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If $f^{\prime}(x)=2 x+3,$ sketch the family of solutions.

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Missouri State University

Baylor University

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.

01:03

If $f^{\prime}(x)=3 / x,$ …

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Suppose $f^{\prime}(x)=(x-…

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Sketch a graph of each equ…

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Sketch the function. $…

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Sketch the graph of the fu…

01:19

Suppose that $f^{\prime}(x…

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Sketch the graphs of the f…

Okay if F prime of X is equal to two X plus three, sketch the family of solutions affects this question is challenging our understanding of basic differential equations specifically, since f prime of X equals two X plus three, we can apply integration or specifically the anti derivative to identify the solution family. So for F prime of X equals to expose three. We integrate or rather take the anti derivative to find a five X. So the derivative gives a five X equals X squared plus three X plus a constant integration. See that's the solution family depends on the value of See if we sketch the solution for C equal zero first, we have the following graph. So F is just parabolas. So if we increase or decrease, see we're interested in how the sketch changes. Thus we can sketch the family of solutions, including C equals negative one and one as follows, C equals two is plotted above equals zero equals negative, two is plotted below.

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