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Find $f(x)$.$$f^{\prime \prime}(x)=3 e^{x}+2, f^{\prime}(0)=-3, f(0)=1$$

$$3 e^{x}+x^{2}-6 x-2$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Missouri State University

Harvey Mudd College

Baylor University

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 $f$ . $f^{\prime}…

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Find $f(x)$.$$f^{\prim…

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Find $f^{\prime}(x)$.$…

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First find $f^{\prime}$ an…

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$$\text { Find } f^{\prime…

Find $f^{\prime \prime}(x)…

01:26

If $f(x)=-x^{3}-x^{2}+3,$ …

All right. We want to find ffx given F double prime of x is equal to three. E V x plus two. For initial conditions, F prime of zero equals negative three and f of zero equals one. This is initial about this is an initial value problem for a second order differential equation of F. Since F double prime is purely a function of X, there's no other variables. We can simply take anti derivatives successively and plugging in our initial conditions to solve the F. So our first anti derivative gives F prime equals three. E. V X plus two X plus the cost of immigration. See plugging in F prime of zero equals negative three gives negative three equals three plus C or C equals negative six. Thus we take the anti derivative again. This gives F equals three. E. V X plus X squared minus six X plus D. D. Is our second cost of immigration plugging in. F zero equals one gives us in the last line, one equals three plus D or D equals negative two. From this, we have our final solution F of x f of X is equal to three E to the X plus X squared minus six x minus two.

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