Is anyone RELATIVELY BRILLIANT at DIFFERENTIAL CALCULUS???

Question

Here's the question, hopefully a math genius will come along or at least point me in the right direction:

--A particle moves on the x-axis so its velocity v at time t>0 is given by v(t) = 1 - tan^-1(e^t). At t=0, the particle is at x = -1. Tan^-1x = arctanx.--

a) Find the acceleration at t = 2.
b) Is the speed of the particle increasing or decreasing at t=2? Why?
c) Find the time when the particle reaches its highest point.
d) Find the position of the particle at t=2. Is it moving toward or away from the origin then? Why?


--Ok.... maybe? Please help if you can! I've been working on it for a while and I just can't get it!-

Answer 1

You have to differentiate the equation, which yields:

1/(tan-1(e^x)*log(e))

Now you can plug in your acceleration points.

Good luck.

Mysstere

Answer 2

for a) you need to take the derivative of the equation and plug in 2 for the t values. For b) the result you get for the accelaration will tell you whether the speed of the particle is decreasing or increasing. If the answer is negative, then it is decreasing. If it is positive, then increasing. For c) once you find the equation for accelaration, set it equal to zero and from there just solve the equation for t. For d) you need to integrate v(t) and once you integrate you will get the equation with +c in the end, so what you need to do is to plug in 0 and since you know that at t= 0, x=-1, then you will find out C. I hope this helps.