# 1.4.293 AP Calculus Exam a(t) and v(t) t=8

• MHB
• karush
In summary, the conversation discusses the goal of having 365 problems in an AP Calculus Exam PDF, with a focus on aligning with first year calculus topics. The conversation also covers the concept of average value of a function over an interval and its relevance to acceleration, velocity, and position. The correct answer to a specific problem is determined to be (B) due to the absence of absolute value.
karush
Gold Member
MHB
View attachment 9508
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.

Last edited:
karush said:
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?

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

assuming c with abs enclosure

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.

Last edited by a moderator:
AP Calculus resource ...

https://apcentral.collegeboard.org/pdf/ap-curricmodcalculusmotion.pdf?course=ap-calculus-bc

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.

karush said:
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

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

## 1. What is the significance of the "t=8" in "1.4.293 AP Calculus Exam a(t) and v(t) t=8"?

The "t=8" represents a specific point in time, in this case, the 8th unit of time. It is used to denote the value of acceleration and velocity at that particular time.

## 2. How are acceleration (a) and velocity (v) related in the context of this problem?

Acceleration and velocity are related by the derivative and integral functions in calculus. The derivative of velocity is acceleration, and the integral of acceleration is velocity.

## 3. Can you explain the difference between instantaneous acceleration and average acceleration?

Instantaneous acceleration refers to the acceleration at a specific moment in time, while average acceleration refers to the overall change in velocity over a period of time. Instantaneous acceleration can be calculated by taking the derivative of velocity, while average acceleration can be calculated by taking the change in velocity over the change in time.

## 4. How can the information provided in "1.4.293 AP Calculus Exam a(t) and v(t) t=8" be applied in real-world scenarios?

The information provided in this problem can be applied in various real-world scenarios, such as calculating the acceleration and velocity of a moving object at a specific time. This can be useful in fields such as physics, engineering, and transportation.

## 5. Is there a formula or equation that can be used to calculate acceleration and velocity at a specific time?

Yes, the formula for calculating acceleration is a= dv/dt, where a is acceleration, v is velocity, and t is time. Similarly, the formula for calculating velocity is v= ∫a(t)dt, where v is velocity, a is acceleration, and t is time.

Replies
1
Views
1K
Replies
1
Views
1K
Replies
6
Views
1K
Replies
4
Views
2K
Replies
1
Views
2K
Replies
7
Views
2K
Replies
22
Views
3K
Replies
4
Views
7K
Replies
8
Views
4K
Replies
4
Views
3K