Download Victoria Howle spr12vhowlem1451s008

Victoria Howle
spr12vhowlem1451s008
WeBWorK assignment number WW01 is due : 01/30/2012 at 02:17pm CST.
The
(* replace with url for the course home page *)
for the course contains the syllabus, grading policy and other information.
This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.
The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making
some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are
having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor for
help. Don’t spend a lot of time guessing – it’s not very efficient or effective.
Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,
you can if you wish enter elementary expressions such as 2 ∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e ∧ (ln(2)) instead of 2,
(2 + tan(3)) ∗ (4 − sin(5)) ∧ 6 − 7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.
You can use the Feedback button on each problem page to send e-mail to the professors.
1. (1 pt) Library/Union/setAlgebraAbsoluteValue/srw1 8 1.pg
Match the statements in the lefthand column with their equivalent statements in the righthand column.
1.
2.
3.
4.
5.
A.
B.
C.
D.
E.
sin( 13π
12 ) =
5π
sin( 12 )=
cos(− 7π
12 )=
cos( 11π
8 )=
|x − 3| = 8
|x − 3| < ∞
|x − 3| < 8
|x − 3| ≤ 8
|x − 3| > 8
Correct Answers:
• -sqrt(2-sqrt(3))/2
• sqrt(2+sqrt(3))/2
• sqrt(2)*(1-sqrt(3))/4
• -sqrt(2-sqrt(2))/2
4.
x ∈ {−5, 11}
x ∈ [−5, 11]
x ∈ (−∞, ∞)
x ∈ (−∞, −5) ∪ (11, ∞)
x ∈ (−5, 11)
The smallest positive number for which
4 cos2 x − 9 cos x + 2 = 0
Correct Answers:
•
•
•
•
•
(1 pt) Library/Utah/Trigonometry/set10 Analytic Trigonometry-
/p7.pg
.
is x =
Hint: Solve the quadratic equation for cosx and then solve for
x.
A
C
E
B
D
Solution: To begin with we think of cos x as our variable. For
clarity let’s denote it by z:
2. (1 pt) Library/Union/setAlgebraAbsoluteValue/p1-abs-d.pg
Solve the following inequality. Enter the answer in interval notation.
|2 − 4x| < 8
Our equation becomes
Answer:
This can be factorized
z = cos x.
4z2 − 9z + 2 = 0.
Correct Answers:
4z2 − 9z + 2 = (4z − 1)(z − 2) = 0.
• (-1.5,2.5)
Thus z = 14 or z = 2. There is no number x whose cosine equals
2, and so the only equation we need to solve is
1
cos x = .
4
The smallest positive solution of that equation is
1
x = arccos ≈ 1.318.
4
3. (1 pt) Library/Rochester/setTrig07Identity/p7.pg
Use a sum or difference formula or a half angle formula to determine the value of the trigonometric functions. Give exact
answers. Do not use decimal numbers. The answer should be
a fraction or an arithmetic expression. If the answer involves a
square root it should be enter as sqrt; e.g. the square root of 2
should be written as sqrt(2).
1
Correct Answers:
• 1.31811607165282
8. (1 pt) Library/Union/setAnalGeomLines/sw2 4 11.pg
Find the equation of the line graphed below. Write the equation
in the form y = mx + b and identify m and b.
5. (1 pt) Library/ASU-topics/setTrigIdentities/p10.pg
For each trigonometric expression A,B,C,D, E, choose the expression from 1,2,3,4,5 that completes a fundamental identity.
Enter the appropriate letter (A,B,C,D, or E) in each blank.
A. (sin x)2 + (cos x)2
sin x
B. cos
x
C. (tan x)2 + 1
cos x
D. sin(x)
E. (sin x)2
1.
2.
3.
4.
5.
1 − (cos x)2
1
(sec x)2
cot x
tan x
Correct Answers:
• E
• A
• C
• D
• B
m =
b
=
Correct Answers:
• -0.25
• 1
6. (1 pt) Library/ASU-topics/setGraphEquations/sw2 2 53.pg
Find an equation of the circle with center at the origin and passing through (−5, −4) in the form of
9.
