Find the velocity, v(t), for an object moving along the x-axis if the acceleration, a(t), is a(t) = cos(t) − sin(t) and v(0) = 3.

a) v(t) = sin(t) + cos(t) +3
b) v(t) = sin(t) + cos(t) +2
c) v(t) = sin(t) - cos(t) +3
d) v(t) = sin(t) - cos(t) +4

To find the velocity function, v(t), we need to integrate the given acceleration function, a(t), with respect to time (t).

Given: a(t) = cos(t) - sin(t)

To find v(t), we integrate a(t) with respect to t:

∫ a(t) dt = ∫ (cos(t) - sin(t)) dt

Using the power rule of integration, we integrate each term separately:

∫ cos(t) dt = sin(t)
∫ -sin(t) dt = cos(t)

Now, putting it all together:

v(t) = sin(t) - cos(t) + C

where C is the constant of integration.

But we are given v(0) = 3. We can use this initial condition to determine the value of C.

When t = 0, v(0) = sin(0) - cos(0) + C = 3

Since sin(0) = 0 and cos(0) = 1, we have:

0 - 1 + C = 3

Simplifying the equation, we find:

C = 4

Therefore, the velocity function v(t) is:

v(t) = sin(t) - cos(t) + 4

So, the correct option is:

d) v(t) = sin(t) - cos(t) + 4