(1) True or False. Explain why or why not. Also, for each of (i)-(iv), graph and (if it exists) on one set of axes. (i) The linear approximation to () = ^2 at = 0 is () = 0. (ii) Linear approximation at = 0 provides a good approximation to () = ||. (iii) If () = + , then the linear approximation to at any point is () = (). (iv) When linear approximation is used to estimate the value of () = ln near = the approximations are underestimates of the true value.

(2) Use linear approximation to estimate (5.1) given that (5) = 10, and (5) = 2.

(3) Given a function () = (1 + ), show that () = 1 + is the linear approximation of at 0.

(4) Consider the function () = 2 cos . (i) Find the linear approximation to the function at = (ii) Graph and on the same set of axes. (iii) Based on the graphs of part (ii), state whether linear approximations to near are underestimates or overestimates. (iv) Compute () to confirm your conclusion.

(5) Use linear approximations to estimate the following quantities. Choose a suitable function and a value of that produces a small error. (i) cuberoot of -7.97 (ii) ^0.02

(6) Differentials. Consider the function () = ln(1 ). (i) Express the relationship between a small change in and the corresponding change in in the form = (). (ii) Use your answer in part (i) to approximate the change in when changes from = 1 to = 1.02.