Need help answering the following questions

4 Consider the piecewise function A15 0 S x S a, 1})(x): 11% agxgb, 0 otherwise, where A, a, and b are constants. a) Plot 1110:) as a function of x. Make sure to label A, a, and b on your plot. b) Without doing any calculus, what is the area under the curve for this function? +00 c) Using calculus, compute the definite integral f 1p(x)dx. Make sure your answer agrees oo with what you got above. 5 Suppose I have a collection of squares with the following side-lengths x: 9cm, 9cm, 10cm, 11cm, 13cm, 15cm. a) What is the expectation value of the side length, (x)? b) What is the expected area in this collection of squares? c) Consider (3:)? Do you expect this value to be the same or different as (x2)? Briey justify your answer. (1) Calculate the standard deviation for the side-lengths of the squares above. (If you use Excel to do this, make sure to calculate the population standard deviation, STDEVP.) Now, calculate the quantity (7 E w/ (x2) (at)? What do you notice about the value you obtain for 0'? Consider now a very large collection of squares, whose side lengths follow a Gaussian distribu- 1 xi 2 tion p(x) = U lzne'i(%) with pa =10c1n and a = 2cm. e) Sketch the function P (x), and label .11 on your plot. f) What is the expectation value of the side length, (x)? Clearly explain your work/ reasoning for how you arrived at this answer. Express your final answer numerically and in terms of 34 and / or 0