[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 41. lim x → 0 z − 1 z 2 ( z + 3 ) z z − 1 z 2 ( z + 3 ) z z − 1 z 2 ( z + 3 ) -0.1 a. 0.1 e. -0.01 b. 0.01 f. -0.001 c. 0.001 g. -0.0001 d. 0.0001 h.
[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 41. lim x → 0 z − 1 z 2 ( z + 3 ) z z − 1 z 2 ( z + 3 ) z z − 1 z 2 ( z + 3 ) -0.1 a. 0.1 e. -0.01 b. 0.01 f. -0.001 c. 0.001 g. -0.0001 d. 0.0001 h.
Graph the following function. Please also graph the asymptote. Thank you.
A ladder 27 feet long leans against a wall and the foot of the ladder is sliding away at a constant rate of 3 feet/sec. Meanwhile, a firefighter is climbing up the ladder at a rate of 2 feet/sec. When the firefighter has climbed up 6 feet of the ladder, the ladder makes an angle of л/3 with the ground. Answer the two related
rates questions below. (Hint: Use two carefully labeled similar right triangles.)
(a) If h is the height of the firefighter above the ground, at the instant the angle of the ladder with the ground is л/3, find dh/dt=
feet/sec.
(b) If w is the horizontal distance from the firefighter to the wall, at the instant the angle of the ladder with the ground is л/3, find dw/dt=
feet/sec.
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate (in mi/h) is the distance between the cars increasing four hours later?
Step 1
Using the diagram of a right triangle given below, the relation between x, y, and z is
z²
= x²+
+12
x
Step 2
We must find dz/dt. Differentiating both sides and simplifying gives us the following.
2z
dz
dt
dx
2x.
+2y
dt
dx
dy
dz
x
+y
dt
dt
dt
2z
dy
dt
×
dx
(x+y
dt
dy
dt
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