# Math 172-1 Final Exam Review Sheet

The final exam covers sections 5.6, 5.7, chapters 6, 7, 8,
10, and 11. No graphing /symbolic

calculators allowed and no notes are allowed. You may bring a simple scientific
calculator.

Don't be late. Bring photo ID.

**Chapter 5: Integration**

**6. Substitution and Area Between Curves:** Be able to
use substitution to calculate

definite integrals. Be familiar with integrals of symmetric functions. Be able
to find the

area between two curves. Note that we can do this using either the usual
vertical strips or

horizontal strips.

**7. The Logarithm Defined as an Integral:** Be able to
evaluate integrals that result

into logarithmic terms. Be able to evaluate integrals of exponential functions
of any base.

## Chapter 6: Applications of Definite Integrals

**1. Volumes by Slicing and Rotation About an Axis: **
Be able to find the volume of

a solid be integrating cross-sections. Know how to find the area of a cross
section. What is a

solid of revolution. Be familiar with the disk method. Be familiar with the
washer method.

Be prepared for any axis of revolution.

**2. Volumes by Cylindrical Shells:** Be able to use the shell method to find
the volume

of a solid of revolution. Be prepared for any axis of revolution.

**3. Lengths of Plane Curves: **Be able to find the length of a parametric
curve. Be able

to find the length of a function y = f(x). Be able to find the length of a
function x = g(y).

**4. Areas of Surfaces of Revolution:** Be able to find the area of a surface
generated

by revolving a curve about either axis.

**5. Exponential Change and Separable Differential Equations:** Be able to
derive

the exponential change function for a differential equation. Be able to solve a
separable

differential equation. Be able to find both the general and a particular
solution to a differ -

ential equation. Be able to show that a given function is indeed a solution to a
differential

equation.

**6. Work:** Be familiar with Hooke's Law. Be able to find the spring
constant. Be able

to find the work to compress or stretch springs. Be able to find the work to
move objects

along a straight line. Be able to find the work required to empty an object of
liquid.

**7. Moments and Centers of Mass:** Be familiar with torque, moments, mass,
and

center of mass. Given a two-dimensional region, be able to find the object's
moment about

either the x or y axis, the object's mass, and the coordinates of its center of
mass. You may

need to use the either vertical or horizontal strips.

## Chapter 7: Techniques of Integration

**1. Integration by Parts:** Be familiar with the
integration by parts formula. Be able

to evaluate both definite and indefinite integrals with exponential terms,
logarithmic terms,

and trigonometric terms. What is we have something like
What about the

reduction formulas?

**2. Trigonometric Integrals:** Be able to evaluate integrals with powers of
sin(x) and/or

cos(x), with powers of tan (x) and/or sec(x), products of sines and cosines, and
square roots

of terms involving trig functions. Know your trig identities, page 26 and 27.

**3. Trigonometric Substitutions: **Be able to convert integrals involving
square roots

and other rational exponents into trigonometric integrals. Know the Pythagorean
Theorem.

Be able to draw the appropriate triangle for each type of substitution. Know
your trig

identities, page 26 and 27.

**4. Integration of Rational Functions by Partial Fractions :** Be able to
express a

rational expression as a sum of its factors, its partial sum. Does it matter if
the factors are

linear or quadratic ? Does it matter if the factors are repeated, i.e. (x - 2)^{4}?
What if a factor

is a quadratic? Be able to do long division. Be able to find the antiderivative
of a rational

function by using partial fractions. What is

**7. Improper Integrals: **Be able to determine if a type I improper integral
(an integral

with infinite limits of integration) converges or diverges. Be able to determine
if a type

II improper integral (an integral where the integrand is discontinuous
(infinite) at a point)

converges or diverges. When does
converge and diverge? We talked about three

methods for doing such things: direct calculation, using the direct comparison
test, and

using the limit comparison test. Be familiar with these.

## Chapter 8: Infinite Sequences and Series

**1. Sequences:** Be familiar with sequences: notation,
identifying patterns, etc. Be able

to determine whether a given sequence converges (and to what) or diverges (and
to what).

Be familiar with the algebra of sequences . Be familiar with recursive sequences.

**2. Infinite Series: **Know what a partial sum is. Know what an infinite
series is. Be

super familiar with the geometric series. When does it converge/diverge and to
what? Be

able to find the sum of a telescoping series. Be able to use the divergence
test. Be familiar

with the algebra of series.

**3. The Integral Test:** Be able to use the integral test to test for
convergence. Is

the resulting integral equal to the sum? Be familiar with the p-series. When do
they

converge/diverge? What does the harmonic series do?