(1
/dist midpoint.pg
pt)
Library/Rochester/setAlgebra07PointsCircles-
Consider the two points (−10, 4) and (1, −5). The distance
between them is:
The x co-ordinate of the midpoint of the line segment that joins
them is:
The y co-ordinate of the midpoint of the line segment that joins
them is:
(x − A)2 + (y − B)2 = C
where A, B,C are constants. Then
A=
B=
C=
Correct Answers:
• 0
• 0
• 41
Correct Answers:
• 14.2126704035519
• -4.5
• -0.5
7. (1 pt) Library/ASU-topics/setGraphEquations/circle3.pg
Find the center (h, k) and the radius r of the circle
10. (1 pt) Library/maCalcDB/setAlgebra14Lines/pts to gen.pg
The equation of the line that goes through the points (−3, 6) and
(5, −10) can be written in general form Ax + By +C = 0 where
A=
B=
C=
4x2 − x + 4y2 − 7y = 0.
h=
k=
r=
Correct Answers:
Correct Answers:
• 0.125
• 0.875
• 0.883883476483184
• 16
• 8
• 0
2
Is the graph symmetric with respect to the origin? Input yes or
no here :
11. (1 pt) Library/maCalcDB/setAlgebra14Lines/sApB 31-36.pg
An equation of a line through (-2, 4) which is perpendicular to
the line y = 4x + 3 has slope:
Correct Answers:
•
•
•
•
•
and y-intercept at:
Correct Answers:
• -0.25
• 3.5
12. (1 pt) Library/Utah/College Algebra/set5 Functions and Their Graphs-
-6, 6
-36
no
yes
no
15. (1 pt) Library/Utah/College Algebra/set6 Polynomial and Rational Functions-
/1050s6p22.pg
/1050s5p8.pg
Suppose
Consider the function f defined by
x+3
f (x) =
x−5
Then the domain of f is the set of all real numbers except x =
.
f (x) = x + 4
and
g(x) = 2x − 5.
Then
Correct Answers:
• 5
.
( f ◦ g)(x) =
−1 (x) =
.
(
f
◦
g)
13. (1 pt) Library/Utah/College Algebra/set5 Functions and Their Graphs- −1 −1
.
(
f
◦
g
)(x)
=
/1050s5p10.pg
−1 ◦ f −1 )(x) =
(g
.
The domain of the function
p
f (x) = 16 − x2
Correct Answers:
is the interval
,
and its range is
,
•
•
•
•
,
.
Correct Answers:
• [
• -4
• 4
• ]
• [
• 0
• 4
• ]
2x-1
(x+1)/2
x/2-3/2
x/2+1/2
16. (1 pt) Library/Utah/College Algebra/set6 Polynomial and Rational Functions/1050s6p28/1050s6p28.pg
14. (1 pt) Library/ASU-topics/setGraphEquations/sw2 2 25.pg
For the graph of the equation y = x2 − 36, draw a sketch of the
graph on a piece of paper. Then answer the following questions:
The x-intercept(s) is (are):
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
The y-intercept(s) is (are):
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
This problem illustrates that we can make statements about
functions without knowing how they are defined. The Figure
shows for functions that form two pairs of functions and their
inverses. Enter the letters b for blue, g for green, r for red, and
y for yellow below, as appropriate.
Is the graph symmetric with respect to the x-axis? Input yes
or no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
3
19. (1 pt) Library/Union/setFunctionComposition/an1
4 47.pg
√
Write the function h(x) = 7x2 + 3 as the composition of two
functions f (g(x)).
f (x) =
g(x) =
Do not use the identity function as one of your answers. That
is, do not use f (x) = x or g(x) = x.
The inverse of the blue function is
.
The inverse of the red function is
.
The inverse of the green function is
.
The inverse of the yellow function is
.
