Find the first four distinct Taylor polynomials about
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- a. Find the linear approximating polynomial for the following function centered at the given point a. b. Find the quadratic approximating polynomial for the following function centered at the given point a. c. Use the polynomials obtained in parts a. and b. to approximate the given quantity. f(x) = 8x³/2, a = 1; approximate 8(1.73/2) a. p₁(x) = b. p₂(x) = c. Using the linear approximating polynomial to estimate, 8 (1.73/2) is approximately (Simplify your answer.) Using the quadratic approximating polynomial to estimate, 8(1.73/2) is approximately. (Simplify your answer.)arrow_forwardThe Lagrange polynomial passes through three data points (3, 11), (9, 10), (26, 21). Find p2(x) at x = 5. Round-off your answer to 3 decimal places.arrow_forwardSuppose if we use a linear Lagrange polynomial, differentiate and evaluate at x = x0 what formula will we get?arrow_forward
- In a hospital, a drug is given to a patient. Let f(t) be the amount of drug (in mg) in the patient's body t minutes after it is administered. The nurses want to predict how the body will absorb the drug (the shape of the graph y = f(t)) to be able to prepare the next course of action. One of them remembers about Taylor polynomials from their Calculus course and explains that they allow to predict the shape of a function by making very accurate measurements of the function at one point. They decide to do this and they find the following details about the patient's absorption of the drug: • f(15) = 140 • f'(15) = -10 • f" (15) = 12 1 000 Use these details to approximate the average amount of drug in the patient's body over the first 2 hours: 120 1 f(e) dt 120 Important: Remember that the measurements were made at t = 15 minutes.arrow_forwardFind the Taylor polynomial of degree 3 for the function f(x) = X T3(x) = + (x − 3) + (x − 3)² + (x − 3)³ about x = 3.arrow_forwardFind the Taylor Polynomials T1, T3, T5, ...Tn where n is a polynomial of degree nn. What do you notice about the relationship between these polynomials and f as n increases?arrow_forward
- Let s(t) be the distance of a truck to an intersection. At time t = 0, the truck is 60 m from the intersection, travels away from it with a velocity of 24 m/s, and begins to slow down with an acceleration of a = −3 m/s2. Determine the second Maclaurin polynomial of s, and use it to estimate the truck’s distance from the intersection after 4 s.arrow_forwardD. Pick two of the given functions and determine up to the 5th degree Taylor polynomial: a. g(x) = 2xe* about x = -1 b. h(x) = cos (2x + n) about x = 0arrow_forwarda. Find the linear approximating polynomial for the following function centered at the given point a. b. Find the quadratic approximating polynomial for the following function centered at the given point a. c. Use the polynomials obtained in parts a. and b. to approximate the given quantity. f(x) = x¹/3, a = 8; approximate 7.5¹/3 a. p₁ (x) = b. p₂(x) = 1/3 c. Using the linear approximating polynomial to estimate, 7.5 is approximately. (Type an integer or decimal rounded to five decimal places as needed.) 1/3 Using the quadratic approximating polynomial to estimate, 7.5 is approximately (Type an integer or decimal rounded to five decimal places as needed.)arrow_forward
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