# Thread: 293 AP Calculus Exam a(t) and v(t) t=8

1. OK, this can only be done by observation so since we have v(t) I chose e
but the eq should have a minus sign.

here the WIP version of the AP Calculus Exam PDF as created in Overleaf

the goal is to have 365 problems that align basically where students are first year calculus
always appreciate comments since it needs to be a group effort.  Reply With Quote

2.

3. Originally Posted by karush OK, this can only be done by observation so since we have v(t) I chose e
but the eq should have a minus sign.
choice (e) works as written only if acceleration remains constant over the indicated interval of time

fyi, the avg value of a function of time over the interval of time $[a,b]$ is $\displaystyle \dfrac{1}{b-a} \int_a^b f(t) \, dt$

which of the other 4 choices matches up?  Reply With Quote

by f(t) do you mean v(t)?

assuming c with abs enclosure  Reply With Quote

5. in general, the average value of a function is ...

$\displaystyle \overline{f(x)} = \dfrac{1}{b-a} \int_a^b f(x) \, dx$

in general, if $f$ is any function of time over the time interval $[a,b]$ ...

$\displaystyle \overline{f(t)} = \dfrac{1}{b-a} \int_a^b f(t) \, dt$

so, specifically ...

average acceleration, $\displaystyle \overline{a(t)} = \dfrac{1}{b-a} \int_a^b a(t) \, dt$

average velocity, $\displaystyle \overline{v(t)} = \dfrac{1}{b-a} \int_a^b v(t) \, dt$

average position, $\displaystyle \overline{x(t)} = \dfrac{1}{b-a} \int_a^b x(t) \, dt$

choice (c) is average speed, not velocity.  Reply With Quote

6.  Reply With Quote

pl I think B is the answer due to absence of absolute value.

sorry just noticed I never replied to this.

that was a lot of help .... kinda confusing at first.  Reply With Quote

8. Originally Posted by karush pl I think B is the answer due to absence of absolute value.

sorry just noticed I never replied to this.

that was a lot of help .... kinda confusing at first.
(B) is correct

$\displaystyle \dfrac{1}{8} \int_0^8 v(t) \, dt = \dfrac{x(8)-x(0)}{8-0} = \dfrac{\Delta x}{\Delta t} = \bar{v}$  Reply With Quote

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