(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
(b) g(x) = x 3 √
x at x = 8
(2) Use the differentiation rules to find the first derivative.
(a) y = 4x3− 1√ x
(b) f (t) = √
(c) r(θ) = sin θ tan2 θ
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(d) y = x + sin x x5 + 2x2
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(e) u = 4 √
(f) f (x) = cot (
(3) Find dy dx .
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(b) x2+y2 = √
(c) x2/3+y2/3 = 1
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(4) Find d2y dx2 in terms of x and y given xy + y2 = 2.
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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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(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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(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?
1knots are nautical miles per hour and one nautical mile is 2000 yds () June 16, 2014 20 / 29
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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 = t4 − 8t3 + 10t2 − 4, t ≥ 0.
(a) Determine the average velocity over the interval [0,1]
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s = t4 − 8t3 + 10t2 − 4, t ≥ 0
(b) Find the velocity of the particle.
(c) Find the acceleration of the particle.
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s = t4 − 8t3 + 10t2 − 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 = t4 − 8t3 + 10t2 − 4, t ≥ 0
(e) At which times is the particle at rest?
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s = t4 − 8t3 + 10t2 − 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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