Quiz 7 Fall 2005 | 2005 - 2007 | Www.ilearnmath.net

AP Calculus – Shubleka Hebron Academy – F all „05 Quiz Seven NAME_____________________ *** PLEASE DO NOT WRITE ANY ANSWERS ON THIS SHEET*** Read the questions carefully. Present your answers neatly and justify each step in your solutions. Show all your work for full credit. NO CALCULATORS ARE ALLOWED. Definition: A function f(x) is differentiable at a if the derivative of f(x) exists at a. A function is said to be differentiable on an open interval (a, b) if it is differentiable everywhere on that interval. Problem 1 Determine where the function f ( x) | x | is differentiable. Problem 2 Prove or disprove with a counterexample: i) If f is differentiable at a, then f is continuous at a. ii) If f is continuous at a, then f is differentiable at a.

f ( x)  f ( a ) f (a  h)  f (a) or f ' (a)  lim . h  0 x a h Hint 2: f is continuous at a if and only lim f ( x)  f (a) . Hint 1: f ' (a)  lim

x a

x a

Problem 3 In each case, find the derivative of the function at a using the definition of derivative. Is each function differentiable everywhere in its domain? Generalize. f ( x)  5 x 2  3x  2

f ( x)  5 x 2  3x  4 What about the general form f ( x)  ax2  bx  c ? f ( x)  x 4 f ( x)  x  x f ( x)  2x 1 f ( x)  5x  What about the general form f ( x)  mx  b where m,b are constant real numbers?

“Proof is the idol before whom the pure mathematician tortures himself.”

Eddington, Sir Arthur (1882-1944) In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.