Correct Answers:
Solution: The Maple code to generate these graphs is
plot([tan(x/2), 2∗arctan(x), log(x), exp(x)], x = −1.8..1.8, y =
−1.8..1.8, discont = true,thickness = 2);
• sqrt(x)
• 7*xˆ2+3
20.
(1
pt)
Library/Rochester/setAlgebra17FunComposition-
/ur fn 2 1.pg
Let f be the linear function (in blue) and let g be the parabolic
function (in red) below.
Correct Answers:
•
•
•
•
y
g
r
b
17. (1 pt) Library/Utah/Business Algebra/set5 Quadratic and Other Special Functions/p01.pg
The first few problems concern inverse functions.
Suppose
f (x) = 3x + 4.
Then
f −1 (x) =
.
Correct Answers:
• (x-4)/3
Note: If the answer does not exist, enter ’DNE’:
18. (1 pt) Library/Utah/College Algebra/set6 Polynomial and Rational Functions1. ( f ◦ g)(2) =
/1050s6p13.pg
2. (g ◦ f )(2) =
The next few problems are exercises in function composition.
3. ( f ◦ f )(2) =
Let
4. (g ◦ g)(2) =
f (x) = 2x + 1 and g(x) = x2 − 1
5. ( f + g)(4) =
6. ( f /g)(2) =
Correct Answers:
Then
A. ( f ◦ f )(x) =
B. (g ◦ g)(x) =
C. ( f ◦ g)(x) =
D. (g ◦ f )(x) =
•
•
•
•
•
•
.
.
.
.
Correct Answers:
•
•
•
•
-1
1
0
4
7
DNE
21. (1 pt) Library/ma117DB/set3/srw2
1 23.pg
2
x + 2x, ifx ≤ −1
Given the function f (x) =
calculate the
x + 5,
ifx > −1
following values:
f (−9) =
f (−1) =
4*x+3
xˆ4-2xˆ2
2xˆ2-1
4xˆ2+4x
4
π
3
π
2
f (2) =
Correct Answers:
• 63
• -1
• 7
22.
π
2π
Correct Answers:
•
•
•
•
•
•
(1 pt) Library/Rochester/setTrig03FunctionsRightAngle-
/ur tr 1 12b.pg
For 0 < θ < π/2, find the values of the trigonometric functions
based on the given one (give your answers with THREE DECIMAL PLACES or as fractions, e.g. you can enter 3/5).
If tan(θ) =
cot(θ) =
sin(θ) =
cos(θ) =
sec(θ) =
csc(θ) =
5
5
then
0.866025403784439
0.707106781186548
0.5
0
-1
1
24. (1 pt) Library/Union/setTrigInverseTrig/srw7 6 1-8b.pg
Evaluate the following expressions. Your answer must be an
angle in radians in the interval [− π2 , π2 ].
1.
sin−1 ( 21 ) =
2.
Correct Answers:
• 1
• 0.707106781186548
• 0.707106781186547
• 1.4142135623731
• 1.41421356237309
3.
sin−1 (−1) =
sin−1 (−
√
2
2 )
=
Correct Answers:
• 0.523599
• -1.5708
• -0.785398
25. (1 pt) Library/ASU-topics/setTrigInverse/srw7 6 12-16c.pg
Evaluate the following expressions. Your answer must be a fraction or an integer. No decimal numbers. If the answer involves
a square root it should be written as sqrt . E.g. the square root
of 2 should be written as sqrt(2).
sin(tan−1 (0))
√
cos(tan−1 ( 33 ))
23. (1 pt) Library/Rochester/setTrig02FunctionsUnitCircle/p4.pg
For each of the following angles (in radian measure), find the
cosine of the angle.
Note: Your answer must be in EXACT form: it cannot contain
trig functions, it must be either an integer or a fraction. If the
answer involves a square root write it as sqrt . For instance, the
square root of 2 should be written as sqrt(2).
Correct Answers:
π
6
π
4
• 0
• sqrt(3)/2
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
5