Question

If F'(x)==" \sqrt{1+x^{3} } " align="absmiddle" class="latex-formula"> and F(1)=5, then F(3)=
(A)1.230
(B)3.585
(C)6.230
(D)8.535
(E)11.230

Answer 1

Answer:

E

Step-by-step explanation:

Since the given expression is an elliptic integral, it's not easy to find F(x) explicitly. However, if we use a linear approximation, we can try to estimate F(3), though it won't be too close, since 3 is not very close to 1.

At the point (1,5) the slope of the tangent line is F'(1) = $\sqrt{2}$

So, using that line,

F(3) = F(1) + F'(1)(3-1) = 5+2$\sqrt{2}$ = 7.828

Well, that didn't work out too well.

So, let's pull out our handy integral calculator, and we find that

$\int\limits^0_1 {\sqrt{1+x^3} } \, dx = 1.111$

But, we know that F(1) = 5. So we need to add C = 3.889

Now, integrating again,

$\int\limits^0_3 \sqrt{1+x^3} \, dx = 7.341 + 3.889 = 11.229$

Oops. I messed up the limits, but you get the idea.