Difference quotients, by hand

Medium0 of 6 parts

Each of these is a difference quotient. Simplify the top until every term carries an $h$ , cancel the $h$ in the denominator, and then answer the question asked. No calculator will help with any of them, and no exam will let you use one.

\frac{f(a+h) - f(a)}{h}

The quantity being simplified in every part below.

Where the answer is in terms of $a$ , write it using the letter $a$ . Where it is in terms of both $a$ and $h$ , use both.

Simplify $\dfrac{(a+h)^2 - a^2}{h}$ for $h \neq 0$ . Give the answer in terms of $a$ and $h$ .

in terms of a, h

Now take it to the limit: $\lim_{h \to 0} \dfrac{(a+h)^2 - a^2}{h}$ , in terms of $a$ .

in terms of a

$f(x) = 3x^2 - 5x$ . Simplify $\dfrac{f(a+h) - f(a)}{h}$ , in terms of $a$ and $h$ .

in terms of a, h

$f(x) = \dfrac{1}{x}$ . Find $\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ , in terms of $a$ .

in terms of a

$f(x) = \sqrt{x}$ . Find $\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ , for $a > 0$ , in terms of $a$ .

in terms of a

$f(x) = 7$ for every $x$ . Find $\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ .