HW 14 Due
P 154 #2 Checkout #1,3
P159 #1 Checkout #3
The following is a repeat:
Checkout 113 #7 on the approximation of the second derivative f”’
f(x0 +h) – 2 f(x0) + f(x0 – h)]/h2 and proceed as follows
Use Thm 4.24 to show
f(x0+h) = f(x0) + f ’(x0)h + f ”(z)h2 for some z in (x0, x0 + h)
Theorem 4.24 on p. 111 is the correct tool for #7, but it is about a function I will call g satisfying
g(x0) = 0 = g’(x0) when n = 2.
Of course g”(x) = f ”(x) for all x.
To get such a function, consider g(x0+h) = f(x0+h) - f(x0) – f ’(x0)h.
Then f(x0+h) - f(x0) – f ’(x0)h = f ”(z)h2 /2 for some z in (x0, x0 + h) and
f(x0-h) - f(x0) + f ’(x0)h = f ”(w)h2 /2 for some w in (x0 -h, x0).
Adding the above two gives a left hand side equal to
f(x0 +h) – 2 f(x0) + f(x0 – h)
Divide both sides by h2 and take limits as h -> 0.
Another approach suggested by Neza Vodopivec (and possibly others) is to use L’Hopital’s rule,
taking the second derivative of numerator and of the denominator at x0.
HW15 Due Friday March 17 Finish reading chapter 6.
This assignment refers to
Intermediate Value Theorem p 62.
Rolle’s Theorem p 103
Mean Value Theorem p 103
Cauchy Mean Value Theorem p 111
Theorem 4.24 p 111
The Lagrange Remainder Theorem p 203
Assume the functions and all derivatives of all orders are continuous.
Prove (and explain which intermediate value(s) are appropriate):
1a Rolle’s Thm. assuming df/dx is constant.
2a Mean Value Thm. assuming df/dx is constant.
3a Cauchy Mean Value Thm. assuming the ratio df/dx / dg/dx is constant.
4a Thm. 4.24 assuming the n-th derivative is constant.
5a The Lagrange Remainder Theorem assuming the n+1-st derivative is constant in the case n=1.
You can use the result that if the derivative of F is a xn, then F(x) = a xn+1 + constant, where a is a constant.
Assume the function and all derivatives of all orders are continuous. Use the intermediate value theorem (applied to appropriate derivatives) to prove
1b Rolle’s Thm.
2b Mean Value Thm.
3b Cauchy Mean Value Thm.
4b Thm. 4.24
5b The Lagrange Remainder Theorem
Do not use earlier b parts of the problem to prove later ones, like 1b to prove 2b.
You may use the fundamental theorems of calculus.
You can also assume that if f(a) = g(a) and df/dx ≥ dg/dx
then f(x) ≥ g(x) for x ≥ a.
Prof Fitzpatrick will lecture on Chapter 5 and possibly Chapter 7 on Monday March 27 and on Monday April 3 & Wed April 5.
The test next week will cover chapters 1-6 with the same 2 part format as test #1
Read Chapter 5
On pp 123-4 do # 1, 8, and checkout 2, 12. For #2, I would use L’Hopital.
In Chapter 6, 172-3 do # 1. For parts a and d, I suggest you use the definition of derivative.
Checkout # 5,6,7.
In 6, I assume there is no simple formula for the integral, so you should show dF/dx > 0 and F -> infinity as x -> infinity.
Checkout: Let H be the function in #3, p. 173. Compute the derivative of H. Take a function like f(x) = constant which you can integrate to make sure your derivative of H is plausible.