Differential Calculus Flashcards
What is the derivative of a function?
The derivative of a function represents the rate of change of the function with respect to its variable.
True or False: The derivative of a constant is zero.
True
Fill in the blank: The notation for the derivative of a function f(x) is ______.
f’(x) or df/dx
What is the power rule for differentiation?
If f(x) = x^n, then f’(x) = n*x^(n-1).
What is the product rule for differentiation?
If u(x) and v(x) are functions, then (uv)’ = u’v + uv’.
What is the quotient rule for differentiation?
If u(x) and v(x) are functions, then (u/v)’ = (u’v - uv’)/v^2.
What does the second derivative of a function indicate?
The second derivative indicates the concavity of the function and can be used to identify inflection points.
What is the chain rule?
The chain rule states that if a function y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).
True or False: The derivative of sin(x) is cos(x).
True
What is the derivative of e^x?
The derivative of e^x is e^x.
What is a critical point?
A critical point occurs where the derivative is zero or undefined.
What is the purpose of finding the derivative in optimization problems?
Finding the derivative helps to identify maximum and minimum values of functions.
Fill in the blank: The derivative of ln(x) is ______.
1/x
What is an inflection point?
An inflection point is where the second derivative changes sign, indicating a change in concavity.
Multiple choice: What is the derivative of x^3? A) 3x^2 B) x^2 C) 3x
A) 3x^2
What is meant by ‘differentiability’?
Differentiability means that a function has a derivative at a particular point.
True or False: A function can be continuous at a point but not differentiable there.
True
What is the relationship between the derivative and the slope of a tangent line?
The derivative at a point gives the slope of the tangent line to the function at that point.
What is the implicit differentiation?
Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation.
Fill in the blank: The derivative of cos(x) is ______.
-sin(x)
What does the Mean Value Theorem state?
The Mean Value Theorem states that if a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f’(c) = (f(b) - f(a)) / (b - a).
What is a tangent line?
A tangent line is a straight line that touches a curve at a single point without crossing it.
Multiple choice: What is the derivative of tan(x)? A) sec^2(x) B) sin(x)/cos(x) C) cos(x)
A) sec^2(x)
What is the first derivative test used for?
The first derivative test is used to determine whether a critical point is a local maximum, minimum, or neither.
Fill in the blank: The derivative of a polynomial function is a ______.
polynomial function
