The function f(x) = x3 + 5x – 4 has a single zero near x0 = 1. Use the backward difference method to estimate f’(1) with a step size of 0.01  and use this value in an approximate version of the Newton-Raphson method to derive one improvement on x0. What is the root x1 of the function after one improvement?

In 8 seconds, a bug travels along the line y=2. The position of the bug along y=2 is x(t)=1/3t^3 - 7/2 t^2 + 10t -2. At what times does the spider turn around? Over what intervals does the spider slow down?

A group of researchers found that people prefer YouTube videos that aren't too long, but not too short either. Suppose these researchers determined that the ratings people gave to a video could be approximated byR(t)=20t/t^2 + 100 where t is the length of the video in minutes and R(t) is its rating. What length of video (in minutes) will get the highest rating?

For what value of m is the value x=-1 a zero of the function f(x)=2x^3+mx^2−2x−1. Please show work!