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Consider the graph of the solutions of the equationy3+xy+2=0

(a) Find all points on the graph with an x-coordinate of x = 1, and then find the slope of the tangent line at each of these points.

(b) Find all points on the graph with a y-coordinate of y = 1, and then find the slope of the tangent line at each of these points.

(c) Find all points where the graph has a horizontal tangent line.

(d) Find all points where the graph has a vertical tangent line.

Short Answer

Expert verified

Part (a): 1,-2isthepointonthegraph.dydx=213

Part (b): -3,1isthepointonthegraph.dydx=-

Part (c): Thereisnopointwheretangentlineishorizontal.

Part (d): Thereisnopointwheretangentlineisvertical.

Step by step solution

01

Step 1. Given information is:

y3+xy+2=0

02

Part (a) Step 1. Calculating dy/dx

Here,y3+xy+2=0dy3+xy+2dx=0(Differentiatingbothsides)3y2dydx+xdydx+ydxdx+0=03y2dydx+xdydx+y=0(3y2+x)dydx+y=0dydx=-y3y2+x

03

Part (a) Step 2. Calculating y and slope 

Substitutingx=1intotheequationy3+xy+2=0:y3+(1)y+2=0y3+y=-2y(y2+1)=-2y=-2Thus,1,-2isthepointonthegraph.Thustheslopeis:x=1,y=-2dydx=-(-2)3(-2)2+1dydx=23(4)+1dydx=213

04

Part (b) Step 1. Calculating x and slope

Substitutingy=1intotheequationy3+xy+2=0:13+x+2=0x=-3Thus,-3,1isthepointonthegraph.Thustheslopeis:whenx=-3,y=1dydx=-13(1)2+(-3)dydx=-10=-

05

Part (c) Step 1. Finding points

Substitutingx=0intotheequationy3+xy+2=0forthetangentlinetobehorizontal:y3+(0)y+2=0y3=-2Thus,thereisnopointwheretangentlineishorizontal.

06

Part (d) Step 1. Finding points

Substitutingy=0intotheequationy3+xy+2=0forthetangentlinetobevertical:03+x(0)+2=02=0Thus,thereisnopointwheretangentlineisvertical.

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