Adding an instance field size of type int to track the number of elements stored in a singly-linked list optimizes the worst-case runtime complexity (from linear time to constant time algorithm) of which of the following operations defined in ListADT? Select all which apply.

Reference Section: Methods from the LinkedNode class used to build a singly linked list:   LinkedNode(T item, LinkedNode next)   Constructs node combining item data with a reference to another node.   T getData()   Accesses the data reference within this node.   void setData(T item)   Mutates (changes) this node’s data to be item.   LinkedNode getNext()   Accesses the next reference within this node.   void setNext(LinkedNode n)   Mutates (changes) this node’s next reference to be n. Question: Which of the following correctly prints out the data associated with the THIRD element (aka the element of index 2) in a singly linked list of LinkedNode objects, where head refers to the first element of the list? Note: As usual in CS300, this implementation does not use a “dummy node”.

The terms of a series are defined recursively by the equations belowa1=2,  an+1=5n+14n+3 an a_1 =2, \quad a_{n+1} = \frac{5n+1} {4n+3} \:a_n Determine if    ∑n=1∞ an   \textstyle \sum_{n=1}^{\infty} \: a_n  is convergent or divergent.

Find the function represented by the power series : f(x)=∑n=0∞ (-1)n2nxn+2n!f(x) = \sum_{n=0}^{\infty} \, \frac{ (-1)^{n}2^nx^{n+2}}{n!}

Determine if the series converges or diverges. Name the test you use for the following series. a. ∑n=1∞ (-4)n+132n \sum_{n=1}^{\infty} \frac{(-4)^{n+1}}{ 3^{2n} } b. ∑n=0∞cosnπn+1 \sum_{n=0}^{\infty} \cos \left( \frac{n \pi}{ n+1} \right) c. ∑n=1∞ lnnn \sum_{n=1}^{\infty} \: \frac{\ln n}{n}

Which series should be compared with ∑n=1∞ 2sinn+3n3/2 \sum_{n=1}^{\infty} \, \frac{2 \sin n +3}{n^{3/2} }

Find the 15th derivative of f(x) f(x) at x=2 x =2 for f(x)=∑n=0∞ (-1)n(x-2)n5n+1 f(x) = \sum_{n=0}^{\infty} \: \frac{(-1)^n (x-2)^n}{ 5^{n+1}}

Find the Maclaurin polynomial T4(x)T_4 (x) for f(x)=ln(1-x2) f(x) = \ln (1-x^2)

Use the geometric series 11-x=∑n=1∞ xn \frac{1}{1-x} =\sum_{n=1}^{\infty} \: x^n and you should not use the table of known Taylor or Maclaurin series for this problem. you should write the answer as the sigma notationa. (3 points) Write the Maclaurin series representation for the function: f(x)=11+x2 f(x) =\frac{1}{1+x^2} b. (4 points) From the expression you found in part (a), derive the Maclaurin series for h(x)=tan-1x h(x) = \tan^{-1} x

Use the graph and answer the following questions. a. (2 points) Find the area of R 1 R_1 b. (4 points) Find the volume of the solid generated by R 1 R_1 over the x-axis c. (4 points) Find the volume of the solid generated by R 2 R_2 over x = – 1 x =-1 Screenshot 2025-07-18 at 12.35.57 PM.png