AP Calculus AB : How to find position

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #1 : How To Find Position

Given the velocity function v(t)=t^3-t+7, find the position at time x=2 with initial position x(0)=2.

Possible Answers:

Correct answer:

Explanation:

First, we find the position function x(t)=\int (t^3-t+7) dt=\frac{t^4}{4}-\frac{t^2}{2}+7t+constant

x(0)=2\Rightarrow constant=2\Rightarrow x(t)=\frac{t^4}{4}-\frac{t^2}{2}+7t+2

Therefore, x(2)=\frac{2^4}{4}-\frac{2^2}{2}+7*2+2=18.

Example Question #2 : How To Find Position

Given the acceleration function is a(t)=2t^2-t+3, intial velocity is 2 m/s, and the displacement at  is .

Find its initial position.

Possible Answers:

Correct answer:

Explanation:

First, we find the velocity function by integrating the acceleration functionv(t)=\int 2t^2-t+3 = \frac{2}{3}t^3-\frac{1}{2}t^2+3t+c

Since the initial velocity is 2 m/s, we have .

Then we integrate again to get the displacement function.

x(t)=\int \frac{2}{3}t^3-\frac{1}{2}t^2+3t+2=\frac{1}{6}t^4-\frac{1}{6}t^3+\frac{3}{2}t^2+2t+k

x(1)=1.5+2+k=3.5+k=4

This implies

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