(1) find the equations of the tangent and the normal lines

(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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