Chapter 3 | Derivatives
317
3.8 EXERCISES For the following exercises, use implicit differentiation to find dy dx . 300. x 2 − y 2 =4 301. 6 x 2 +3 y 2 =12 302. x 2 y = y −7
316. [T] The graph of a folium of Descartes with equation 2 x 3 +2 y 3 −9 xy =0 is given in the following graph.
303. 3 x 3 +9 xy 2 =5 x 3 304. xy −cos( xy ) =1 305. y x +4= xy +8 306. − xy −2= x 7 307. y sin( xy ) = y 2 +2 308. ( xy ) 2 +3 x = y 2 309. x 3 y + xy 3 =−8
a. Find the equation of the tangent line at the point (2, 1). Graph the tangent line along with the folium. b. Find the equation of the normal line to the tangent line in a. at the point (2, 1). 317. For the equation x 2 +2 xy −3 y 2 =0, a. Find the equation of the normal to the tangent line at the point (1, 1). b. At what other point does the normal line in a. intersect the graph of the equation? 318. Find all points on the graph of y 3 −27 y = x 2 −90 at which the tangent line is vertical. 319. For the equation x 2 + xy + y 2 =7, a. Find the x -intercept(s). b. Find the slope of the tangent line(s) at the x -intercept(s). c. What does the value(s) in b. indicate about the tangent line(s)? 320. Find the equation of the tangent line to the graph of the equation sin −1 x +sin −1 y = π 6 at the point ⎛ ⎝ 0, 1 2 ⎞ ⎠ . 321. Find the equation of the tangent line to the graph of the equation tan −1 ( x + y ) = x 2 + π 4 at the point (0, 1). 322. Find y ′ and y ″ for x 2 +6 xy −2 y 2 =3.
For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.
310. [T] x 4 y − xy 3 = −2, (−1, −1) 311. [T] x 2 y 2 +5 xy =14, (2, 1)
312. [T] tan( xy ) = y , ⎛ ⎝ π ⎞ ⎠ 313. [T] xy 2 +sin( πy )−2 x 2 = 10, (2, −3) 314. [T] x y +5 x −7= − 3 4 y , (1, 2) 4 , 1
315. [T] xy +sin( x ) =1, ⎛ ⎝ π
⎞ ⎠
2 , 0
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