BC Calc Test 10
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Please box all answers
Name:
All problems are to be solved algebraically.
No Calculator
Multiple Choice- Write the letter of your choice in the space at right.
1)________
1) If f is a vector-valued function defined by f (t )  ln(t )i  t j, then f (1) 
2)________
1
A) i  j
4
1
B) i  j
4
1
C) i  j
2
1
D) i  j
2
1
E) i + j
2
3)________
4)________
2) If x(t )  t 2  t and y (t )  3t  1, then
A)
1
2
B)
1
2
C)
3) Find the value of the integral
A)

B) 1
2
C)
3
4

1
D) 1
2
0
5)________
dy
at t  1 is
dx
E) 2
1
dx.
1  4 x2

D)
8
1
2
ln 2
E)

4
4) Which of the following gives the best approximation for sin(1) ?
1 1
A) 1  
3 5
1 1
B) 1  
6 120
C)
1 1
1
 
2 24 720
D)
1 1 1
 
2 4 6
5) A unit vector normal to 4, 3 is
A)
4
5
, 53
B)
4
5
, 53
C)
3
5
, 54
D)
3
5
, 54
E)
5
4
, 53
E)

2
6) A particle has position ( x(t ), y (t )) at any time t. The position of the particle at time t  1 is
1
 1
(2, 6), and the velocity vector at any time t  0 is given by 1  2 , 2  2  .
t 
 t
a) Find the acceleration vector at time t  3.
b) Find the position of the particle at time t  3.
c) For what time t  0 does the line tangent to the path of the particle at ( x(t ), y (t )) have a slope
of 8?
d) The particle approaches a line as t  . Find the slope of this line. Show the work that leads
to your conclusion.
BC Calc Test 10
Show all work
Please box all answers
Name:
All problems are to be solved algebraically.
Calculator Allowed
Multiple Choice- Write the letter of your choice in the space at right.
3

. The slope of the curve at   is
7) A polar curve is given by r 
2  cos
2
A) 0
B)
1
2
C)
3
4
D)
3
4
E) not defined
7)________
8)________
9)________
8) Let R be the region bounded by the curves f ( x)  x3  x 2 and g ( x)  2 x for x  0. Let W
represent the volume of the solid formed when R is rotated about the y-axis. There is a line x  c
that divides R such that the volume of the region bounded by f, g, and x  c is half of W.
What is the value of c ?
A) 1.238
B) 1
C) 1.122
D) 0.907
E) 1.311
9)  sin 1 (5 x)dx 
A) Hey
B) x sin 1 (5x) 
1  25x 2
C
5
C) Dummy,
D) Choose
E) Answer B
10) Let R be the region inside the graph of the polar curve r  2 and outside the graph of the
polar curve r  2(1  sin  ).
a) Sketch the two polar curves in the xy-plane provided below and shade the region R. Label all
x- and y- intercepts.
b) Find the area of R.
c) Find the slope of the curve r  2(1  sin  ) at the point where  
leads to your answer.
d) Find the perimeter of the region R.

4
. Show the work that
x
4
and G be the function given by G( x)   f (t )dt.
2
0
1 t
a) Find the first four nonzero terms and the general term for the power series expansion of f (t )
about t = 0.
11) Let f be the function given by f (t ) 
b) Find the first four nonzero terms and the general term for the power series expansion of G ( x)
about x = 0.
c) Find the interval of convergence of the power series in part b). (Your solution must include an
analysis that justifies your answer.)