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DBSCAN with a single eps merges two nearby tight clusters into one blob whenever eps is set large enough to also capture a separate, diffuse cluster elsewhere in the data. Which algorithm is specifically designed to fix this failure mode?

DBSCAN is run with min_samples = 5. A point lies within eps of 6 other points (7 total, including itself). This point should be classified as:

Evaluate the limit using algebraic or trigonometric methods when applicable. If the limit is infinite, write  or .  If the limit does not exist and is not infinite, write DNE. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper. Find   using algebraic and trigonometric methods.

Determine whether Mean Value Theorem can be applied to on the closed interval [a, b] by answering the questions below.  If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) that satisfy the conclusion of the Mean Value Theorem. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper. Make sure to label your answers.   a.) Continuous on [-5, 5]? Why or why not? b.) Differentiable on (-5, 5)? Why or why not? c.) Can we apply the Mean Value Theorem? d.) If so, find the values of c in [-5,5] for which .

Find the limit. If the limit is infinite, write  or .  If the limit does not exist and is not infinite, write DNE. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper. This is worth up to 3 bonus points.

Use the graph of the function given below.  If the limit is infinite, write  or .  If the limit does not exist and is not infinite, write DNE. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper.                                                                                          What type of discontinuity is at x = 2?                    

Use the graph of the function given below.  If the limit is infinite, write or .  If the limit does not exist and is not infinite, write DNE. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper.                                                                                          Find .            

Use the graph of the function given below.  If the limit is infinite, write  or .  If the limit does not exist and is not infinite, write DNE. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper.                                                                                          Find .      

Find the limit. If the limit is infinite, write ∞ or −∞.  If the limit does not exist and is not infinite, write DNE. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper.

True or false: ChatGPT or other AI writing programs are considered a form of academic dishonesty. Use of these tools on any assignment will result in automatic failure of the assignment and may be reported to the Honor Code Office.