MHB 1.8.4 AP Calculus Exam Integral of piece-wise function

karush
Gold Member
MHB
Messages
3,240
Reaction score
5
View attachment 9418
image due to macros in Overleaf

ok I think (a) could just be done by observation by just adding up obvious areas

but (b) and (c) are a litte ?

sorry had to post this before the lab closes
 

Attachments

  • Capture.PNG
    Capture.PNG
    13.7 KB · Views: 153
Physics news on Phys.org
Is this the intro to the problem you're trying to cite ?

rvfGV0-1024x461.png
 

Attachments

  • rvfGV0-1024x461.png
    rvfGV0-1024x461.png
    41 KB · Views: 117
Last edited by a moderator:
skeeter said:
Is this the intro to the problem you're trying to cite ?

Yeah I have seen it in several AP books but never did it.
 
karush said:
image due to macros in Overleaf

ok I think (a) could just be done by observation by just adding up obvious areas

but (b) and (c) are a litte ?

sorry had to post this before the lab closes

correction ... $\displaystyle g(x) = \int_1^x f(t) \,dt$

(a) $\displaystyle g(2) = \int_1^2 f(t) \,dt = \dfrac{1}{4}$

$\displaystyle g(-2) = \int_1^{-2} f(t) \,dt = -\int_{-2}^1 f(t) \,dt = \dfrac{\pi - 3}{2}$

(b) from the FTC, $g'(x) = f(x) \implies g''(x) = f'(x)$

$g'(3) = f(3) = -1$, $g''(3) = f'(3) = -\dfrac{1}{2}$

(c) horizontal tangents to $g \implies g' = f = 0$

$f = 0$ at $x = \pm 1$ ... relative max at $x=-1$ because $g'$ changes sign from (+) to (-)

neither a max or min at $x=1$ because $g' = f$ does not change sign.

(d) inflection points occur where $g''=f'=0$ or is undefined and $g''=f'$ changes sign.

$g'' = f'$ equals 0 and is undefined at $x = \pm 1$ ... inflection point at $x=1$

$g'' = f'$ is undefined at $x=-2$ ... inflection point there, also because $g''=f'$ changes sign.
 
mahalo:cool:

That was a great help...

I did look at some other free responses, but it was messy...:confused:
 
Back
Top