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Find $f(x)$.$$f^{\prime \prime}(x)=x-2, f^{\prime}(2)=1, f(1)=-1$$

$$\frac{x^{3}}{6}-x^{2}+3 x-\frac{19}{6}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Missouri State University

Campbell University

University of Nottingham

Boston College

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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Find $f(x)$.$$f^{\prim…

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find $f^{\prime \prime}(2)…

01:10

Find $ f $.

$ f&qu…

01:55

Find $f^{\prime}(x)$ and $…

01:30

Find $f^{\prime}(0)$ and $…

00:46

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00:35

Find $f(-1)$ if $f(x)=2 x^…

mhm. We want to find ffx given F double prime of x equals x minus two. Were given the initial conditions at prime two equals 1.5 of one equals negative one. This question is asking us to solve a second order differential equation for F given initial conditions a Kia It is an initial value problems for a second order to thank you to solve because F double private access simply a function of X will take anti directive successively until we land fx. So first the derivative of F double prime gives F prime equals expert over two minus two X plus C. The cost of immigration plugging in F prime of two equals one gives one equals two minus four plus C. Or C equals three. Thus we have F prime equals X squared minus two X plus three. We take a second anti derivative to obtain F equals X cubed over six minus X squared plus three X plus D. The second class of immigration plug in F one equals negative one gives negative one equals +16 minus one plus three plus D, or D equals negative 1936. Thus we have final solution, F equals X cubed over six minus X squared plus three X minus 19/6.

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