t re Flashcards
What is the derivative of a function?
The derivative of a function represents the rate of change of the function with respect to a variable.
True or False: The derivative can be interpreted as the slope of the tangent line to the graph of the function.
True
Fill in the blank: The derivative of f(x) = x^2 is ______.
2x
What rule is used to find the derivative of the product of two functions?
The Product Rule
If f(x) = x^3, what is f’(2)?
12
What is the quotient rule used for?
To find the derivative of the quotient of two functions.
True or False: The second derivative indicates whether a function is concave up or concave down.
True
What does a positive second derivative indicate?
The function is concave up.
What is the chain rule used for?
To differentiate composite functions.
If g(x) = sin(x), what is g’(x)?
cos(x)
What does it mean if a derivative is zero?
The function has a critical point at that location.
True or False: A function can have more than one local maximum.
True
Fill in the blank: The derivative of f(x) = e^x is ______.
e^x
What is implicit differentiation?
A technique to find the derivative of a function defined implicitly.
If h(x) = ln(x), what is h’(x)?
1/x
What is the purpose of finding critical points?
To identify local maxima, local minima, and points of inflection.
True or False: The Mean Value Theorem guarantees at least one point where the derivative equals the average rate of change.
True
What does the Fundamental Theorem of Calculus relate?
It relates differentiation and integration.
Fill in the blank: The integral of f’(x) gives you ______.
f(x) + C
How do you find the maximum and minimum values of a function on a closed interval?
Evaluate the function at critical points and endpoints.
What is the purpose of the first derivative test?
To determine whether a critical point is a local maximum or minimum.
True or False: A function can be increasing and decreasing at the same time.
False
What is the relationship between a function’s concavity and its second derivative?
If the second derivative is positive, the function is concave up; if negative, concave down.
What does a horizontal tangent line indicate about a function’s derivative?
The derivative is zero at that point.
