Chapter 3 | Derivatives
285
3.5 EXERCISES
dy dx
192. y = sin x cos x
For the following exercises, find
for the given
functions. 175. y = x 2 −sec x +1
193. y = x − 1
x
2 sin
194. y = 1 x +tan x 195. y =2csc x 196. y = sec 2 x
176. y =3csc x + 5 x 177. y = x 2 cot x 178. y = x − x 3 sin x 179. y = sec x x 180. y = sin x tan x 181. y = ( x +cos x )(1−sin x ) 182. y = tan x 1−sec x 183. y = 1−cot x 1+cot x 184. y =cos x (1+csc x )
x
197. values on the graph of f ( x ) =−3sin x cos x where the tangent line is horizontal. 198. Find all x values on the graph of f ( x ) = x −2cos x for 0< x <2 π where the tangent line has slope 2. Find all 199. Let f ( x ) =cot x . Determine the points on the graph of f for 0< x <2 π where the tangent line(s) is (are) parallel to the line y =−2 x . 200. [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function s ( t ) =−6cos t where s is measured in inches and t is measured in seconds. Find the rate at which the spring is oscillating at t =5 s. 201. Let the position of a swinging pendulum in simple harmonic motion be given by s ( t ) = a cos t + b sin t where a and b are constants, t measures time in seconds, and s measures position in centimeters. If the position is 0 cm and the velocity is 3 cm/s when t =0 , find the values of a and b . 202. After a diver jumps off a diving board, the edge of the board oscillates with position given by s ( t ) =−5cos t cm at t seconds after the jump. a. Sketch one period of the position function for t ≥0. b. Find the velocity function. c. Sketch one period of the velocity function for t ≥0. d. Determine the times when the velocity is 0 over one period. e. Find the acceleration function. f. Sketch one period of the acceleration function for t ≥0.
For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x . Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.
185. [T] f ( x ) =−sin x , x =0 186. [T] f ( x ) =csc x , x = π 2 187. [T] f ( x ) =1+cos x , x = 3 π 2 188. [T] f ( x ) = sec x , x = π 4 189. [T] f ( x ) = x 2 −tan x , x =0 190. [T] f ( x ) =5cot x , x = π 4
d 2 y dx 2
For the following exercises, find
for the given
functions. 191. y = x sin x −cos x
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