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$y=c x^{2}$ represents a one-parameter family of parabolas. Find the equation of another one-parameter family of curves with the property that at any point(x, $y$ ) where the curves from both families pass through the point, their tangent lines at this point are perpendicular to each other. (Show that the slope of the tangent lines at any point not on the $y$ -axis to the original family of parabolas is $2 y / x$

$$2 y^{2}=-x^{2}+c_{2}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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10:57

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