Suppose we have a counter that stores an arbitrary number of bits and counts in binary. It always begin at 0. The only mutator operation it can perform is to increment, adding one to the current count. This changes one or more bits. Show that if we start at 0 and perform k increment operations, a total of O(k) bits will change. Hint 1: If you are stuck, count to 20 in binary on paper. After each increment, write how many bits have changed. See if you notice a pattern.
Hint 2: Whenever you add one, give it some credit, similar to the analysis of expansion in lecture 7.

In a heat transfer course, we can derive the equation for the temperature distribution in a flat rectangular plate. in this example, we will look at a plate at steadystate with three sides being held at t1, and one side held at t2. the temperature for any location on the plate, t(x, y), can be calculated by where create a function (prob3_5) that will take inputs of vectors x and y in feet, scalar n, scalars l and w in feet and scalars t1 and t2 in degrees fahrenheit. it will output a matrix t which is the temperature of each x and y locations. t will have the number of columns equal to the number of elements in x and rows equal to the number of elements in y. though this can be done without loops (perhaps more efficiently), your program must use a nested loop.