Download Group work 02:03

1. Group Work
Linearization.
(1) Find the linearization L(x) of the function at a = 3
2
f (x) = √
x2 − 5
(2) Verify the given linear approximation at a = 0
√
1
4
1 + 2x ≈ 1 + x
2
(3) Find the linear approximation
√ of the
√ function f (x) =
approximate the numbers .9 and .99
√
1 − x at a = 0 and use it to
(4) (Hard) Use a linear approximation to estimate the given number
√
3
1001
Hint: Think about what f (x) should be and then write down your line as L(x) =
f (a) + f 0 (a)(x − a). Finally evaluate at the point that you are interested in.
Advanced Related Rates.
(a) A lighthouse is located on a small island 3km away from the nearest point P on a straight
line shoreline and its light makes 4 revolutions per minute. How fast is the beam of light
moving along the shoreline when it is 1 km from P
(b) (Hard) A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks
away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of
the shadow moving when he is 40 ft from the pole.
Hint: Let x be the distance between the pole and the man and y be the distance
between the pole and the tip of his shadow. Then use similar triangles.
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(c) (Extra Challenge) A particle moves along the curve y = 2 sin(πx/2).√As the particle
passes through the point ( 31 , 1), its x-coordinate increases at a rate of 10 cm/s. How
fast is the distance between the particle and the origin changing at this instant.
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