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.
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)
a) To find the value of t in the open interval (0,2) for which the tangent line at (t, f(t)) is parallel to the line through (0,0) and (2, 4pi), we need to find the derivative of f(t) and then solve the equation for the slope of the tangent line.
1. Take the derivative of f(t) with respect to t:
f'(t) = 2*pi + 2*pi*cos(2*pi*t)
2. The tangent line at (t, f(t)) will have the same slope as the line through (0,0) and (2, 4pi). Let's find the slope of that line:
Slope of the line = (4pi - 0)/(2 - 0) = 2pi
3. Set the slope of the tangent line equal to the slope of the given line:
2*pi + 2*pi*cos(2*pi*t) = 2pi
4. Solve the equation for t:
2*pi*cos(2*pi*t) = 0
cos(2*pi*t) = 0
5. To find the values of t, we need to find where cos(2*pi*t) equals zero. The cosine function equals zero at integer multiples of pi/2.
For t = 1/4, 3/4, 5/4, 7/4, etc., the equation is satisfied.
So, the values of t in the open interval (0,2) for which the tangent line at (t, f(t)) is parallel to the line through (0,0) and (2, 4pi) are t = 1/4 and t = 3/4.
b) To find the average velocity of the particle over the interval 0 <= t <= 2, we need to calculate the displacement of the particle and divide it by the time taken.
1. The displacement of the particle is the change in its position from t = 0 to t = 2, which is given by f(2) - f(0):
Displacement = f(2) - f(0) = (2*pi*2 + sin(2*pi*2)) - (2*pi*0 + sin(2*pi*0))
2. Simplify the expression:
Displacement = 4*pi + sin(4*pi) - 0 - sin(0)
Displacement = 4*pi + sin(4*pi)
3. The time taken is 2 - 0 = 2.
4. Calculate the average velocity:
Average Velocity = Displacement / Time Taken
= (4*pi + sin(4*pi)) / 2
c) To determine the velocity and acceleration of the particle at t = 1, we need to find the derivative of f(t) and evaluate it at t = 1.
1. The derivative of f(t) is given by f'(t) = 2*pi + 2*pi*cos(2*pi*t).
2. Substitute t = 1 into f'(t):
f'(1) = 2*pi + 2*pi*cos(2*pi*1)
3. Simplify the expression:
f'(1) = 2*pi + 2*pi*cos(2*pi)
f'(1) = 2*pi + 2*pi*cos(0)
f'(1) = 2*pi + 2*pi*1
f'(1) = 4*pi
So, the velocity of the particle at t = 1 is 4*pi, and the acceleration is equal to the derivative of the velocity, which is constant and equals zero.