Problem 1 (10pt). Compute f(x² sin x + y²)dx + (2xy + y³e³)dy where C is the unit circle with counterclockwise orientation.

Please give me two scenarios of two different families who are seeking a financial advise to invest thier money and prepare their children's college education. i must use logarithms to calculate interest . the two families must have two different quotes of  :  1)salaries, savings , anything else that will help in saving for their kids future college tuition fees.  2) we need a currebt interest compund annually on different invsetments 3) logarithmic function that expresses the investment that im advising them to do 4) a graph that displays for each family of the interest compound annually and the amount of time it will take to reach their goal. thanks so much.

Consider the differential equation y'(t)- 2y(t) = 11, y(0) = 3. a. Find a power series for the solution of the differential equation. b. Identify the function represented by the power series. a. Choose the correct answer below. OA. y(t)=3-17t+ 17t 34 (-1)"17.2"-1 3 th + n! OB. y(t)=3+17t+17t 3 34 3 17-27-1 +.. n! OC. y(t) 11-17t+171 = 171² - 3413+ (-1) 17.2"-1 ++ -t t + OD. y(t) = 11+17t+ 17t + b. y(t) = ☐ n! 34 17.2"-1 3 + -t + n!

K Consider the function f(x) = 4 cos x sx. a. Differentiate the Taylor series centered at 0 for f(x). b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. a. Choose the correct answer. 23 35 11 24x 24x OA. -24x 3! 51 17 29 5 24x 24x OC. -24x 21 4! 41 24x 61 b. The differentiated series converges to ☐ (Type an expression using x as the variable.) c. The interval of convergence of the power series for the derivative is (Type your answer in interval notation.) OB G1 EBC 03 TAB 04 CAPS 右邊 CTAL SHIFT 1 This test: 11 point(s) possible This question: 1 point(s) possible 17 29 41 24x OB. -24x5 24x 24x + +... 21 4! 6! OD. - 24x11+ 24x23 31 24x 35 +... 51 ENTER B

Find a power series representation centered at 0 for the following function using known power series. Give the interval of convergence for the resulting series. 4 10x4 f(x) = (1+x5) 2 Which of the following is the power series representation for f(x)? OA. (-1)*2k 5k-1 k=1 00 5k-1 ○ c. Σ (− 1)k+12k•x5k O k=1 The interval of convergence is (Simplify your answer. Type your answer in interval notation.)

Use the Ratio Test or the Root Test to determine if the following series converges absolutely or diverges. 00 Σ (-17)* k=1 k! Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer in simplified form.) OA. The series converges absolutely by the Ratio Test because r = O B. The series diverges by the Root Test because p= OC. Both tests are inconclusive because r = and p=