A Flashcards
What is the derivative of a constant function?
The derivative of a constant function is zero.
True or False: The limit of a function as x approaches a value can be different from the function’s value at that point.
True.
Fill in the blank: The Fundamental Theorem of Calculus links __________ and __________.
differentiation, integration.
What is the definition of a limit?
A limit is the value that a function approaches as the input approaches a certain point.
What is the power rule for differentiation?
If f(x) = x^n, then f’(x) = n*x^(n-1).
True or False: The second derivative provides information about the concavity of a function.
True.
What is the integral of x with respect to x?
The integral of x with respect to x is (1/2)x^2 + C.
Multiple Choice: What is the derivative of sin(x)? A) cos(x) B) -sin(x) C) tan(x) D) sec^2(x)
A) cos(x).
What does it mean for a function to be continuous at a point?
A function is continuous at a point if the limit as x approaches that point equals the function’s value at that point.
Fill in the blank: The area under a curve can be found using __________.
integration.
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 e^x is e^x.
True.
Multiple Choice: Which of the following represents the limit definition of a derivative? A) f(a+h) - f(a) B) lim(h->0) [f(a+h) - f(a)]/h C) ∫ f(x) dx D) f’(x) = f(x+1) - f(x)
B) lim(h->0) [f(a+h) - f(a)]/h.
What is an asymptote?
An asymptote is a line that a graph approaches but never touches.
Fill in the blank: The derivative of ln(x) is __________.
1/x.
What is the difference between definite and indefinite integrals?
Definite integrals have upper and lower limits and yield a numerical value, while indefinite integrals represent a family of functions and include a constant of integration.
True or False: A local maximum is a point where a function’s value is higher than all nearby points.
True.
What is the Product Rule for differentiation?
If u and v are functions of x, then (uv)’ = u’v + uv’.
Multiple Choice: What is the derivative of tan(x)? A) sec^2(x) B) cos(x) C) sin(x) D) -csc^2(x)
A) sec^2(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).
Fill in the blank: The limit of (1/x) as x approaches infinity is __________.
0.
What is a critical point?
A critical point is a point on a graph where the derivative is zero or undefined.
True or False: The integral of a function represents the accumulation of its values over an interval.
True.
What is the difference quotient?
The difference quotient is defined as [f(x+h) - f(x)] / h and is used to find the derivative.
Multiple Choice: Which of the following is a method for finding limits? A) Substitution B) Factorization C) Rationalization D) All of the above
D) All of the above.
