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rozin
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Problem:Find the differential equation satisfied (i) by the equation of the family of tangents to y=x^2 and (ii) by the equation of the family of normals to y=x^2.
A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to represent the rate of change of a function at a given point.
The tangent line to y=x^2 at a specific point is a line that touches the curve y=x^2 at that point and has the same slope as the curve at that point. It represents the instantaneous rate of change of y=x^2 at that point.
To find the equation of the tangent line to y=x^2 at a specific point, you need to find the slope of the curve at that point using the derivative of y=x^2. Then, you can use the point-slope formula to find the equation of the tangent line.
The normal line to y=x^2 at a specific point is a line that is perpendicular to the tangent line at that point. It represents the rate of change of the tangent line at that point.
To find the equation of the normal line to y=x^2 at a specific point, you can first find the slope of the tangent line at that point using the derivative of y=x^2. Then, you can use the negative reciprocal of this slope to find the slope of the normal line. Finally, you can use the point-slope formula to find the equation of the normal line.