Question

Q1:
f is differentiable at c, prove that:
1) f′(c)=limh→0((f(c+h)−f(c)0\h)
2)f′(c)=limh→0(9f(c+h)−f(c−h))\2h)


Q2:
Let f:R→R .The function f is even if f(−x)=f(x) for all x∈R, and odd if f(−x)=−f(x) for all x∈R. if f is differentiable, prove that f′ is odd when f is even, and when f is odd.

Answer 1

Answer on Question #54820 – Math – Real Analysis

Question 1. f is differentiable at c, prove that:

1) $f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$.

2) $f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c - h)}{2h}$.

Solution

1) Since f is differentiable at c, we may write $f(c + h) = f(c) + f'(c)h + \partial(h)$. Thus we obtain

$$\lim_{h \to 0} \frac{f(c + h) - f(c)}{h} = \lim_{h \to 0} \frac{f(c) + f'(c)h + \partial(h) - f(c)}{h} = f'(c) + \lim_{h \to 0} \frac{\partial(h)}{h} = f'(c) + 0 = f'(c)$$

2) Since f is differentiable at c, we may write also $f(c - h) = f(c) + f'(c)(-h) + \partial(h)$. Thus we obtain

$$\lim_{h \to 0} \frac{f(c + h) - f(c - h)}{2h} = \lim_{h \to 0} \frac{f(c) + f'(c)h + \partial(h) - (f(c) - f'(c)h + \partial(h))}{2h} = f'(c) + \lim_{h \to 0} \frac{\partial(h)}{h} = f'(c) + 0 = f'(c)$$

Question 2. Let $f: \mathbb{R} \to \mathbb{R}$. The function f is even if $f(-x) = f(x)$ for all $x \in \mathbb{R}$, and odd if $f(-x) = -f(x)$ for all $x \in \mathbb{R}$ if f is differentiable. Prove that 1) $f'(x)$ is odd when f is even, and 2) $f'(x)$ is even when f is odd.

Solution

1) Let f be even. This means that $f(-x) = f(x)$.

Denote $(-x) = t$ and rewrite

$$f(t) = f(-t).$$

Differentiate the previous equality (1) by $t$:

$$f'(t) = \left(f(-t)\right)' = f'(-t) \cdot (-t)' = -f'(-t).$$

We got

$$f'(t) = -f'(-t), \text{ or}$$ $$f'(-x) = -f'(x).$$

The last expression (2) means that $f'(x)$ is odd by the definition.

2) Let f be odd. This means that $f(-x) = -f(x)$.

Denote $(-x) = t$ and rewrite

$$f(t) = -f(-t).$$

Differentiate the previous equality (3) by $t$:

$$f'(t) = \left(-f(-t)\right)' = -f'(-t) \cdot (-t)' = f'(-t).$$

We got

$$f'(t) = f'(-t), \text{ or}$$ $$f'(-x) = f'(x).$$

The last expression (4) means that $f'(x)$ is even by the definition.

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