Question

Find the antiderivative for a function f(x) = 6x3÷4 +6
that satisfies condition F(4) = 63
Evaluate F(x) at point x=2​

Answer 1

Answer:

$F(2)=-39$

Step-by-step explanation:

We are given: $f(x)=\frac{6}{4}x^3+6$

First, simplify:

$f(x)=\frac{3}{2}x^3+6$

Then, find the anti-derivative (integrate). Thus...

$F(x)=\int (f(x)) dx= \int \frac{3}{2}x^3dx+6 dx$

Simplify:

$\frac{3}{2} \int x^3dx+6\int 1dx$

Use Power Rule.

Simplify:

$\frac{3}{2} (\frac{1}{4}x^4)+6x+C$

$F(x)= \frac{3}{8}x^4 +6x+C$

Now, determine C.

$F(4)=63=\frac{3}{8}(4)^4 +6(4)+C$

$C=-57$

Thus, we have:

$F(x)= \frac{3}{8}x^4 +6x-57$

Now, plug in 2.

$F(2)=\frac{3}{8}(2)^4 +6(2)-57$

$F(2)=-39$