Please answer them with details and explaination! Thank you so much! 1. Given the polynomial function f(2)=1+2x+3x²+42²³² + 5x¹.a. Find the Taylor polynomial of degree 3 approximating f(z) for near 0.b. Find the Taylor polynomial of degree 3 approximating f(z) for z near 1.c. Are the Taylor polynomials obtained in parts (a) and (b) the same? Explain.2. The water consumption (in millions of liters per year) in the country A is given by r-f(t), where tin years and t-0 is the start of 2015.isa. Write a definite integral representing the total amount of water consumed between the start of2017 and the start of 2022.b. Since the start of 2016 (t-1), the rate was modeled by r-Using a left Riemann sum withfive subdivisions, find an approximate value for the definite integral in part (a).e. Compute the exact value of the definite integral in part (a) using a substitution.d. Find the error of the approximation in part (b). Do you think it was a good approximation?3. An avid runner starts from home at t-0, and runs back and forth along a straight east-west road.The velocity of the runner, v(t) (given in km/hour) is a function of time t (given in hours). The graphof the runner's velocity is given by e(t)- 10sin(t) with t counted in radians.a. How far is the runner from home afterhours?b. What is the total running distance afterhours?e. What is the farthest distance the runner can be away from home? Explain.d. If the runner keeps running, how many times will the runner pass by home? Explain.4. The graph of the function f(z) is given below.fuxa. Draw the anti-derivative F(2) of f(1) such that F(0) — 0b. Draw the anti-derivative G(z) of f(z) such that G(0) - 1.c. Find F(2), F(5) and G(5).d. For what values of z, the function F(z) is concave up?5. An oil tank contains 100 liters of oil an initial time t-Q. Someone starts pumping oil into the tankat a constant rate of 10 liters per second. Simultaneously, oil starts flowing out through the outlet inthe base at the rate ² (in liters per second) at time t.a. Find the differential equation that would describe the volume of the oil in the tank at time t (inseconds).b. Find the volume of the oil in the tank at timet (in seconds).c. Will the tank ever be empty? Explain.9 10

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