The period T of a pendulum of length L is given by  where g is the acceleration due to gravity. A pendulum is moved from the Canal Zone, where g = 32.09 feet per second per second, to Greenland, where g = 32.24 feet per second per second. Because of the change in temperature, the length of the pendulum changes from 2.5 feet to 2.48 feet. Approximate the change in the period of the pendulum.

Find the tangent line approximation for near x = 0. What is the tangent line approximation to ex near x = 9? Find the tangent line approximation to 2/x near x = 2. What is the local linearization of near x = 1? The equation has a solution near x = 0. By replacing the left side of the equation by its linearization, find an approximate value for the solution. Writing g for the acceleration due to gravity, the period, T, of a pendulum of length I is given by Show that if the length of the pendulum changes by delta I, the change in the period, delta T, is given by If the length of the pendulum increases by 2%, by what percent does the period change? Suppose f'(x) is a differentiable decreasing function for all x. In each of the following pairs, which number is the larger?

Consider a pendulum with length L (meters) and let the angle theta (radians) be the angle from the vertical to the pendulum. It can be shown that theta as a function of time satisfies the differential equation: d2theta/dt2 + g/L sin theta = 0 where g = 9.8 m/sec/sec is the acceleration due to gravity. For small values of theta we can use the approximation sin(theta) ~ theta, and with that substitution, the differential equation becomes linear. Determine the equation of motion of a pendulum with length 0.5 meters and initial angle 0.4 radians and initial angular velocity d theta/dt 0.4 radians/sec. _____ At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer) _____ What is the maximum angle (in radians) from vertical? _____ How long after reaching its maximum angle until the pendulum reaches maximum deflection in the other direction? (Hint: where is the next critical point?) _____ What is the period of the pendulum, that is the time for one swing back and forth? _____