dt - mlgibbons
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Transcript dt - mlgibbons
25. A police car traveling south toward Sioux Falls at 160 km/h pursues
a truck traveling east away from Sioux Falls, Iowa, at 140 km/h. At time
t = 0, the police car is 20 km north and the truck is 30 km east of Sioux
Falls. Calculate the rate at which the distance between the vehicles is
changing:
dl
l 20 30 1300
2
dl
dt
dy
dx
160 &
140
dt
dt
y 20 160t & x 30 140t
l 2 30 140t 20 160t
2
2l
2
dt
27.735 km/h
t 0
@ t 1/12 h, l 42.1966 km
112.963 km/h
t 1/12
2
dl
dl
2 30 140t 140 2 20 160t 160 l 4200 19600t 3200 25, 600t
dt
dt
dl 1000 45, 200t
dt
l
30. A particle moves counterclockwise around the ellipse with
equation 9x2 + 16y2 = 25.
(a)
In which of the four quadrants is dx/dt > 0? Explain.
(b) Find a relation between dx/dt and dy/dt.
(c) At what rate is the x-coordinate changing when the particle passes
the point (1, 1) if its y-coordinate is increasing at a rate of 6 m/s?
(d) Find dy/dt when the particle is at the top and bottom of the
ellipse.
dx
x w/ respect to t
dt
dx
is positive in quadrants III & IV
dt
x is moving to the right .
9 x 2 16 y 2 25
18 x
dx
dy
32 y
0
dt
dt
dx
dy
32 y
0
dt
dt
dy
dx
1,1 & 6 18 32 6 0
dt
dt
dx
32 6
32
m/s
dt
18
3
18 x
dy
0
dt
41. A roller coaster has the shape of the graph. Show that
when the roller coaster passes the point (x, f (x)), the
vertical velocity of the roller coaster is equal to f '(x) times
its horizontal velocity.
dy
dx
y f x
f ' x
dt
dt
In other words, vertical velocity equals
dx
f ' x or f ' x horizontal velocity.
dt
y f x