SOLVE THE HIGHLIGHTED QUEST 1. (a) Use Formula (1) to obtain the local linear approximation 7. tan x N X 8. -NI-x of x at x0 = 1. 1+ x (b) Use Formula (2) to rewrite the approximation obtained 9. JuItx 10. In(I + x) ~ x in part (a) in terms of Ax. 11-16 Confirm that the stated formula is the local linear ap- (c) Use the result obtained in part (a) to approximate proximation of f at x0 = 1, where Ar = x - 1. (1.02), and confirm that the formula obtained in part (b) produces the same result. 11. fur) =x4: (1 + Ax) @ 1 +4Ax 2. (a) Use Formula (I) to obtain the local linear approxima- 12. f(x) = x: VI+ ArNI+ +Ax tion of 1/x at x0 = 2. (b) Use Formula (2) to rewrite the approximation obtained 13. f(x) = 7 2+x 3+ Ar in part (a) in terms of Ar. 14. f(x) = (4 + x): (5+ Ax) = 125 +75Ax (c) Use the result obtained in part (a) to approximate 1/2.05, and confirm that the formula obtained in part 15. tan x; tan (1 + Ax) + + ;Ax (b) produces the same result. FOCUS ON CONCEPTS in ' (=): sin ' (7+ 7 Ax) ~ + 3. (a) Find the local linear approximation of the function 17-20 Confirm that the formula is the local linear approxima- f(x) = VI + x atxo = 0, and use it to approximate tion at xo = 0, and use a graphing utility to estimate an interval 10.9 and VI.I. of x-values on which the error is at most 40.1. (b) Graph f and its tangent line at xp together, and use 17. Vx+ 3 ~ V3+ 18. the graphs to illustrate the relationship between the exact values and the approximations of v0.9 and VI.I. 19. tan 2r ~ 2x 20. I- 10x -2.x)- 4. A student claims that whenever a local linear approxima- 21. (a) Use the local linear approximation of sin x at x0 = 0 tion is used to approximate the square root of a number, the approximation is too large. obtained in Example 2 to approximate sin | ", and com- (a) Write a few sentences that make the student's claim pare the approximation to the result produced directly precise, and justify this claim geometrically. by your calculating device. (b) How would you choose to to approximate sin 44? (b) Verify the student's claim algebraically using ap- (c) Approximate sin 44"; compare the approximation to the proximation (1). result produced directly by your calculating device. 5-10 Confirm that the stated formula is the local linear approx- 22. (a) Use the local linear approximation of tan x at x0 = 0 to imation at x0 = 0. approximate tan 2, and compare the approximation to the result produced directly by your calculating device. 5. (1+x) 1+ 15x 6 . (b) How would you choose to to approximate tan 61"