Chapter 3 | Derivatives
307
292. f ⎛
⎞ ⎠ = 1 2 ,
f ′ ⎛
⎞ ⎠ = 2 3 ,
b. find the equation of the tangent line to the graph of f −1 at the indicated point.
⎝ 3
⎝ 3
a = 1 2
293. f (1) =−3, f ′(1) =10, a =−3 294. f (1) =0, f ′(1) =−2, a =0 295. [T] The position of a moving hockey puck after t seconds is s ( t ) = tan −1 t where s is in meters. a. Find the velocity of the hockey puck at any time t . b. Find the acceleration of the puck at any time t . c. Evaluate a. and b. for t =2, 4, and 6 seconds. d. What conclusion can be drawn from the results in c.? 296. [T] A building that is 225 feet tall casts a shadow of various lengths x as the day goes by. An angle of elevation θ is formed by lines from the top and bottom of the building to the tip of the shadow, as seen in the following figure. Find the rate of change of the angle of elevation dθ dx when x =272 feet.
274. f ( x ) = 4
, P (2, 1)
1+ x 2
275. f ( x ) = x −4, P (2, 8)
4
⎛ ⎝ x 3 +1
⎞ ⎠
276. f ( x ) =
, P (16, 1)
277. f ( x ) =− x 3 − x +2, P (−8, 2) 278. f ( x ) = x 5 +3 x 3 −4 x −8, P (−8, 1)
dy dx
For the following exercises, find
for the given
function. 279. y = sin −1 ⎛ ⎞ ⎠ 280. y =cos −1 ( x ) 281. y = sec −1 ⎛ ⎝ 1 x ⎞ ⎠ ⎝ x 2
282. y = csc −1 x
3
⎛ ⎝ 1+tan −1 x ⎞ ⎠
283. y =
284. y =cos −1 (2 x ) · sin −1 (2 x )
297. [T] A pole stands 75 feet tall. An angle θ is formed when wires of various lengths of x feet are attached from the ground to the top of the pole, as shown in the following figure. Find the rate of change of the angle dθ dx whenawire of length 90 feet is attached.
285. y = 1 tan −1 ( x ) 286. y = sec −1 (− x ) 287. y =cot −1 4− x 2 288. y = x ·csc −1 x
For the following exercises, use the given values to find ⎛ ⎝ f −1 ⎞ ⎠ ′( a ).
289. f ( π ) =0, f ′( π ) =−1, a =0
290. f (6) =2, f ′(6) = 1 3 ,
a =2
⎛ ⎝ 1 3
⎞ ⎠ =−8, f ′ ⎛
⎞ ⎠ =2, a =−8
⎝ 1 3
291. f
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