Category: Differential Equations, Integral Calculus, Analytic Geometry, Algebra
"Published in Newark, California, USA"
Find the equation of the curve for which y" = 2, and which has a slope of -2 at its point of inflection (1, 3).
Solution:
The concavity of a curve is equal to the second derivative of a curve with respect to x. In this case, y" = d²y/ dx². Let's consider the given concavity of a curve
We can rewrite the above equation as follows
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
The point of inflection is a point where the direction of the concavity of a curve will start to change. In this case (1, 3) is the point of inflection of a curve. Since it is also included in the curve, then we can use it to substitute the value of x and y later in the problem.
Substitute the value of the given slope and the point of inflection to the above equation in order to solve for the value of a constant, we have
Hence, the above equation becomes
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
In order to solve for the value of a constant, substitute the value of x and y from the coordinates of a given point of inflection to the above equation, we have
Therefore, the equation of a curve is
Category: Differential Equations, Integral Calculus, Analytic Geometry, Algebra
"Published in Newark, California, USA"
Find the equation of the curve for which y" = 6x², and which passes through the points (0, 2) and (-1, 3).
Solution:
The concavity of a curve is equal to the second derivative of a curve with respect to x. In this case, y" = d²y/ dx². Let's consider the given concavity of a curve
We can rewrite the above equation as follows
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
Since the slope of a curve is not given in the problem but the two points are given, then we have to continue the integration until we get an equation in terms of x and y. Let's consider the equation above
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
In
order to get the value of arbitrary constants, we need to use the coordinates of two points so that we can form the two equations, two unknowns.
By using the point (0, 2), substitute the value of x and y to the above equation, we have
By using the point (-1, 3), substitute the value of x and y to the above equation, we have
but
then the above equation becomes
Therefore, the equation of a curve is
