Find the velocity, v(t), for an object moving along the x-axis in 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

a = cost - sint = dv/dt

so, v = sint + cost + C
now solve v(0) = 3 to find C

so will be answer A, right?

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

Given:
Acceleration, a(t) = cos(t) - sin(t)
Initial velocity, v(0) = 3

To find the velocity function, we integrate the acceleration function with respect to time:

∫a(t) dt = ∫(cos(t) - sin(t)) dt

Integrating cos(t) with respect to t gives:
∫cos(t) dt = sin(t) + C1

Integrating -sin(t) with respect to t gives:
∫-sin(t) dt = cos(t) + C2

Now we can find the constant of integration. Since we are given that v(0) = 3, we substitute t = 0 into the velocity function:

v(0) = sin(0) + cos(0) + C1 + C2 = 0 + 1 + C1 + C2 = 3

Therefore, C1 + C2 = 2 (equation 1)

Now we can write the velocity function by combining the integrals and the constant terms:

v(t) = sin(t) + cos(t) + C1 + C2

Substituting the value of C1 + C2 from equation 1, we get:

v(t) = sin(t) + cos(t) + 2

Therefore, the correct option is:
b) v(t) = sin(t) + cos(t) + 2

NO!

sin(0) + cos(0) + C = 3