MATH 1172
NAME : 
SPRING 13
QUIZ 1 HSOLUTIONSL

INSTRUCTIONS :
Show your work ! Answers without supporting
written work will not receive full credit.
1 H4 ptsL
Find the area of the region in the
first quadrant bounded by the graphs of
2
y=
and y = 2.
4
Davis Buenger
Math 1172 11.1
1. Rewrite the parametric equation as y equals
a function of x.
5. (Test Problem) Find an equation of the line
tangent to the curve at the point corresponding
to the given value of t.
x = t3 + t, y = t4 t; t = 1
Solution:Recal
Davis Buenger
Math 1172 12.7
March 26, 2014
1. Consider the position function of an ob 3. Consider an object red with an initial
ject:
velocity 1, 2, 4 ft/s and position 3, 2, 0 ft.
Assume that the only force acting on the obr(t) = 3 cos(2t), 3 sin(2t) ,
Davis Buenger
Math 1172 12.3
1.
Compute the dot product between
u = i 3j + 4k and v = 2i + 4j + 2k
Solution:
2. Let u = 5, 0, 15 and v = 4, 1, 1 . Calculate projv u ( the orthogonal projection of u onto
v) and scalv u the scalar component of u in the
dire
Davis Buenger
Math 1172 12.6
March 26, 2014
1 Find the rst and second derivative of the 3. Compute the indenite integral of the funcfollowing function
tion
1
r(t) = tet , t3 sin(t4 ),
6 1
r(t) = 3 t, 2t ,
1 + t2
t
Solution: To compute the integral of a ve
Davis Buenger
Math 1172 12.4
Solution: Let u be the vector from A to B
and let v be the vector from A to C. Then
u = 1, 4, 0 and v = 3, 2, 1 , and the area
of the triangle is given by: Area = 1 u v.
2
Let u = u1 , u2 , u3 and v = v1 , v2 , v3 . Then
we
Davis Buenger
Math 1172 12.8
March 26, 2014
Solution: Let f (t) = 2et sin(t), g(t) = 2et ,
h(t) = 2et cos(t). Then
Arc length for vector functions: Consider
the dierentiable curve r(t) = f (t), g(t), h(t) ,
i.e. f (t), g(t), and h(t) have continuous deriv
Math 1172 (Buenger)
Section 6.2
December 30, 2013
3.
Usub review:
tan(7x) ln cos(7x) dx
1.
2x sin(x2 )dx
4.
2.
ex cos(ex )dx
1
x5
dx
6x2 22
Area between two curves: Sketch the re 7. (Past Test question) Find the area of the
gion bounded by the followi
Davis Buenger
Math 1172 6.4
January 16, 2014
Table 1: Shell vs Washer Method
Axis of rotation Method Variable of integration
Formula
b
x
Washer
x
a out(x)2 in(x)2 dx
x
Shell
y
2
y
Washer
y
y
Shell
x
2
b
y(top(y) bot(y)dy
a
b
out(y)2 in(y)2 dy
a
b
x(top(x
Davis Buenger
Math 1172 6.7
Mass of an linear object of variable
density: Suppose we have a thin strait object like a bar or a wire with variable density. Further suppose that we can model
the object by a line segment on the interval
a x b with density fu
Davis Buenger
Math 1172 12.5
March 26, 2014
1. Find an equation of the line segment 3. Determine the domain of the vectored valued
joining the points (2, 2, 4)
and
(1, 3, 5). function
Solution: The vector joining our rst point to
1
t, 49 t2 , 2
r(t) =
the
Davis Buenger
Math 1172 12.1& 12.2
March 26, 2014
1.
Let u = 4, 2 .
Find a vec 5. Give a geometric description of the set of
tor parallel to u with magnitude 4 times u. points satisfying the condition
Solution:
X 2 + Y 2 10Y + Z 2 9
2. Let u = 3, 4 and L
Math 1172 (Buenger)
Section 9.2
February 5, 2014
1.
Give an example of a non Solution:
increasing
sequence
with
a
limit.
n
lim
Solution: Consider the sequence an with
2 + 2n + 1
n
4n
1
an = n . The sequence is non increasing
1
n
1
1
since an = n > n+1 =
Davis Buenger
Let
Math 1172 9.5 Solutions
ak be an innite series. Assume that r = limk
February 5, 2014
ak+1 
ak  .
1. If 0 r < 1 then the series converges.
2. If r > 1 the series diverges.
3. If r = 1 the test is inconclusive.
Use the ratio test to d
Davis Buenger
February 5, 2014
Math 1172 9.3
Geometric series Let a R and r s.t
r < 1
n1
1 rn
ark = a
1r
k=0
Solution: Write the given series as
k=3
= 3
1
ark = a
1r
k=0
k=3
= 3
Evaluate the following geometric sums or
state that they diverge
k=0
3
4
4
Davis Buenger
Math 1172 10.1
February 17, 2014
1. Find linear and quadratic approximations to f (x) = x centered at the point a = 9. Use these
polynomials to approximate the value of the function at x = 8.9.
