# Let #F(x)=int_0^x e^(-5t^4)dt#. Find the MacLaurin polynomial of degree 5 for F(x)?

Applying the Calculus Fundamental Theorem:

By definition:

Including:

And:

By evaluating an entire series of derivatives and computing as the Maclaurin Series, you can obtain the same result:

However, I wouldn't advise it.

By signing up, you agree to our Terms of Service and Privacy Policy

The MacLaurin polynomial of degree 5 for ( F(x) = \int_0^x e^{-5t^4} dt ) is:

[ P_5(x) = \sum_{n=0}^{5} \frac{F^{(n)}(0)}{n!}x^n ]

To find the coefficients, ( \frac{F^{(n)}(0)}{n!} ), we need to find the derivatives of ( F(x) ) up to the fifth order evaluated at ( x = 0 ). Then, substitute these values into the polynomial formula.

First, let's find the derivatives of ( F(x) ): [ F'(x) = e^{-5x^4} ] [ F''(x) = -20x^3 e^{-5x^4} ] [ F'''(x) = -60x^2(1-20x^8)e^{-5x^4} ] [ F^{(4)}(x) = -240x(1-20x^8)e^{-5x^4} ] [ F^{(5)}(x) = -240(1-20x^8-160x^4)e^{-5x^4} ]

Now, evaluate these derivatives at ( x = 0 ): [ F(0) = 0 ] [ F'(0) = 1 ] [ F''(0) = 0 ] [ F'''(0) = 0 ] [ F^{(4)}(0) = 0 ] [ F^{(5)}(0) = -240 ]

Substitute these values into the formula for ( P_5(x) ): [ P_5(x) = 0 + \frac{1}{1!}x^1 + 0 + 0 + 0 - \frac{240}{5!}x^5 ] [ P_5(x) = x - \frac{4}{3}x^5 ]

Thus, the MacLaurin polynomial of degree 5 for ( F(x) ) is ( P_5(x) = x - \frac{4}{3}x^5 ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the power series of #f(x)= ln(5-x)^2#? What is its radius of convergence?
- How do you find the power series representation for the function #f(x)=e^(x^2)# ?
- How do you find the radius of convergence #Sigma x^(3n)/5^n# from #n=[1,oo)#?
- What is the interval of convergence of #sum_1^oo xsin((pi*n)/2)/n #?
- How do you find the Maclaurin Series for # f(x)= x sinx#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7