 # 1. (Going Downhill). Chapter 3.3 problem 8. 2. (A Differential Squ...

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1. (Going Downhill). Chapter 3.3 problem 8. 2. (A Differential Squeeze). Let f(x) < g(x) < h(x) for x € (a,b). Suppose f(x) and h(I) are differentiable at x = c € (a,b) and f(c) = h(c). a) Prove f'(c) = h'(c). (Hint: consider D(x) = h(I) - f(x). What kind of point is c for the function D(x)? Use Fermat's theorem.) b) Prove that g(x) is differentiable at x = c and that f'(c) = o' (c) = h'(c). (Hint: This uses the Squeeze Theorem. Use f(x) < g(x) < h(x), subtract f(c) < g(c) < h(c) and then divide through by x - c. There will be two cases corresponding to the one-sided limits of the derivative.) 3. (If a function is differentiable then is its derivative continuous?). Do problem 13 on page 105. (For part c you needn't complete problem 2.2.12, just use the result.) 4. (A Tangential Error). On page 114 do problem 13. 5. (A Second order Extension of MVT). Let f(x) have first and second order derivatives on [a,b]. (So f'(x)andf"(x) all exist). Consider the function = - - a) Verify that H(a) = 0 and compute the value of k that makes H(b) = 0. Then verify that you can use Rolle's Theorem to find a c € (a,6) so that H'(c) = 0. b) Now verify that you can apply Rolle's to H'(x) on [a,c]. Use it to find a C1 E (a,c) so that H"(c1) = 0. c) Use this to show that k = (C1). Conclude that for C1 E (a,b) we have 6. (Cartheodory differentiability). The standard definition of the derivative is lim f(x)-f(a) - x a The mathematician Cartheodory developed an equivalent definition with no reference to a limit! How could this be? What happened to the limit? Definition: A function is differentiable at x = a iff there is a function o(I) which is continuous at x = a so that f(x) - f(a)=(x-a)-o(x). f(x) - f(a) a) Prove that if f(x) - f(a) = (x - a) d(I) with (()) continuous at a then lim exists. I a What is the value of that limit? b) Now prove the converse, i.e. that if lim f(x) - f(a) = f'(a) then we can find a o(I) so that x->a x a f(x) (Hint: the formula f(x) - f(a) = (x - a) o(x) determines what (() must be at all points except x = a. What value must it have at x = a in order to be continuous?) c) Cartheodory's alternative definition provides a slick way to prove the chain rule. The function f differentiable at y = g(a) means f(y) - f(g(a)) = b(y) . (y - g(a)) and g differentiable at x = a means that g(x) - g(a) = b(x) . (x - a). Put these two expressions together to write f(g(x)) - f(g(a)) = C(x)(2 - a) where C(x) is an expression composed of o and 6. d) Verify that C(x) is continuous at x = a and find its value in terms of fl and g'.

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