Question

let f(t) = 2*pi*t + sin(2*pi*t)
a) find value of t in open interval (0,2) for which the tangent line at (t, f(t)) is parallel to the line through (0,0( and (2, 4pi)

b) suppose the given funtion describes the position of a particle on the x-axis for 0<= t <= 2. What is the average velocity of the particle over that interval?

c) determine the velocity and the acceleration of the particle at t=1.

Answer 1

f(t) = 2πt + sin(2πt)
f'(t) = 2π + 2π cos(2πt)

slope of line = (4π-0)/(2-0) = 2π

then 2π + 2π cos(2πt) = 2π
cos (2πt) = 0
2πt = π/2 or 2πt = 3π/2 or 2πt = 5π/2 or 7π.2
t = π/4 or t = 3/4 or t = 5/4 or t = 7/4

b) find f'(0) and f'(2) and take the average

c) take second derivative to find acceleration
evaluate f'(1) and f''(1)