Consider the following for loops in R. For each for loop, list the values (in order) that the variable i takes on in the body of the loop. Briefly (in no more than a few sentences) explain why. a) for(i in 1+2:3.4*5) { } b) for(i in dim(matrix(0, nr = 7, nc = 8))) { } c) for(i in rnorm(3)) { } d) for(i in iris[1:3,3]) { } e) for(j in c(1, 2, 3, 4, 5)) { } f) for(i in (function(x) x*x)(c(1, 2, 3))) { } g) for(i in NULL) { } h) for(i in strsplit(as.character(4*atan(1)),’’) [[1]][1:10]) { }

The following data are given: x= [1,3,5,7]; y=[ln (1), In (3), ln (5), ln (7)] 3. a) Use the interpld function from scipy.interpolate library to interpolate the above data using the spline interpolation method. b) Estimate the logarithm of 2 for linear, quadratic and cubic splines. Print out the true value, interpolating values and the true relative errors in % for linear, quadratic and cubic splines. c) Show on a graph the true function In(x), interpolating linear spline, interpolating quadratic spline, interpolating cubic spline, and data points. Round numbers to four significant digits.

es remaining 8. Consider the function f:NxN-N defined recursively by: 1) Base case: Let meN and define (0,m) = 0 2) Recursive case: For any x,meN, x>0, define f(x,m) = (x-1,m) + (m+m) Prove the following theorem holds using proof by induction: Thereom: For any n,meN, m>0 we have (n.m) I m = n+n Fill in your answer here 9 Help BIU X, x L - ɔE =N E X Format

Let 2= (0,1). Compute the language of the regular expression. ((01 +10) (1(11)")

d) for (k0; k=0; t = t-1) printf("B"); for-t: TC1iter) = Change of var: Σ/repetitions, Closed form: for-k: TC1iter) =. Change of var: Σ/repetitions Closed form: dependent / independent of loop variable dependent / independent of loop variable e(

If A={8,12,16,20} and B={12,16,20,24,28}, what is the value of P (A and B)?

Describe an efficient way of putting a VECTOR rep­resenting a deck of n cards into random order in C++. You may use the function randomlnt(n), which returns a random number between 0 and n-1, inclusive. Your method should guarantee that every possible ordering is equally likely. What is the run­ning time of your function?