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Friday, February 28, 2014

Second Derivative Problems - Product Formula

Category: Differential Calculus, Algebra

"Published in Vacaville, California, USA"

Find a formula for


Solution:

The first thing that we have to do is to get the derivative of uv with respect to x where u and v are functions of x. Apply the derivative by product formula, we have




Take the derivative again of the above equation with respect to x by product formula, we have









where:




 

Thursday, February 27, 2014

Angle Between Two Curves, 4

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. If you subtract the first equation from the second equation, the left side of the equation will be equal to zero. Hence, the resulting equation is



The value of y is






The point of intersection of the given two curves is P(1, ½).

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 = 1 and y = ½ 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 = 1 and y = ½ 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






 

 
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