1
Solution: f (x) = 1 x 2 . Thus the slope of t
Davis Buenger
Math 1172 10.2
February 17, 2014
Determine the radius of convergence of the 2.
following power series. Test the end points to
determine the interval of convergence
1.
k=1
k=0
Solution: Fix x and let us perform the ratio
test on the resulting
Davis Buenger
Math 1172 11.3
1. Find the slope of the line tangent to the
curve r = 1 + 2 sin(2) at the point (3, /4).
Then nd an equation for this line.
Back ground: Recall that to nd the slope of
a tangent line to an equation in polar coordinates, we co
Davis Buenger
Math 1172 10.3 & 10.4
1. (Test Problem) True or false: Suppose
f (x) = x3 + 2x2 . Then the 3rd order Taylor
polynomial for f centered at
a = 1 is p3 = 3 + 7(x 1) + 10(x 1)2 + 6(x 1)3 .
False:
f (x) = x3 + 2x2
Solution: The taylor series for
Davis Buenger
Math 1172 11.2
1. Graph the following its in polar coordinates:
March 3, 2014
Solution: Multiplying through by r we have
r2 = 6r cos()
Since r2 = x2 + y 2 and x = r cos(), we have
(3, /4)
(1, 5/6)
(2, /2)
(3, 7/4)
x2 + y 2 = 6x.
This is an a
Davis Buenger
Math 1172 6.5
January 13, 2014
Arc Length: Let f (x) have a continuous derivative on
[a, b]. The length of the curve from (a, f (a) to (b, f (b) is
b
1 + f (x)2 dx.
L=
a
1. Find the arc length of the curve y =
x4
4
+
1
8x2
on the interval [1
Math 1172 (Buenger)
Section 7.2
January 20, 2014
3.
1.
3x
x sin x cos x dx
2xe dx
4.
2.
tan1 xdx
x ln x dx
1
5.
7. (Test Question) The region under the
graph y = sin(2x) for 0 x is revolved
2
about the yaxis. Find the volume of the
resulting solid.
ln 3
x
Davis Buenger
Math 1172 10.3 & 10.4
February 17, 2014
1. (Test Problem) True or false: Suppose
f (x) = x3 + 2x2 . Then the 3rd order Taylor
polynomial for f centered at a = 1 is
p3 = 3 + 7(x 1) + 10(x 1)2 + 6(x 1)3 .
3. Find the rst four nonzero terms of
Davis Buenger
Math 1172 11.1
March 3, 2014
3. Consider the parametric x = t, y = 4t.
Find dy/dx in terms of t and evaluate the
derivative at t = 3.
1. Rewrite the parametric equation as y equals
a function of x.
x = t2 + 6, y = 4t; 4 t 4
4. (Test Problem)
Davis Buenger
Math 1172 11.2
1. Graph the following its in polar coordinates:
March 3, 2014
3. (2, 2) is a pt in cartesian coordinates.
Express the pt in polar coordinates.
(3, /4) (1, 5/6) (2, /2) (3, 7/4)
4. Consider the polar function r = 6 cos().
Conv
Davis Buenger
Math 1172 11.3
1. Find the slope of the line tangent to the
curve r = 1 + 2 sin(2) at the point (3, /4).
Then nd an equation for this line.
March 3, 2014
3. Make a sketch of the limacon r = 6 + 3 cos
and nd the area of the inner region.
4.
Davis Buenger
Math 1172 12.1 & 12.2
March 26, 2014
1. Let u = 4, 2 . Find a vector parallel to u
with magnitude 4 times u.
3. Sketch the plane parallel to the xyplane
containing the point (2, 3, 1).
2. Let u = 3, 4 and Let v = 1, 1 . Which
has greater ma
Davis Buenger
Math 1172 12.3
March 26, 2014
1. Compute the dot product between u =
Recall that when a constant force is applied
i 3j + 4k and v = 2i + 4j + 2k
to an object resulting in a displacement d, where
is the angle between d and F then the work
do
Davis Buenger
Math 1172 12.4
March 26, 2014
2. Consider the points A = (5, 6, 1), B =
(6, 2, 1), and C = (2, 4, 2) and the triangle they
form. What is the area of the triangle they form?
Let u = u1 , u2 , u3 and v = v1 , v2 , v3 . Then
we dene the cross p
Name JELSN M/I/hf Math 1172 Quiz 6
Problem 1
a) (1.5 pts) Below on the left is a portion of the Cartesian graph of r = 2  2 cos(6) plotted
on the 7, 0 plane. Sketch the polar graph in the m, y plane below on the right.
1'
l
b) (1.5 pts) Convert the pol
SCAN/70143
Name $0 M M , Z/w Math 1172 Quiz 4
Problem 1 (6 pts)
Determine whether the series converges or diverges. Be sure to explain your reasoning.
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