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Wednesday, February 26, 2014

Angle Between Two Curves, 3

Category: Differential Calculus, Analytic Geometry, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the angle of intersection between the two curves:



Solution:

Consider the given pair of two curves above 

  

The first thing that we need to do is to get their point of intersection by solving the systems of equation as follows




Substitute this value of y to the second given equation, we have








The value of y is





The point of intersection of the given two curves is P(0, 1). 

Consider the first given equation of a curve 


Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have







The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve.

To get the value of the slope of a curve at their point of intersection, substitute x = 0 and y = 1 at the equation above, we have





The slope of a curve at their point of intersection is equal to the slope of tangent line that passes thru also at their point of intersection. Hence, 


Consider the second given equation of a curve


Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have






The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve.

To get the value of the slope of a curve at their point of intersection, substitute x = 0 and y = 1 at the equation above, we have
 
 
 
 
   
The slope of a curve at their point of intersection is equal to the slope of tangent line that passes thru also at their point of intersection. Hence, 


Therefore, the angle between two curves at their point of intersection is








or

 

Tuesday, February 25, 2014

Angle Between Two Curves, 2

Category: Differential Calculus, Analytic Geometry, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the angle of intersection between the two curves:



Solution:

Consider the given pair of two curves above



The first thing that we need to do is to get their point of intersection by solving the systems of equation as follows


but


Hence, the above equation becomes





The value of y is




The point of intersection of the given two curves is P(1, 2).

Consider the first given equation of a curve


Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have 




The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve. Actually, the first curve is a straight line. 

Since the right side of the equation contains coefficient only, the slope of the first curve is 


Consider the second given equation of a curve 


Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have 








The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve.  

To get the value of the slope of a curve at their point of intersection, substitute x = 1 and y = 2 at the equation above, we have 





The slope of a curve at their point of intersection is equal to the slope of tangent line that passes thru also at their point of intersection. Hence,  


Therefore, the angle between two curves at their point of intersection is 








or