Mathematics 115 F2013 / Exam 4 Some Practice Problems
(1) Find each of the following denite and indenite integrals; one part of one of the following cannot
be solved - which one?
ex ex
dx
ex + ex
(a)
2
(t3 t) dt
(b)
2
(c)
x ln x dx ;
9
(d)
1
x ln x10 dx ;
-Review of pre-calculus before we start working on calculus
I. The concept of the function is the most fundamental to calculus.
a. Functions model real life systems
b. The simplest function is a constant function
b.i. Ie. F(x) = b
c. The linear function i
I. Rational Functions
a. The quotient of two polynomials is a rational function.
b. The domain of a rational function is all x except where f(x) = 0.
c. Wolfram Alpha is an online app for checking graphs of functions.
d. In the graph of tan(x), the domain
I. Quiz 1 Tuesday: Topics
a. Trig
b. Limits
c. No calculators
II. Limits
a. A function f(x) has limit L as x approaches a.
a.i. Denoted limx approaches a f(x) = L
a.ii. If all values of f(x) are close to L for x arbitrarily close to a but not
equal to a.
I. Polynomial Functions
a. A quadriatic function f is given by f(x) = ax2 +bx +c, where a does not
equal 0.
b. The vertex of a parabola is given by the equation x = -b/2a.
b.i. Once one finds the x coordinate, it must be plugged into the
function to obtai
I. Slope and Linear Functions
a. The definition of calculus evolves as an individuals knowledge of the
subject grows.
b. Y=mx+b is called the slope intercept equation of a line.
c. A constant function occurs when slope is equal to zero.
d. The point-slope
MATH 115 000/001 LIFE SCIENCES CALCULUS 1
SYLLABUS FALL 2012
Following are an overview of the course and a detailed week-by-week syllabus with
exam and quiz dates. Please note that this syllabus is tentative. It may be
necessary to make changes to the top
I. Trigonometric Functions
a. A positive angle is counterclockwise.
b. A negative angle is clockwise.
c. The rotating ray is called the terminal side of the angle while the positive
side of the x-axis is considered the initial side.
d. Even if angles are
I. 2.1 Limits and Continuity: Numerically and Graphically
a. Absolute zero is the limit scientists approach as they develop more
advanced cooling methods. This limit cannot be reached.
b. In order for a limit to exist, both the limit approaching from the
I. 2.2 Limits: Algebraically
a. Limit Principles (let c be a constant and suppose that lim of f(x) is L and
lim of g(x) is M as x approaches a.
a.i. Lim cf(x) = cL as x approaches a.
a.ii. Lim (f(x)+g(x) = L + M and (f(x)-g(x) = L-M as x approaches a.
a.i
Mathematics 115 F2011 - Example Problem
Let f (x) =
x.
(1) Use the denition of the derivative to nd f (x):
From the denition
f ( x + h) f ( x )
f (x) = lim
= lim
h 0
h 0
h
x+h
h
x
provided the limit exists. Simplify the dierence quotient:
x+h
h
x
x+h x
x+
Maria Agnesi: (May 16, 1718 - January 9, 1799)
Born in Milan
Age 5- spoke French and Italian
Age 9- gave an academic speech about womens rights on education
Age 10- spoke 7 languages
Wrote book on differential and integral calculus
Discussed the curve kno
Math 115 - Exam 1 - Sample Problems - Fall 2013
Here are some sample problems to give you an idea of the type of problems that will appear on
Exam 1. Some of the problems are a bit long for an exam but are included here because they
test knowledge of the
Mathematics 115 F2013 / Exam 2 Some Practice Problems
(1) Find the derivatives of each of the following functions. Show all steps, reasoning and calculations. Name the main dierentiation rules used.
(a) y = cos(tan(x)
2x
x2
(c) y = log3 (3x x)
(b) g (x) =
Math 115 Solutions to Exam 4 Practice Problems Fall 2013
The following solutions are not complete answers in call cases; there is enough here for you to start
your solutions and to check your nal answers.
(1) The antiderivatives and denite integrals:
ex e
Mathematics 115 F2013 / Exam 3 Some Practice Problems
(1) #22, p. 235 is a typical max-min problem.
(2) #26, p. 235 is a very good max-min problem.
(3) #34, p. 254 is a typical related rate problem.
(4) #39, p. 255 is a more challenging related rate probl
Mathematics 115 F2013 / Exam 5 Some Practice Problems
(1) Solve the higher-order initial-value problem: f (x) = 2x; f (1) = 2 and f (0) = 1
(2) Do Problems #35, #36 on p. 550 and add one more part for each question:
(c) Use the applet at http:/www.scottsa
my,
again-y-
. (15pts)
Consider the two functions
f(:c)=as2+x+3 g(:c)=3a:+2
(i) Find the value a: = b Where the two functions intersect.
(ii) Compute the area of the region enclosed by the graphs of the two functions between
as = 0 and the value com
Mathematics 115 Fall 2013: Practice Problems for the Final Examination
When doing these practice problems, write out complete careful solutions, including all work,
calculations and reasoning. It helps to practice writing clear, concise solutions start at
Mathematics 115 Fall 2011
Max/Min, Exponential Growth/Decay
Heating/Cooling, and Limited Growth Problems
When answering these questions, make sure to show all work and reasoning, and use the correct
units. These problems are not to be handed in, but are a
Math 115 / Fall 2011
Math 115 Life Science Calculus 1 Precalculus Review Fall 2011
The following exercises review the basic functions we will use in this course.
(A) Basic properties of functions, composition, inverses and graphs.
(1) Let f (x) = 2x + 3 o
Mathematics 115 Fall 2011: A Comprehensive Example
Please note: You should try to do the following problem before you read the solution. It requires accurate
sketches of two graphs, nding points of intersection of graphs, and integration by parts [see you
Mathematics 115 F2011: Syllabus for Exam 1
A. The best ways to prepare for the test are to:
read your notes from the lectures and labs;
read all the sections listed below, being careful to do as many of the examples as possible, particularly those liste
Math 115 / Fall 2011
Math 115 Life Science Calculus 1 Precalculus Review Fall 2011
The following exercises review the basic functions we will use in this course.
(A) Basic properties of functions, composition, inverses and graphs.
(1) Let f (x) = 2x + 3 o