# Help on 2 previous AP Calc AB test questions

Sympathy
Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at t=0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 <(or = to) t <(or = to) 80 seconds, as shown in the table below.

t (seconds) 0 10 20 30 40 50 60 70 80
v(t) ft/sec 5 14 22 29 35 40 44 47 49

a) Find the average acceleration of rocket A over the time interval 0 <(or = to) t <(or = to) 80 . Indicate units of measure.
(I did this one :D)

b) using correct units, explain the meaning of "the integral of 10 to 70 of v(t)dt" in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate "the integral of 10 to 70 of v(t)dt".

I honestly have no idea how part 2 of this works, and if you could, explain pt 1 of this question also.

c) Rocket B is launched upward with an acceleration of a(t) = 3/sq root(t + 1) feet per second per second. At time t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 ft / second. Which of the 2 rockets is traveling faster at time t = 80 seconds? explain your answer.

yeah clueless on part c as well, help would be much appreciated =)

## Answers and Replies

Sympathy
and second problem:

The twice differentiable function f is defined for all real numbers and stisfied the following conditions:

f(0) = 2 f'(0) = -4 f''(0) = 3

a) the function g is given by g(x) = e^(ax) + f(x) for all real numbers, where a is a constant. Find g'(0) and g''(0) in terms of a. Show the work that leads to your answers.

b) The function h is given by h(x) = cos(kx)f(x) for all real numbers, where k is a constant. Find h'(x) and write an equation for the line tangent to the graph of h at x = 0.

Homework Helper
Ahh guys, Sympathy has already personally asked me about this thread on Msn Messenger...we managed to do a and most of b before sympathy went offline...
so I think its just c and 2nd post left...

Staff Emeritus
Sympathy, welcome to PF. Firstly, in future please post these sorts of questions in the dedicated homework forums. Please note that before we can help you, you must show your working. Gib Z seems to think that you have done the first question, but for the second one, please post your thoughts on the question.

Homework Helper
Ahh Sympathy came back online, fine with all the posted questions but needs this one, I can't do it:

$$\frac{dy}{dx}=\frac{1+y}{x}$$.

Graph that with slope fields.

Sympathy
I got it, its not very hard, thanks guys!

Homework Helper
O I think I got it too :D

x/dx = 1+y, integrate both sides.
ln x= ln y+1, or

y=x-1 :D

jimjimmonk
hey guys, can anyone here show how to work that first problem that was posted?
"Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at t=0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 <(or = to) t <(or = to) 80 seconds, as shown in the table below.

t (seconds) 0 10 20 30 40 50 60 70 80
v(t) ft/sec 5 14 22 29 35 40 44 47 49

a) Find the average acceleration of rocket A over the time interval 0 <(or = to) t <(or = to) 80 . Indicate units of measure.

b) using correct units, explain the meaning of "the integral of 10 to 70 of v(t)dt" in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate "the integral of 10 to 70 of v(t)dt".

c) Rocket B is launched upward with an acceleration of a(t) = 3/sq root(t + 1) feet per second per second. At time t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 ft / second. Which of the 2 rockets is traveling faster at time t = 80 seconds? explain your answer."

we got this ORQ in class a few days ago and had the whole period to work on it but we were only able to figure out part a.
((ending velocity)-(initial velocity))/(time passed)=((49)-(5))/(80)=(44)/(80)=11/20

Part B completely confuses me so that would be a huge help if someone could help me with that.

And I think I understand how to do part C, just didn't want to skip b. But you would just integrate a(t)=3/sq root(t + 1) to change it to velocity and plug in t=80 and find out which one has a higher velocity at 80 seconds correct?