(1) Find the equations of the tangent and the normallines to the graph of the given function at the indicatedx value.

(a) f (x) = sec(x

4

)at x = π

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(b) g(x) = x 3√

x at x = 8

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(2) Use the differentiation rules to find the firstderivative.

(a) y = 4x3− 1√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 xx5 + 2x2

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(e) u = 4√

x sec(2x)

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(f) f (x) = cot(

3x

)

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(3) Find dydx .

(a) xy+cos y = tan x

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(b) x2+y2 =√

x − 2y

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(c) x2/3+y2/3 = 1

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(4) Find d2ydx2 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 ofx 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 of1200 cm3/min at the instant that its edges are 20 cmlong? At what rate are the lengths of the edgeschanging at that instant?

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(7) A stone dropped into a still pond sends out acircular ripple whose radius increases at a constantrate of 3 ft/sec. How fast is the area enclosed by theripple increasing at the end of 10 sec?

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(8)Two commercial airplanes are flying at 40,000 ft along straight-linecourses that intersect at right angles. Plane A is approaching theintersection point at a speed of 442 knots1 while plane B isapproaching the intersection point at 481 knots. At what rate is thedistance between the planes changing when A is 5 and B is 12nautical 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 itsposition 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 overwhich 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 betweent = 0 and t = 7 sec, the particle must be at the origin.

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