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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 9 December 2018
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 Fall 2017 paper test #6 : 31.51 perfect = 45 See future grade prospects?

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[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 (just after the integral sign) 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 + . . .

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