Question

7. (5 points) Find the linear approximation for f(x) = tan(2x) at a = 0 and use it to approximate the value of tan(0.002). Hint: The linear approximation is just the tangent line to the curve at a = 2. 8. (5 points) Use the Mean Value Theorem for derivatives to find the value of x = c for f(x) = Vx on the interval (1,9). 9. (5 points) The acceleration of an object moving along the number line at time t is given by a(t) = 2t +2. Given that the initial velocity of the object is v(0) = 5 and the initial position of the object is x(0) -- = -4, use integration to find the velocity function v(t) = S a(t)dt and the position function x(t)- S v(t)dt. Use the initial conditions to solve for any arbitrary constants.

Answer 1

fint94 (20) at a= f'im) - 2 See'lon) trol-0, trole 2011=2 By linear apponimation of fino at azo is C(n): f(x) + b) 49 fin - 0+ ( 1-0). (2) finlaan Put x=0.000 fro.oup) = 2 (0.001) =tan(0.002) = 0.002 (82 fini-54 on [119] clearly this continuous an E193 and differentiable on (119). it is so, by mean value theorem their enista such that numbercin (199) f'(c) f(b)-fa) ba - Frea fca1-fly 9-1 - - 3 - 6 + 1 =) 256-4 =) 5c =2 --Hel119) Henee c=4

VH-2t2 (9) a ( +22, VO)=5: Mo--4 we have V(+1= Saltid VH1 = S(2+ +2) at VC이는 어어C szc VC41524 +5 also attie = Sucts ar 째 HEJ (+²+2+45) at uit = 북 라다 +st td acol= o tototd 4d uu t +5t -4