Math 2020, Section 004
Practice Problems for Midterm 1
1. A particle is travelling along the x-axis. At time t (sec.), its velocity
is given by v(t) = t3 1 meters/second.
a. Use the left endpoint Riemann sum with six subdivisions to estimate the distance
Math 2020, Section 001
Practice Problems for Midterm 2
1. Write down the sums to approximate the integral with the specified
valuesR of n. (You do not have to evaluate it with the calculator.)
5
a. 2 1+e2 x2 dx, with the Simpsons rule, n = 6
We have that
MAT 2020 - TEST #1
[Sample]
Please write all solutions NEATLY. You are required to show all your work, provide complete
justifications of your responses, and present all steps clearly in a logically coherent manner.
Failure to do so will cause losing the
MAT 2020 - TEST #2
[Sample]
Please write all solutions NEATLY. You are required to show all your work, provide complete justifications of your
responses, and present all steps clearly in a logically coherent manner. Failure to do so will cause losing the
1.
2.
(a)
x = t 2 + 1,
y = t 2 1,
From the first equation : t 2 = x 1.
Subsituting in the second equation we get
y = (x 1) + 1
y = x 2,
x 1
which is an equation of a part of a straight line in the xy - plane.
As t increases from to , the particle moves do
1.
ln( 5 x )
dx
x
1
ln( 5 x ) dx
x
u du
=
=
u = ln( 5 x )
du = 1 5 dx
5x
du = 1 dx
x
u2
=
+C
2
[ln( 5 x ) ]2 + C
=
2
2.
=
=
=
7
x 3 x 2 + 1 dx
0
7
0
8
u 1 du
2
3
1
1
2
x 2 + 1 xdx
3
8
u
1
1
8
1 u
= 2
4
3 1
4
3
= u u
3
8
3
]
[16 1]
=
3
8
=
45
8
du
MAT 2020 - TEST #3
[Sample]
Please write all solutions NEATLY. You are required to show all your work, provide complete justifications of your
responses, and present all steps clearly in a logically coherent manner. Failure to do so will cause losing the
MCS 1424 - Test #1 (Answers)
1.
To find the lower and upper limits of the integratio we set thefunctionsequal to each other
n,
3 3x2 = 3x 3
f ( x) = kx
3x 2 3x 6 = 0
2.
x2 x 2 = 0
2
[
]
1
We conclude that
0.10
kxdx
0
x2
1.5 = k
2 0
f ( x) = 300 x
3 =
MAT 2020 - Test #2 (Answers)
2.
1.
3.
f ( x) =
Let
ex
, the area is
x
4
A=
ex
( 4 1) / 6
dx S 6 =
[ f (1) + 4 f (1.5) + 2 f (2) + 4 f (2.5) + 2 f (3) + 4 f (3.5) + f (4)]
x
3
1
=
1 e1
e1.5
e2
e 2.5
e3
e 3 .5 e 4
+ 2 + 4
+ 2 + 4
+
+ 4
61
1 .5
2
2 .5
3
3
10/26/2013
Polar Coordinates
Polar Coordinate System: In this system a point P in
the plane is located by giving directed distance r from
a fixed point called the pole , and angle from a fixed
ray called the polar axis. We take the polar axis to be
the po
Math 2020, Section 001
Practice Problems for Midterm 2
1. Write down the sums to approximate the integral with the specified
valuesR of n. (You do not have to evaluate it with the calculator.)
5
a. 2 1+e2 x2 dx, with the Simpsons rule, n = 6
R3
2
b. 1 ex
Math 2020, Section 004
Final Exam Practice Problems
1. Find the area of of the region enclosed by the curves y 2 = x and
x + y = 6.
2. Find the volume of the following solids:
a. the solid obtained by rotating the region bounded by the curves
y = x2 and y
Math 2020, Section 004
Practice Problems for Test 3
1. Sketch each of the following parametrized curves, and eliminate the
parameter to give an equation for the curve containing only x and y.
a. x = 3 cos(t), y = 31 sin(t)
We use the Pythagorean theorem
c
Math 2020, Section 002
Final Exam Practice Problems
1. Find the area of of the region enclosed by the curves y 2 = x and
x + y = 6.
We slice up the region horizontally, and set up the integral with
respect to y. The two curves are x = y 2 and x = 6 y, wit
Math 2020, Section 004
Practice Problems for Test 1
1. A particle is travelling along the x-axis. At time t sec., its velocity
is given by v(t) = t3 1 meters per second.
a. Use the left endpoint Riemann sum with 6 subdivisions to estimate the distance tra
Math 2020, Section 004
Practice Problems for Test 3
1. Sketch each of the following parametrized curves, and eliminate the
parameter to give an equation for the curve containing only x and y.
a. x = 3 cos(t), y = 31 sin(t)
b. x = t21 . y = 3t 2
2. For eac
Integration Of
Rational Functions By
Partial Fractions
Polynomial Factorization Theorem:
The factorization of any polynomial Q(x) with
real coefficients is the product of polynomials
of the form
(ax + b) i
and
(ax2 + bx + c) j
where i and j are nonnegativ