In the town of Lakemont (population = 10,000), 200 people we…

In the town of Lakemont (population = 10,000), 200 people were living with chronic hepatitis B at the beginning of 2025.During the year 2025, 50 new cases of hepatitis B were diagnosed. Calculate the incidence rate of hepatitis B for 2025. (You may use the online calculator here in the exam.  Express your answers as percentages or per 1,000 people.)

Question 1 : (4 points)The power series converges on [-2.4][…

Question 1 : (4 points)The power series converges on [-2.4][-2.4] where the coefficient should be 1. Find the center and radius of convergence. Write the power series in standard form with the correct radius of convergence.Question 2:  (4 points) You are given the Maclaurin Series ,  11+x2=∑n=0∞(-1)nx2n \frac{1}{1+x^2}= \sum_{n=0}^{\infty}(-1)^nx^{2n} , |x|0Show that bnb_{n} is decreasing sequence. Find the limit of bn b_n State whether the series converges or diverges. Determine whether the convergence is absolute or conditional.Question 8: (8 points)Find the radius and interval of convergence of the series ∑n=0∞n!(x+1)n7n \sum_{n =0}^{\infty} \frac{n! (x+1)^n}{7^n} Question 9: (8 points)Consider the series f(x)=x2e-2x f(x) = x^2 e^{-2x}  . Using ex=∑n=0∞xnn! e^x = \sum_{n =0}^{\infty} \frac{x^n}{n!} Find the Maclaurin series for f(x)f(x) Compute f(10)(0) f^{(10)} (0) and simplify your answer.Question 10: (6 points)You are given the Maclaurin series for cosx \cos x : sinx=∑n=0∞(-1)n(2n+1)!x2n+1 \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n }{(2n+1)!} x^{2n+1} Differentiate the given series term by term.Identify the function the new series represents.Find the sum of the series. ∑n=0∞(-1)n22n(2n+1)! \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n}}{(2n+1)!} Question 11: (6 points) Find the area under the curve y=cosx y =\cos x from to x=0 x =0 to x=πx=\pi (6 points) Find the volume of the solid generated by rotating the region bounded by y=cosx y =\cos x , x=0 x =0 and y=0y =0 about the x-axis.(8 points) Find the volume of the solid generated by rotating the region bounded by y=cosx y =\cos x , x=0 x =0 and y=0y =0 about the y-axis.

Short answer question: Pretend that you are a self-identifie…

Short answer question: Pretend that you are a self-identified male older adult shopping for OTC medicines to restock in your home for common, minor health issues. You have high blood pressure and an enlarged prostate (benign prostatic hyperplasia, or BPH), and you’ve had several falls recently. Based on your health conditions and fall history, what is one (1) OTC medicine or ingredient you should avoid, and why could it be risky for you? (one-half point for the medicine/ingredient & one-half point for “why”; 1 point total possible)