(1) Find the equations of the tangent and the normal
lines to the graph of the given function at the indicated
x value.
(a) f (x) = sec
()
x 4
at x = π
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()
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√
(b) g(x) = x 3 x
()
at x = 8
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()
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(2) Use the differentiation rules to find the first
derivative.
1
(a) y = 4x 3 − √
x
(b) f (t) =
()
√
cos t
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(c) r (θ) = sin θ tan2 θ
()
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(d) y =
x + sin x
x 5 + 2x 2
()
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(e) u =
√
4
()
x sec(2x)
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3
(f) f (x) = cot
x
()
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(3) Find
dy
dx .
(a) xy+cos y = tan x
()
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(b) x 2 +y 2 =
()
p
x − 2y
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(c) x 2/3 +y 2/3 = 1
()
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(4) Find
d 2y
dx 2
()
in terms of x and y given xy + y 2 = 2.
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()
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(5) Find the equation of the line tangent to the curve of
x sin 2y = y cos 2x at the point (π/4, π/2).
()
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(6) The volume of a cube is increasing at a rate of
1200 cm3 /min at the instant that its edges are 20 cm
long? At what rate are the lengths of the edges
changing at that instant?
()
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()
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(7) A stone dropped into a still pond sends out a
circular ripple whose radius increases at a constant
rate of 3 ft/sec. How fast is the area enclosed by the
ripple increasing at the end of 10 sec?
()
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()
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(8)
Two commercial airplanes are flying at 40,000 ft along straight-line
courses that intersect at right angles. Plane A is approaching the
intersection point at a speed of 442 knots1 while plane B is
approaching the intersection point at 481 knots. At what rate is the
distance between the planes changing when A is 5 and B is 12
nautical miles from the intersection?
1
knots are nautical miles per hour and one nautical mile is 2000 yds
()
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()
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()
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(9) A particle is moving along the x-axis so that its
position s in feet at time t in seconds satisfies
s = t 4 − 8t 3 + 10t 2 − 4,
t ≥ 0.
(a) Determine the average velocity over the interval [0, 1]
()
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s = t 4 − 8t 3 + 10t 2 − 4,
t ≥0
(b) Find the velocity of the particle.
(c) Find the acceleration of the particle.
()
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s = t 4 − 8t 3 + 10t 2 − 4,
t ≥0
(d) Over which intervals is the particle moving to the left, and over
which is it moving to the right?
()
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s = t 4 − 8t 3 + 10t 2 − 4,
t ≥0
(e) At which times is the particle at rest?
()
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s = t 4 − 8t 3 + 10t 2 − 4,
t ≥0
Argue, with some solid mathematics, that at some moment between
t = 0 and t = 7 sec, the particle must be at the origin.
()
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()
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()
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