AMA3007 Assignment 1 Solved

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Exercise 5.2.3. (a) Use Definition 5.2.1 to produce the proper formula for the derivative of h(x) = 1/x.

• Combine the result in part (a) with the Chain Rule (Theorem2.5) to supply a proof for part (iv) of Theorem 5.2.4.
• Supply a direct proof of Theorem2.4 (iv) by algebraically manipulat­ing the difference quotient for (fig) in a style similar to the proof of Theorem 5.2.4 (iii).

 

Exercise 5.3.1. Recall from Exercise 4.4.9 that a function Lipschitz on A if there exists an M > 0 such that

f (s) — 1(Y)
x — y

 

for all x y in A.

• Show that if f is differentiable on a closed interval [a, b] and if f’ is con­tinuous on [a, b], then f is Lipschitz on [a, b].
• Review the definition of a contractive function in Exercise3.11. If we add the assumption that Ifⁱ(x)1 < 1 on [a, bb does it follow that f is contractive on this set?

Exercise 5.3.2. Let f be differentiable on an interval A. If f(x)        0 on A,

show that f is one-to-one on A. Provide an example to show that the converse statement need not be true.

Exercise 6.2.2. (a) Define a sequence of functions on R by

1 if = 1, _D ^fn(x) = 0 otherwise

and let f be the pointwise limit of f_n.

Is each f,, continuous at zero? Does f_r, —> f uniformly on R? Is f continuous at zero?

Exercise 6.3.1. Consider the sequence of functions defined by

_Xⁿ

gm(s) = _n

• Show (y_n) converges uniformly on [0,1] and find g = lim g_ri. Show that g is differentiable and compute g'(x) for all x E [0, 1].
• Now, show that (4) converges on [0,1]. Is the convergence uniform? Set h = lirng’_n and compare h and g’. Are they the same?

 

Exercise 6.4.5. (a) Prove that h(x) =                                                 x 4-s n2 71=1 is continuous on [-1, 1]. (b) The series

2 X 3                     _A X + 4 + 9 + 16 +

cc

X²X³ X⁴

f(x) =           .x+ _ __  4 _±

n                                                                            2       3

converges for every x in the half-open interval [-1, 1) but does not converge when x = 1. For a fixed xo E (-1, 1), explain how we can still use the Weierstrass M-Test to prove that f is continuous at xo.

Exercise 6.6.5. (a) Generate the Taylor coefficients for the exponential func­tion f (x) = ex, and then prove that the corresponding Taylor series con­verges uniformly to ex on any interval of the form [-R,

• Verify the formula f (x) = ex.
• Use a substitution to generate the series for e^–x, and then informally calculate ex e^–x by multiplying together the two series and collecting common powers of x.

Exercise 6.6.6. Review the proof that g'(0) = 0 for the function

g (x) =

e-1/x2         for x      0, 0                  for x = O.

 

introduced at the end of this section.

• Compute g’ (x) for x Then use the definition of the derivative to find g”(0).
• Compute g” (x) and g^m(x) for x L Use these observations and in­vent whatever notation is needed to give a general description for the nth derivative g(n) (x) at points different from zero.
• Construct a general argument for why g(n)(0) = 0 for all n E