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MATH 2300 review problems for Exam 2
1. A metal plate of constant density (in gm/cm2 ) has a shape bounded by the curve y =
xaxis, and the line x = 1.
x, the
(a) Find the mass of the plate. Include units.
(b) Find the center of mass of the plate. Include
Math 2300: Calculus II
Introduction to comparison tests for series
Worksheet Purpose: A few weeks ago we saw that a given improper integral converges if its
integrand is less than the integrand of another integral known to converge. Similarly a given
impr
Math 2300
Recursive Sequences
Goal: An introduction to the idea of recursively defined sequences, meaning: sequences where each
term is defined by a formula involving the previous term (or terms).
1. Sometimes sequences can be described recursively in add
Math 2300
Separable Differential Equations and Toricellis Law
1. Warmup (Review of related rates from Calc 1): A cylindrical keg with radius 10 cm and
height 25 cm has a drain hole at the bottom with an area of 2 cm2 . The keg is draining at a
cm3
consta
Math 2300
Building new Taylor Series from known Taylor Series
Background: Weve calculated the Taylor Series centered about a = 0 for some important functions.
For each of these functions, we know that if it has a power series representation, then it must
Math 2300 Calculus II University of Colorado
Final exam review problems
1. A slope field for the differential equation y 0 = y ex is shown. Sketch the graphs of the solutions that
satisfy the given initial conditions. Make sure to label each sketched grap
Math 2300: Calculus II
Taylor Polynomials
1. Recall the idea of linear approximation, which we will use to approximate the numbers e.1
and e. Let f (x) = ex . We want to find a linear function L(x) = C0 + C1 x whose value and
whose derivative at x = 0 mat
Math 2300: Calculus II
The error in Taylor Polynomial approximations
Background knowledge:
In the following statement, f (x) is a function, Tn (x) is its nthdegree Taylor polynomial centered
at a, and the remainder Rn (x) = f (x) Tn (x).
Taylors Inequali
Math 2300: Calculus II
Series: the Big Picture
Developing your intuition: For each of the following series, guess if it diverges, converges
conditionally or converges absolutely. Keep in mind that you must answer two separate questions:
1. Does the series
MATH 2300 review problems for Exam 1 ANSWERS
1. Evaluate the integral
leave it to you.
R
sin x cos x dx in each of the following ways: This one is selfexplanatory; we
(a) Integrate by parts, with u = sin x and dv = cos x dx. The integral you get on the r
Math 2300: Calculus II
Numerical Integration using Power Series
Background for the example:
In probability it is important to be able to find areas under the bell curve. The bell curve
is formally known as the normal distribution, and the function definin
MATH 2300 review problems for Exam 3, part 1
1. Find the radius of convergence and interval of convergence for each of these power series:
(a)
X
(x + 5)n
n=2
2n ln n
Solution:
Strategy: use the ratio test to determine that the radius of convergence is 2,
Math 2300: Calculus II
The SIR Model for Disease Epidemiology
This worksheet will analyze the spread of Ebola through interaction between infected and
susceptible people. Ebola is an infectious and extremely lethal viral disease that first surfaced
in hum
Math 2300: Calculus II
Review
1. Evaluate the following integral:
Z
x2
dx
1 x2
Let x = sin . Then dx = cos d, and 1 x2 = cos . Substituting, we have
Z
Z
Z
sin2
x2
dx =
cos d = sin2 d.
cos
1 x2
By the double angle formula, sin2 =
1cos 2
.
2
Integrating,
Math 2300: Calculus II
Antiderivative Review
Below is a list of indefinite integrals that you know how to find from your Calculus I class. Evaluate
these integrals.
Z
x3
1.
x2 dx =
+C
3
Z
xn+1
2.
xn dx
for n 6= 1 =
+C
n
Z
1
3.
da = ln a + C
a
Z
4.
e d =
Math 2300 Calculus II University of Colorado
Final exam review problems
1. A slope field for the differential equation y 0 = y ex is shown. Sketch the graphs of the solutions that
satisfy the given initial conditions. Make sure to label each sketched grap
Math 2300: Calculus II
Comparison tests for series: the Big Picture
P
For each of the
an below, we can determine convergence
by comparing to
Ppositiveterm series
P
another series bn . Choose the sequence bn to compare to, say whether bn converges or dive
Math 2300
Practice Modeling with Differential Equations
1. In Yellowstone park there were 21 bison in 1902, and 250 in 1915. Using the model that the
rate of change of the population is proportional to the population itself, set up an initial value
proble
MATH 2300 review problems for Exam 3, part 1
1. Find the radius of convergence and interval of convergence for each of these power series:
(a)
(b)
(c)
(d)
(e)
X
(x + 5)n
n=2
X
n=0
X
n=0
X
n=0
X
n=1
2n ln n
n(x 1)n
4n
n!(3x + 1)n
(2)n+1 xn
n3 + 1
ln nxn
n!
Math 2300: Calculus II
Tables, Computer Algebra Systems, and alternate solutions
Even when we stay within the realm of elementary functions, the general families of integrals that
come up can be quite varied. So when integrating, it will often be practica
Math 2300
Flow In, Flow out, Calc I style
Flow in: If there are V0 mL of fluid in a tank at time t = 0, and fluid is flowing in at a rate
of P (t) mL/min, then at time t the volume of the fluid in the tank is
Z t
V (t) =
V0 +
P (s) ds
0
Flow in and flow
Math 2300: Calculus II
Identifying Integral Substitutions
Goal: To identify what (if any) usubstitutions are necessary to compute an integral and to practice
making such substitutions.
For each problem, identify what (if any) usubstitution(s) need to be
MATH 2300 review problems for Exam 3, part 2
1. Shown below are the slope fields of three differential equations, A, B, and C. For each slope
field, the axes intersect at the origin.
3
2
3
3
3
2
2
2
1
1
1
1
2
3
3
2
1
1
1
2
3
3
2
1
1
1
1
1
2

Series  summing it all up
Heres a list of all of the convergence tests for series that you know so far:
Divergence test (a.k.a. nth term test)
Geometric series test
Telescoping series
Integral test
pseries (including harmonic series)
Termsize co