minutes
1. Write down the denition of the derivative at the poi
University of Oregon: Math 246
Quiz 3, Jan. 31, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 15 minutes
1. Dierentiate the following functions.
(a) f (x) =
4x2
University of Oregon: Math 246
Quiz 4, March 1, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 15 minutes
1. Let f (x) =
x2
.
x3
(a) Find the domain of f . Locat
University of Oregon: Math 246
Test 1, Feb. 3, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 50 minutes
1. (6 points) State carefully the following denitions.
(
University of Oregon: Math 246
Test 2, Feb. 17, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 50 minutes
1. (30 points) Dierentiate the following functions. No
University of Oregon: Math 246
Test 3, March 3, 2011
No Calculators, Notes, or Formula Sheets. Show all your work or you may get no credit!
Do all your work on the paper provided. You have 50 minutes
1. (25 points) Evaluate the following limits.
(a) lim
x
Name:
Math 246 Quiz 1 10/2/12
Directions: Read each question carefully, and write your answers in the space provided. Show your work in
a clear and organized fashion when appropriate.
1. (3pts each) Consider the function P (t) = t2 5t + 7, which gives the
Name:
Math 246 Quiz 2 10/9/12
Directions: Read each question carefully, and write your answers in the space provided. Show your work in
a clear and organized fashion when appropriate.
f (x + x) f (x)
x 0
x
Use the limit denition to compute f (3) where f (
Name:
Math 246 Exam 1 10/23/12
Directions: Read each question carefully, and write your answers in the space provided. Show your work in
a clear and organized fashion when appropriate.
Part I: Multiple Choice. (3 points each) You do not need to show any w