1. The equationz2ry y2 3 defines a point on a curve called an ellipse. Show that

Question

1. The equationz2ry y2 3 defines a point on a curve called an ellipse. Show that (1, 1) is on the ellipse, then find the slope of the tangent line to the curve at (1,1) 2. Suppose that a model rocket is launched from a 5 foot tall stand. The day is calm, and the rocket travels straight up from its starting position. A five foot tall observer stands 10 feet away from the stand. Let be the angle formed by the line between the observer and the stand and the line between the observer and the model rocket. The observer is able to track at each point in time with a sextant. At one point, the observer notices that the angle has a value of 1.373 radians, and the angle is i stand, and how quickly is the rocket going upwards? at about .3244 radians per second. At this point in time, how high is the rocket above the

Explanation / Answer

1. x2+xy+y2=3

Substituting point (1,1)

12+1*1+12 =1=1+1=3

Therefore point lies on the ellipse

x2+xy+y2=3

2xdx +xdy+ydx+2ydy=0

dx(2x+y)+dy(2y+x)=0

dy/dx=-(2x+y)/(2y+x)

At point (1,1) , dy/dx=-(2+1)/(2+1)=-1

So slope of the tangent at point (1,1) is -1

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