BC 3 Name:
Show all appropriate work clearly for full credit. No calculators allowed.
1(6 pts). Let u

3 i

4 j and v

2 i

3 j . Find the following: a.
2 u v b.
2 u v c.
A unit vector in the direction of u
2 (2 pts each)
Given the slope field shown at the right, sketch possible solution curves that pass through the points A and B listed below. Be sure to label each curve. a. A(0, –2) b. B (–2, 2)
3 2 1
3
2
1
1
2
1
3
3(4 pts). Suppose the function y
  
is a solution to the differential equation y
Determine the equation of the tangent line to the graph of f at the point (6, –2).
2 x
2 y x y
1 .
3
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IMSA
4(4pts). Match each differential equation listed below with the one of the 3 slope fields shown at the right. a. y
 
 x
2 
4
 y
2 
4

8 b. y
 
5
2 sin
 
2 x
 sin
 
2 y

Slope Field #1
Slope Field #2
Slope Field #3
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5(8 pts). Let y
  xy
2
, with y
 
2 .
2 a. Use Euler’s Method starting at x = –1, with step size =
1
2
Show steps clearly.
to estimate the value of y (0). b. Now solve the given IVP for y . (5 pts)
6(6 pts) Suppose that v (1)

0, 0 a

1
3
, 6 gives the acceleration of an object at time t  0 . If r (1)

0, 0 t
, find the position vector
 
.
and
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7(9 pts). An object moving along a curve in the xy -plane is at position ( ( ), ( )) at time t , where for t

0 . At time t dx dt
 tan
 
= 0, the object is at position
 t and
(1, 1) dy
.
dt
 sec
Give exact answers – your answers may involve quantities such as t a n (1 ) o r s e c . e a.
Find the slope of the line tangent to the curve at the position (1, 1) . b.
Write an expression that gives the speed of the object at time t = 0. c.
Write an integral expression that gives the total distance traveled by the object over the time period 1 t 2 . d.
Is there a time t

0 at which the object is on the y -axis? Explain.
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8(3 pts) Find the length of the graph of the cardioid r
  
.
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