Question

AP CALC or CALC 1 help. I give medals

please help me. the problem is attached.

Answer 1

@phi @Loser66

Answer 2

@jim_thompson5910

Answer 3

Q is not full shown in fig

Answer 4

the last part of the questions says " evaluate f(0)="

Answer 5

On the interval \(\color{#000000 }{ x\in [2,4] }\) the graph of \(\color{#000000 }{ \displaystyle f'(x) }\) is simply,

\(\color{#000000 }{ \displaystyle f'(x)=-|x-2|+2 }\)

Answer 6

Oh, I meant to start the interval from -2, not 2. Sorry

Answer 7

oh, okay it makes sense now.

Answer 8

So, you have a piece wise \(\color{#000000 }{ \displaystyle f'(x) }\).

\(\color{#000000}{f'(x)=\begin{cases} -x, ~~~~ x\in [-2,2) \\ x, ~~~~~~~ x\in (-2,4] \end{cases} }\)

(point x=2 is not any important)

Answer 9

Note, that the +C (when you integrate) is the same,
for both parts of the piece wise function.

You are given, that f(4)=6.
So, you can use second part of piece wise to find C.

\(\color{#000000 }{ \displaystyle \int x dx=\frac{x^2}{2} +C}\)

\(\color{#000000 }{ \displaystyle f(x)=\frac{x^2}{2} +C\quad x\in[2,4]}\)

\(\color{#000000 }{ \displaystyle f(4)=\frac{4^2}{2} +C=6\quad \Longrightarrow \quad C=-2 }\)

Answer 10

Now, the f on the interval \(x\in [-2,2)\) is:
\(\color{#000000 }{ \displaystyle f(x)=\int (-x) dx=\frac{-x^2}{2} +C}\)

We know the C from the second part of the piece wise where x=[2,4],
C=-2.

\(\color{#000000 }{ \displaystyle f(x)=\frac{-x^2}{2} -2}\)
(this is the function, for x=[-2,2] ...)

Answer 11

So from there you just need to evaluate f(0).

Answer 12

-2

Answer 13

Yes.

Answer 14

okay I understand now. first you found the function of the derivative, then you integrated the function to convert it to a function. Okay I understand. thank you so much.

Answer 15

there might be a mistake in what I said.

Answer 16

yes, there is a mistake.

Answer 17

The f' indeed is an absolute value function,
f'(x)=-|x-2|-2,

for, x-2>0 ---> x>2 for, x-2>0 ---> x>2
then, f'(x)=-x+4 then, f'(x)=x

therefore,

\(\color{#000000}{f'(x)=\begin{cases} &x,~~~~~~~~~~~~~ x\in [-2,2) \\ & -x+4~~~~~ x\in [2,4] \end{cases} }\)

and you are given that f(4)=6.

Answer 18

my piece wise f' ws wrong initially. Apologize for that mistake.

Answer 19

So, you can integrate the (-x+4) to find f(x) on the interval [2,4].
(add the integration constant +C)

Then, use the given information: f(4)=6, to solve for C.

Next, integrate x to find the f(x) on the interval [-2,2).
Then, plug in the integration constant C that you found previously,
instead of +C. And that is the f(x) on the interval [-2,2).

Lastly, use this f(x) on the interval [-2,2), to find f(0).