AP Calculus BC Flashcards: Defining the Derivative of a Function and Using Derivative Notation
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Review key ideas with interactive flashcards. This set includes 11 cards to help you master important concepts.
List three common notations for the derivative of a function y = f(x).
Three common notations for the derivative are dy/dx, f'(x), and y'.
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List three common notations for the derivative of a function y = f(x).
Three common notations for the derivative are dy/dx, f'(x), and y'.
Define the notation dy/dx.
The notation dy/dx, also known as Leibniz notation, represents the derivative of the function y with respect to the variable x.
Explain the relationship between the value of f'(a) and the graph of f(x) at x=a.
The value of f'(a) is the slope of the line that is tangent to the graph of f(x) at the point where x=a.
What is the formal limit definition of the derivative of a function f(x)?
The derivative of f is the function whose value at x is defined as f'(x) = lim(h→0) [f(x+h) - f(x)] / h, provided this limit exists.
Using limit notation, how would you set up the problem to find the derivative of the function g(t)?
You would set up the limit of the difference quotient as lim(h→0) [g(t+h) - g(t)] / h.
What is the name of the expression [f(x+h) - f(x)] / h used in the definition of a derivative?
This expression is called the difference quotient.
What does the derivative of a function at a point represent graphically?
The derivative of a function at a point represents the slope of the line tangent to the graph of the function at that specific point.
What are the four ways a derivative can be represented?
A derivative can be represented graphically, numerically, analytically, and verbally.
How is the derivative of a function formally represented in terms of a limit?
The derivative of a function is formally represented as the limit of a difference quotient.
To determine the equation of a line tangent to a curve at a given point, what key value must be found using the derivative?
To find the equation of a tangent line, you must use the derivative to find the slope of the line at the given point.
If a problem asks for the 'slope of the curve' at a point, what is it asking you to find?
It is asking you to find the value of the derivative of the function at that point, which is equivalent to the slope of the tangent line.