E3_2 (6474623)
Due:
Fri Oct 24 2014 06:00 PM AKDT
Question
1.
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9 10 11 12
Question Details
SCalc7 6.7.017. [1874237]
Prove the identity.
tanh(ln x) =
x2 − 1
x2 + 1
=
tanh(ln x) =
eln x −
/2
(eln x + e−ln x)/2
cosh(ln x)
x − (eln x)−1
− x−1
=
=
x + x−1
=
x − 1/x
=
x + 1/x
/x
(x2 + 1)/x
=
Solution or Explanation
tanh(ln x) =
sinh(ln x)
(eln x − e−ln x)/2
x − (eln x)−1
x − x−1
x − 1/x
(x2 − 1)/x
x2 − 1
=
=
=
=
=
=
ln
x
−ln
x
ln
x
−1
−1
2
cosh(ln x)
x + 1/x
(e
+e
)/2
x + (e
)
x+x
(x + 1)/x
x2 + 1
-
2.
Question Details
SCalc7 6.7.023. [1873309]
Use the definitions of the hyperbolic functions to find each of the following limits.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
lim tanh x
x→∞
lim
x → −∞
tanh x
lim sinh x
x→∞
lim
x → −∞
sinh x
lim sech x
x→∞
lim coth x
x→∞
lim
coth x
lim
coth x
lim
csch x
x → 0+
x → 0−
x → −∞
Solution or Explanation
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-
3.
Question Details
SCalc7 6.7.027. [1873818]
-
Prove this equation using the method of this example and a previous equation with x replaced by y.
(a) using the method of the example
Let y = tanh−1 x. Then
= tanh y =
sinh y
=
cosh y
y
2y
· e = e −1
(ey + e−y)/2
ey
e2y + 1
xe2y + x = e2y − 1
− xe2y
1+x=
1 + x = e2y(1 − x)
1+x
e2y =
2y = ln
1+x
1−x
y=
.
(b) using the method of a previous equation
1 + tanh y
Let y = tanh−1 x. Then x =
2y = ln
y=
, so we have e2y =
=
1+x
1−x
1+x
1−x
.
Solution or Explanation
(a) Let y = tanh−1 x. Then
y
2y
sinh y
(ey − e−y)/2
x = tanh y =
=
· e = e −1
xe2y + x = e2y − 1
cosh y
(ey + e−y)/2
ey
e2y + 1
1+x
1+x
1
1+x
e2y(1 − x)
e2y =
2y = ln
y=
ln
.
1−x
1−x
2
1−x
1 + x = e2y − xe2y
1+x=
(b) Let y = tanh−1 x. Then x = tanh y, so from Exercise 18 we have
1 + tanh y
1+x
1+x
1
1+x
e2y =
=
2y = ln
y=
ln
.
1 − tanh y
1−x
1−x
2
1−x
4.
Question Details
Find the derivative. Simplify where possible.
g(x) = cosh(ln x)
g'(x) =
Solution or Explanation
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SCalc7 6.7.032. [1874177]
-
5.
Question Details
SCalc7 6.7.034. [2535420]
-
SCalc7 2.9.011. [1884373]
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SCalc7 2.9.004. [1875047]
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SCalc7 2.9.021. [1679520]
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Find the derivative. Simplify where possible.
y = x coth(6 + x2)
y'(x) =
Solution or Explanation
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6.
Question Details
Find the differential of each function.
y = x2 sin 4x
(a)
dy =
(b)
y=
5 + t2
dy =
Solution or Explanation
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7.
Question Details
Find the linearization L(x) of the function at a.
f(x) = x1/2, a = 25
L(x) =
Solution or Explanation
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8.
Question Details
Compute Δy and dy for the given values of x and dx = Δx. (Round your answers to three decimal places.)
y=
6
, x = 4, Δx = 1
x
Δy =
dy =
Sketch a diagram showing the line segments with lengths dx, dy, and Δy.
Solution or Explanation
y = f(x) = 6/x, x = 4, Δx = 1
Δy = f(5) − f(4) = 6 − 6 = −0.3
5
4
6
6
dy = − 2 , dx = − 2 (1) = −0.375
x
4
9.
Question Details
SCalc7 2.9.034. [1680683]
-
Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome
with diameter 58 m. (Round your answer to two decimal places.)
m3
Solution or Explanation
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10.
Question Details
SCalc7 2.9.501.XP. [1874656]
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SCalc7 2.8.024.MI. [1874479]
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Find the linearization L(x) of the function at a.
f(x) = cos x, a = 5π/2
L(x) =
Solution or Explanation
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11.
Question Details
A trough is 12 ft long and its ends have the shape of isosceles triangles that are 4 ft across at the top and have a height of
1 ft. If the trough is being filled with water at a rate of 9 ft3/min, how fast is the water level rising when the water is 9
inches deep?
ft/min
Solution or Explanation
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12.
Question Details
SCalc7 2.8.038. [1874875]
Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the
floor h = 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/s. How fast
is cart B moving toward Q at the instant when cart A is 5 ft from Q? (Round your answer to two decimal places.)
ft/s
Solution or Explanation
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Assignment Details
Name (AID): E3_2 (6474623)
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Author: Frith, Russell ( [email protected] )
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