AP Calculus BC Flashcards: Defining Average and Instantaneous Rates of Change at a Point
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Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What condition must be met for the derivative, $f'(a)$, to exist?
For the derivative $f'(a)$ to exist, the limit of the difference quotient (e.g., $\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$) must exist.
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What condition must be met for the derivative, $f'(a)$, to exist?
For the derivative $f'(a)$ to exist, the limit of the difference quotient (e.g., $\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$) must exist.
What do the expressions $\frac{f(a+h)-f(a)}{h}$ and $\frac{f(x)-f(a)}{x-a}$ represent?
These difference quotients express the average rate of change of a function over an interval.
What is the relationship between the average rate of change and the instantaneous rate of change?
The instantaneous rate of change at a point is the limit of the average rate of change over an interval as the interval's length approaches zero.
Which expression represents the average rate of change of a function $f$ on the interval from $a$ to $a+h$?
The expression $\frac{f(a+h)-f(a)}{h}$ represents the average rate of change of $f$ over the interval from $a$ to $a+h$.
Define instantaneous rate of change at a point $x=a$.
The instantaneous rate of change at $x=a$ is the limit of the average rate of change as the interval around $a$ shrinks to zero, and it is equivalent to the derivative $f'(a)$.
What is a difference quotient used to determine?
A difference quotient is used to determine the average rate of change of a function over an interval.
Write the two equivalent limit forms for the instantaneous rate of change of a function $f$ at $x=a$.
The instantaneous rate of change can be expressed as $\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$ or $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$, provided the limit exists.
How would you set up the limit to find the derivative of a function $f$ at the point $x=a$ using the form that involves $x$ approaching $a$?
You would set up the limit as $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$.
How is the derivative of a function represented using a limit?
The derivative of a function is represented as the limit of a difference quotient.
What is the notation for the derivative of a function $f$ at the point $x=a$?
The notation for the derivative at $x=a$, which represents the instantaneous rate of change, is $f'(a)$.