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If $f$ is a differentiable function and $a$ is in the domain of $f$, then $f'(a)$ is given by which of the following expressions?
• 1. $\lim _{h \to 0} \dfrac {f(a+h)-f(a)}{h}$
• 2. $\lim _{x \to a} \dfrac {f(x)-f(a)}{x-a}$
• 3. $\lim _{h \to 0} \dfrac {f(x+h)-f(x)}{x-h}$
I only II only I and II only I and III only I, II, and III
The expression $\lim _{h \to 0} \dfrac {2(x+h)^7-5(x+h)+8-(2x^7-5x+8)}{h}$ is the derivative of what function?
$f(x) = 2(x+h)^7-5(x+h)+8$ $f(x) = 2(x+h)^7-5(x+h)+8-(2x^7-5x+8)$ $f(x) = 14x^7-5$ $f(x) = 2x^7-5x+8$ $f(x) = \dfrac {2(x+h)^7-5(x+h)+8-(2x^7-5x+8)}{h}$
Which of the following statements represent the meaning of $f'(4.2) = 1.7$?
• 1. The slope of the line tangent to the graph of $y = f(x)$ at $x = 4.2$ is $1.7$.
• 2. As $x$ increases by $1$ from $x = 4.2$, $f(x)$ increases by $1.7$.
• 3. The limiting value of the average rate of change of $f$ over the interval $[4.2, 4.2+\Delta x]$ as $\Delta x$ approaches zero is $1.7$.
I only II only III only I and III only I, II, and III