Project #7706 - calculus1

1)  Find all points on the graph of the function 

f(x) = 2 cos x + cos2 x

 at which the tangent line is horizontal. (Use n as your arbitrary integer.)

(xy)  = 
  
 
 (smaller y-value)
(xy)  = 
  
 
 (larger y-value)

2)  Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

y sin 12x = x cos 2y,    (π/2, π/4)
 
y=
 
 
3)  Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
x2 + y2 = (2x2 + 4y2 − x)2
(0, 0.25)
(cardioid)
y =
 
 
 
 
 
4)  Find 
y''
 by implicit differentiation.
7x2 + y2 = 3
y'' =  

 

5)   Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.

If the circle has radius r, its equation is 
x2 + y2 = r2
 implies 
 + 2yy' = 0
 implies 
y' =  ,
 so the slope of the tangent line at 
P(x0y0)
 is  . The negative reciprocal of that slope is  , which is the slope ofOP, so the tangent line at P is perpendicular to the radius OP.

Subject Mathematics
Due By (Pacific Time) 06/11/2013 11:12 pm
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