5. Suppose you want to model the length L(x) of an adult salamander, measured in cm, as a function of the ambient concentration of oxygen x, measure in mg/L, when it undergoes meta- morphosis. Direct measurement shows that ƒ(8) = 15, ƒ'(8) = 0.5, and ƒ"(8) = −2. Use the second-degree Taylor polynomial centered at 8 to predict the length that would result from a dissolved oxygen concentration of 7.5 mg/L.

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consider the polynomial f (x) = 7x4- (2 + 2x + 6x2 + 5x3) -8x5. In x = 0.7, use the approximation in divided differences centered for the second derivative of F (x). Use a rack of Δx = 0.2 step.   F '' (x) ≈

Suppose that a continuous income stream has an annual rate of flow  f(t) = 500e−0.02t2  (in thousands of dollars per year). Use Simpson's Rule with  n = 4  to approximate the total income (in dollars) from this stream over the next 4 years. (Round your answer to the nearest dollar.)

Please answer all parts.

Which of the following are true for all polynomial functions? (1) If f'(x) = g'(x), then f(x) = g(x) %3D (2) If f(x) = g(x), then f'(x) = g'(x) %3D Select one: O a. Only (2) O b. Neither (1) or (2) O c. Only (1) O d. Both (1) and (2)

Calculate L6 for f(x) = \6x + 1 over [1,3]. (Give your answer to three decimal places.)

5. Find the average rate of growth of the function f(r) = x² + 2x + 3 from x = -5 to r= -3.

3- Determine the Lagrange polynomial that interpolates the data in the following table 2 4 6. y 1 -1 3 4 4- Let f (x) = 2x² e* +1. Construct a Lagrange polynomial of degree two or less using x, = 0, x, = 0.5 and x, = 1. Approximate f (0.8) 5- Determine the polynomial of degree <5, using Newton's divided- differences, that interpolate the data in the following table 1.0 2.0 3.0 4.0 5.0 6.0 f (x) 14.5 19.5 30.5 53.5 94.5 159.5 Use the resulting polynomial to estimate the value of f (4.5). Compare with the exact value, f (4.5) = 71.375. 6- To investigate the relationship between yield of potatoes, y, and level of fertilizer application, x, an experimenter divided a field into 5 pots of equal size and applied differing amounts of fertilizer to each. The data recorded for each plot are given by the table (in Kgs). X 1.0 2.0 3.0 4.0 5.0 y 22 23 25 30 28 - 1 - a) Find an interpolation polynomial for this table. b) According to the interpolating polynomial, approximately how many Kgs would you…