Finding f(x) Given f''(x) and f'(x)

In summary, an anti derivative in calculus is the reverse process of differentiation, where the original function is found from a given function. Its purpose is to solve indefinite and definite integrals, allowing for the calculation of total accumulated change over a given interval. Anti derivatives can be found using methods such as the power rule, substitution, integration by parts, and partial fractions, depending on the complexity of the function. The main difference between an anti derivative and a derivative is that the former gives the original function while the latter gives a new function. Finally, an anti derivative can have multiple solutions due to the loss of the constant term during differentiation, which can be differentiated back to the same given function.
  • #1
kevinr
32
0

Homework Statement



f''(x) = SQRT(x) - 2 cos(x)

What is f(x)? (all possible solutions)

Homework Equations



-

The Attempt at a Solution



Well here is what i got:

f'(x) = (2/3)x^(3/2) - 2 sin(x) + C
f(x) = (2/3)(2/5)x^(5/2) + 2 cos(x) + C + D

thus final answer is:

f(x) = (4/15)x^(5/2) + 2 cos(x) + C + D

I hope that's right!

Thanks for your time! :)
 
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  • #2
It's correct except for one thing. When you integrate the second time, you're integrating a constant C. What should C have after the second integration?
 
  • #3
Cx ?

o you thank you!
 

Related to Finding f(x) Given f''(x) and f'(x)

1. What is an anti derivative?

An anti derivative is the reverse process of differentiation in calculus. It involves finding the original function that was differentiated to obtain a given function.

2. What is the purpose of finding an anti derivative?

The purpose of finding an anti derivative is to solve indefinite integrals and evaluate definite integrals in calculus. It helps in finding the total accumulated change over a given interval.

3. How do you find an anti derivative?

To find an anti derivative, you can use various methods such as the power rule, substitution, integration by parts, and partial fractions. The method used depends on the complexity of the function.

4. What is the difference between an anti derivative and a derivative?

An anti derivative is the reverse process of differentiation, while a derivative is the rate of change of a function at a given point. In other words, an anti derivative gives the original function, while a derivative gives a new function.

5. Can an anti derivative have multiple solutions?

Yes, an anti derivative can have multiple solutions. This is because when differentiating a function, the constant term is lost, leading to multiple possible original functions. These solutions can be differentiated back to the same given function.

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