(3.2) Find the second order Taylor formula for the following functions at x0 = 1) f(x, y) = tan(2x + 3y) 2) f(x, y) = ex-y² (0,0
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- YOUR TURN Suppose x are y are both functions of t and x3+2xy+y2=1. If x=1, y=2, and dxdt=6, then find dydt.Assume x and y are functions of t. Evaluate dydtfor each of the following. 2xy5x+3y3=51; dxdt=6,x=3,y=2Q2. Find y' (a) by applying the Product Rule (b) by multiplying the factors to produce a sum of simpler terms to differentiate. y = (x² + 1)(x + 5 st.
- 3. (Straightforward) Solve the separable ODEs: (a) dy/dx = y/x². (b) dx + sec x dy = 0. (c) x dy/dx %3Dу?. y². dP (d) — Р— Р3. - dtEvaluate xy dx + (x+ y)dy along the curve y = x² from (-3,9) to (- 1,1). xy dx + (x + y)dy = (Type an integer or a simplified fraction.)Approximate f' (1) using the given data f(0) = 1, f(1) = 2, f(2) = 1, and f(3) = 10 and the Lagrange interpolation polynomial. -1 O -2 O 14 O 6 O 17
- 80. The function f has derivatives of all orders for all real numbers with f(2) = -1, f'(2) = 4, ƒ"(2) = 6, and f(2) = 12. Using the third-degree Taylor polynomial for f about x = 2, what is the approximation of f(2.1)? (A) -0.570 (B) -0.568 (C) -0.566 (D) -0.5289. Let f be a function that has derivatives of all orders on the interval (-1,1). Assume f(0) =1, (¹) (x) ≤ 6 for all x in the interval (−1,1). ƒ'(0) = *(0) = -1, " (0)=, and (a) Find the third-degree Taylor polynomial about x = 0 for the function f.