**4. The Comparison Tests:** Be able to use the (direct) comparison test to
test for

convergence. Be able to use the limit comparison test to test for convergence.

**5. The ratio and Root Tests:** Be familiar with both the ratio tests. For
some

problems, either test works, but one may be easier to implement then the other.
Be prepared

for factorials and n-th powers. You may need to use
Rule . Remember how to

evaluate infinite limits of ratios.

**6. Alternating Series, Absolute and Conditional Convergence: **Know what an

alternating series is. Be able to use the alternating series test. Be able to
justify the three

conditions. What does the alternating-harmonic series do? Know the difference
between

absolute and conditional convergence. Does one imply the other? If so, which
one?

**7. Power Series: **Know what a power series is. Know the basic power series

Be able to find the sum of a power series by combining geometric series with
substitution.

Be able to find where a power series converges by using either the ratio or root
tests. Be

able to find a power series radius and interval of convergence. Be able to both
differentiate

and integrate power series.

**8. Taylor and Maclaurin Series: **Know what Taylor and Maclaurin series are
and

how to find them. If given a function with continuous derivatives up to order n ,
be able to

find its Taylor polynomials up to order n about x = a.

**9. Convergence of Taylor Series: **Be able to find the interval of
convergence of a

Taylor series. Be able to find a Taylor series by using the basic known Taylor
series in

combination with some algebra and substitution. Be able to use Taylor series to
evaluate

integrals and limits.

Note: Know the Taylor series for

,
sin(x), cos(x).

**10. The Binomial Series :** Know what a binomial series is. Be able to
compute

combinations. Be able to use the basic form of the binomial series along with
substitution

to find other binomial series.

## Chapter 10: Vectors and the Geometry of Space

**1. Three-Dimensional Coordinate Systems :** Know what
space is. Be able to both

plot and identify points, simple lines, and simple planes. Be able to identify
regions in space.

Know the distance formula. Know what a sphere is, even when not centered at the
origin.

**2. Vectors:** In both the plane and space, know what a vector is. Be able
to find the

vector between two points. Be able to sketch vectors. Be able to find the length
of a vector.

Know both vector addition and scalar multiplication . Graphically, understand how
vector

addition and substraction relate to the sides and diagonal of a parallelogram.
Be able to

find unit vectors. Know what the standard unit vectors are. Know the midpoint
formula.

Be familiar with vector arithmetic.

**3. The Dot Product: **Be able to find the dot product. Be able to find the
angle

between two vectors. Be able to find angles. What is the dot product is zero?
When are two

vectors orthogonal? Know the properties of dot products. Be able to find vector
projections.

Be able to find effective force in a given direction. Be able to find work.

**4. The Cross Product:** Know the definition of the cross product. When are
two vectors

parallel? What does a zero cross product tell us? Know the properties of the
cross product.

Be able to find the area of a parallelogram. Be able to use determinants to
calculate cross

products. Be able to find the area of a triangle by using the cross product. Be
able to find

unit normal vectors. Be able to find torque. Be able to use the triple scalar or
box product

to find the volume of a parallelpiped.

**5. Lines and Vectors in Space:** Be able to formulate a line in space. What
is needed?

Be able to formulate a plane in space. What is needed?

**6. Cylinders and Quadric Surfaces: **Know what a cylinder in space is. Be
able

to sketch simple ones. Be familiar with the six-quadric surfaces. Know the
equations for

ellipsoids , elliptic paraboloids, cones, etc. for various orientations.

## Chapter 11: Vector-Valued Functions and Motion in Space

**1. Vector Functions and Their Derivatives:** Know
what vector-valued functions

are. When is one differentiable? How do we find velocity, speed, acceleration,
unit tangent

vector, and how do we write velocity as a product of speed and direction?

**2. Integrals of Vector Functions:** Be able to integrate vector-valued
fucntions. Be

able to solve an initial-value problem. Know the ins and outs of ideal
projectile motion.

What two values do we use for gravity g?

**3. Arc Length in Space:** Be able to find arc length in space.

**4. Curvature of a Curve:** Given a curve in the plane or space, be able to
find its

curvature at a point. Be able to find the prncipal unit normal vector of a
curve.

**5. Tangential and Normal Componets of Acceleration:** If given a curve, be
able

to find tangential and normal scalar components of acceleration without finding
T and N.

Be able to find the binormal vector. Should torsion be bonus?

Look over this list and the sample exams. Don't study
material that we haven't covered

or not on the list. Then bring questions regarding those to the review.

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