The above information is from your computer : be sure it is correct.
For grade credit, submit over the internet by 11:59 PM (= 23:59 hours) on Sunday 10 December 2017

Math 76O-250 Calculus for Business
Self-marking Practice quiz 6

(Areas , functions of 2 variables, and Series)
Submit over the internet by 11:59 PM Sunday 10 December 2017
Maximum value toward semester grade is 4 pts

Methods will be discussed in class;
CAUTION: Prof McFarland makes new tests each semester.
Average grade on Spring 2017 paper test #6 : 32.31 perfect = 45 See future grade prospects?

Click or type the appropriate answer for each question.
When you are sure of your answers, send them to your Professor:

Enter your name above

Enter preferred email address;
include @ and whatever follows

Enter student I.D.
[1]  Find 6x

x2 + 1
dx = y(x) + C ;  
As you use the Substitution Method, you will alter the integrand so that u' appears as a factor. Identify your substitution variables u and y and their derivatives in the table at the right.
Finally, let C = 0, x = 1, and enter   y(1) =
Be accurate to within 0.01
Scoring : 3 pts for y(1) above ; 2 pts for for table (right)
For example, if this question were to find 6x2 dx ,
then y(x) = 2x3 + C , and you would enter   y(1) = 2
u =
  x
x2
x2 + 1
(x2 + 1)-1
6x
du
dx
=
1
2x
6
-2x(x2 + 1)-2
2u
y =
  (½)u2
3x2
u
3 u
u-1
dy
du
=
u
6x
u-1
-u-2
3u-1

[2] (a)(1 pt) On the graph at the right, click those boxes located in the region R bounded by the curve y = x3 - x (red) and the curve y = 2 - 2x2 (green). On paper tests, you must shade these areas






  (b)(3 pts) Enter integers below (such as 4, 0, or -3) to express the size of the region R (in question [2a] above) as one definite integral:
It is possible to use 2 integrals, but here, use only one
Scoring: Limits worth 1 point, integrand (in parenthses) worth 2 points
 



( x3 + x2 + x + ) dx
  (c)(1 pt) Find the size of the region you drew in [2a] above by evaluating the integral which you wrote as an answer in (2b) above.
(size of R) =

[3] (a)(3 pts) If the value (in $ billions) of "dot-com" stocks on the NASDAQ is V(t) = 4 + e-x, where x is the time in years since 1 Jan 2000, then enter the average value of these stocks between 1 Jan 2000 and 1 Jan 2002.   $ billion
 
(b)(2 pts) For the geometric series

n=1
3n

4n+1
, write (for 1 pt) the sum of its first 3 terms :
and (for the 2nd point) the sum of all its terms : (Write decimals accurate to within 0.01)

[4]
Complete the GENERAL SOLUTION (below) for the differential equation y' = x

y + 1
  Your answer will contain a constant symbol "C" or "K". Thus, for example, if your original GENERAL SOLUTION was "y = x2 + K", you would re-write the answer here as "1x2 + 0y2 + 0x -1y + K = 0", and enter the red numbers below.
Scoring : First 4 numbers worth 4 points (no part credit) ; last entry worth a separate 1 point
  x2 + y2 + x + y + =0

[5] (a)(3 pts) Enter the value of
 
2
1

x x
 ) dx =
Review integration methods ?
Review improper integrals ?
(Above) Use decimal accurate to 0.01, or enter u if the area is infinite or undefined.
(b)(2 pts) Write the repeating decimal N = 0.151515.... as a reduced fraction of integers N =

Scoring : 2 points (above) for correct reduced fraction, one point for an equal but unreduced fraction

[6] Enter decimal numbers (accurate to within 0.01) to form the line which best fits the data points using the method of least-squares. If you must consult your text while answering this question, then you are at risk when writing your paper test later.
Scoring : 2½ points (above) for each coefficient (m and b)
 
Data Points : (-1,1) , (1,3) , (2,3), (4,5)
  y = x +

[7] (a)(2 pts) Find fx , fy , fxx , fxy , and fyy where f(x,y) = x3 - 3xy + y3 .
Enter integers below, such as 7, 1, 0, or -3
Scoring : ½ point for each of top 2 derivatives ;
1 point (total) for the bottom 3 derivatives (no part credit)
fx = x2 + y2 + x + y +
fy = x2 + y2 + x + y +
fxx = x + y +
fxy = x + y +
fyy = x + y +
Remember to hold one variable fixed or constant
(b) (1 pt) Use fx and fy above to find the two critical points (CPs) for f(x,y) in (7a) above; enter these CPs in the table below.
(c)(2 pts) Complete the table below using your results in (7a) and (7b) above:
Scoring : ½ point for each CP and each "type" ; 1 point (total) for the middle 4 table columns. Your must get full credit in (7a) and (7b) before credit in (7c) is awarded
CPs ( ½ pt each) A = fxx B = fxy C = fyy D = AC - B2 Type of CP
( ½ pt each)
( , )
smaller x-coordinate
Max
Min
Saddle Point
( , )
larger x-coordinate
Max
Min
Saddle Point

[8](a)(3 pts) Evaluate (showing work)
y
 
1
(y - 2xy) dx = x3 + y3 + x2 + y2 + xy
(Above and below) Enter integers to complete the integrals. The algebra and calculus are fairly easy.
   (b)(2 pts) Enter the value of the double integral:
1
 
0
y
 
1
(y - 2xy) dxdy =
Accuracy to within 0.001 is needed above

[9]
(a)(3 pts)Fill in the table at the right for the function
f(x) = e-x
 
Enter decimals accurate to within 0.001
in both (9a) and (9b)
Scoring : 1 point for column 1 only
2 points for all but right column
3 points for all 3 columns
n function: f (n)(x)   value at  
x = 0
    an    
0 e-x
1 e-x
2 e-x
3 e-x
 
Help with Taylor Series ?
(b)(2 pts) Use table entries (above) to write the first 4 terms of the Taylor series for the function f(x) = e-x
f(x) = e-x = + x + x2 + x3 + . . .

Do you want a "Least Squares" question (like [6] above) on next Tuesday's test? ( answer yes or no)
If a majority answers "no" to this question, then problem [6] next Tuesday will come from a different section of the text.


HOW COULD THIS WEB QUIZ AFFECT MY COURSE GRADE?
When your score is as high as you can make it, submit your test to Professor HERE.
Be sure your email address is correct above: your graded quiz will be immediately copied there!

Re-load page for new fortune