4.7 Exercises Exercises 1-4 Answer true or false. 1. The linearization of a differentiable function f at c=0 is the same as the first order Maclaurin poly- nomial of f. 2. The graph of the linearization of f at x=c is the tangent line off at (c, f(c)). 3. If f(2) = g(2) and f'(2) g'(2), then it follows that the second order Taylor polynomials of ƒ and g at c 2 are identical. If f(2) = g(2) and f'(2) = g'(2) then it follows that the linearizations of ƒ and g at c = 2 are identical. Exercises 5-10 Find the linearization L(x) of the given function at the indicated value. ⑤. (2) 5. (r) = 3-2r2 +5 at c = 2 ⑥f(x) = 1-2 at c=0 f(x) = In(x + 1) at c = 0 ⑧f(x)= cos z at c=-1 9. f(x)=(x-2)e at c=0 10. f(x) = r sin(r) at c= Exercises 11-14 Find the linearization L(x) at c for the given function f and use it to complete the table. 11. f(x) √√, c=4 f(4.1) L(4.1) 12. f(x) =, c-2 f(-2.5) L(-2.5) = 13. f(x) tanr, c = Absolute error Absolute error f() L() Absolute error 14. f(x) e2, c=0 f() L() Absolute error Exercises 15-16 Find the linearization of the function f at c and use a graphing utility to find the maximum absolute error over the given intervals. 15 f(x) = x²; c=1 a) [0,2] b) (0.9, 1.1] c) (0.99, 1.01] 16. f(x)=√√r; c=8 a) [7,9] b) [7.9,8.1] Exercises 21-26 Use an appropriately chosen linearization to estimate the given value. 21. 122 23. 102 22. 26 24. 0.1 26. In(0.95) 1 2 3 45. f(x)= 25. tan(0.24) Exercises 27-32 Find the nth-order Taylor polynomial centered at c for the function f. 27. f(x) = x³-3x²+x+2, c = 1, n = 3 28. f(x)=√√, c=4, n = 3 n P 0 1 2 30 46. f(x) = x n Pr 29. f(x)=re, c = 0, n = 2 30. (r) = Inx, c = 1, n = 4 31. f(x) = cos x, c = 1, n = 4 32. f(x) = tanx, c = 1, n = 2 Exercises 33-36 Find the nth-order Maclaurin polynomial for the function f. 33 f(x)=sinx, n = 3 (34. f(x) = e², n = 4 35. f(x)= 1 1-2 , n = 4 36. f(x) In(x+1), n = 4 37. f(x)=arctanr, n = 3 38. f(x)=re, n = 3 Exercises 39-40 Find the 4th-order Maclaurin polynomial and use it to guess the form of the 5th- and 6th-order Maclaurin polynomials. 0 1 2 3 Applications 47. Electric Di the electric charge q is charge q is -q is place- the strength a point P lo 39. f(x)=cos r 40. f(x)=-