Homework Answers

{F} test the series : infinity sigma n=1 (-1) ^n-1 sin nx/n√n for absolute and conditional convergence

{F} The equation for a displacement 𝑠(𝑚), at a time 𝑡(𝑠) by an object starting at a displacement of 𝑠0 (𝑚), with an initial velocity 𝑢(𝑚𝑠 −1 ) and uniform acceleration 𝑎(𝑚𝑠 −2 ) is: 𝑠 = 𝑠0 + 𝑢𝑡 + 1 2 𝑎𝑡 2 A projectile is launched from a cliff with 𝑠0 = 30 𝑚, 𝑢 = 55 𝑚𝑠 −1 and 𝑎 = −10 𝑚𝑠 −2 . The tasks are to: a) Plot a graph of distance (𝑠) vs time (𝑡) for the first 10s of motion. b) Determine the gradient of the graph at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. c) Differentiate the equation to find the functions for: i) Velocity (𝑣 = 𝑑𝑠 𝑑𝑡) ii) Acceleration (𝑎 = 𝑑𝑣 𝑑𝑡 = 𝑑 2 𝑠 𝑑𝑡2 ) d) Use your results from part c to calculate the velocity at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. e) Compare your results for part b) and part d). f) Find the turning point of the equation for the displacement 𝑠 and using the second derivative verify whether it is a maximum, minimum or point of inflection. g) Compare your results from f) with the graph you produced in a).

Let F(x,y)=2x^2+3sin y and let G(x,y)=e^x-2y.Compute the Jacobian determinant d(F ,G)/d(x,y)