Problem 1: Let f XR be differentiable at the point a = int(X). Prove that ƒ(a + h) − f(a − h) lim = f'(a). h→0 2h Is there a function that satisfies the above limit but is not differentiable at a? Problem 2: Let f : I → R satisfy |f(y) − f(x)| ≤c|xy|a with a > 1, cЄ R, and x, y Є I arbitrary. Prove that f is constant. Problem 3: Let I be an interval centered at 0. A function f : I→ R is called even when f(x) = f(x) and odd when f(x) = f(x), for all x € I. If ƒ is even, its even-order derivatives (when they exist) are even functions, and its odd-order derivatives are odd functions. In particular, these last ones vanish at point 0. State the analogous result for odd f. Problem 4: Let f [a, b] R be a function continuously differentiable up to order n. Show that if f has n + 1 distinct roots in [a, b], then f(n) has at least one root in [a, b]. Problem 5: Let f, g: IR be twice differentiable at the point a = int(I). If ƒ (a) = g(a), f'(a) = g'(a), and f(x)g(x) for all x € I, prove that f" (a) > g" (a).

Problem 1: Let f XR be differentiable at the point a = int(X). Prove that ƒ(a + h) − f(a − h) lim = f'(a). h→0 2h Is there a function that satisfies the above limit but is not differentiable at a? Problem 2: Let f : I → R satisfy |f(y) − f(x)| ≤c|xy|a with a > 1, cЄ R, and x, y Є I arbitrary. Prove that f is constant. Problem 3: Let I be an interval centered at 0. A function f : I→ R is called even when f(x) = f(x) and odd when f(x) = f(x), for all x € I. If ƒ is even, its even-order derivatives (when they exist) are even functions, and its odd-order derivatives are odd functions. In particular, these last ones vanish at point 0. State the analogous result for odd f. Problem 4: Let f [a, b] R be a function continuously differentiable up to order n. Show that if f has n + 1 distinct roots in [a, b], then f(n) has at least one root in [a, b]. Problem 5: Let f, g: IR be twice differentiable at the point a = int(I). If ƒ (a) = g(a), f'(a) = g'(a), and f(x)g(x) for all x € I, prove that f" (a) > g" (a).

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter4: Calculating The Derivative

Section4.3: The Chain Rule

Problem 52E: Margy and Nate are working on taking the derivative of fx=23x+14. Margy uses the quotient rule and...

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