Calculus Basics Flashcards
What is the derivative of a constant?
0
What is the power rule for differentiation?
If f(x) = x^n, then f’(x) = n*x^(n-1)
True or False: The derivative of sin(x) is cos(x).
True
What is the integral of x^n with respect to x?
(x^(n+1))/(n+1) + C, where n ≠ -1
Fill in the blank: The limit of a function as x approaches a point must exist for the function to be ________ at that point.
continuous
What is the derivative of e^x?
e^x
What is the integral of cos(x) with respect to x?
sin(x) + C
What does it mean for a function to be continuous?
A function is continuous if it is defined at a point, the limit exists, and the limit equals the function’s value at that point.
True or False: The sum rule of differentiation states that the derivative of a sum is the sum of the derivatives.
True
What is the chain rule in differentiation?
If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x)
What is the integral of 1/x with respect to x?
ln|x| + C
What is the derivative of tan(x)?
sec^2(x)
Fill in the blank: The Fundamental Theorem of Calculus connects ________ and ________.
differentiation, integration
What is the product rule in differentiation?
If u and v are functions, then (uv)’ = u’v + uv’.
True or False: The derivative of ln(x) is 1/x.
True
What is the integral of sin(x) with respect to x?
-cos(x) + C
What is the quotient rule in differentiation?
If u and v are functions, then (u/v)’ = (u’v - uv’)/v^2.
What does the mean value theorem state?
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 the second derivative test used for?
To determine the concavity of a function and to identify local maxima and minima.
What is the integral of sec(x)tan(x) with respect to x?
sec(x) + C
True or False: The limit of a function can be infinite.
True
What is the derivative of x^3?
3x^2
Fill in the blank: The derivative represents the ________ of a function at a point.
slope
What is the integral of e^(ax) with respect to x?
(1/a)e^(ax) + C
What is the derivative of a sum of functions?
The derivative of the sum is the sum of the derivatives.
What is the integral of a constant k with respect to x?
kx + C
True or False: The derivative of a product of two functions requires the use of the product rule.
True
What is the limit definition of the derivative?
f’(x) = lim(h -> 0) [(f(x + h) - f(x))/h]
What does the term ‘critical point’ refer to?
A critical point is where the derivative is zero or undefined.
What is the integral of (1 + x^2) with respect to x?
x + (1/3)x^3 + C
What is the derivative of sin(x^2)?
2x cos(x^2)
Fill in the blank: A function is ________ if it has a derivative at every point in its domain.
differentiable
What is the integral of tan(x) with respect to x?
-ln|cos(x)| + C
True or False: The limit of a function at a point can be found using substitution.
True
What is the derivative of cos(x)?
-sin(x)
What is the integral of a constant multiplied by a function?
k * ∫f(x)dx = k * F(x) + C
What is the difference between definite and indefinite integrals?
Definite integrals have limits and yield a numerical value, while indefinite integrals do not have limits and yield a function.
What is the derivative of ln(x^2)?
2/x
What is the integral of (ax + b)^n with respect to x?
(1/(a(n+1)))(ax + b)^(n+1) + C
True or False: The second derivative of a function gives information about its concavity.
True
What is the derivative of the inverse function f^-1(x)?
If y = f(x), then dy/dx = 1/(f’(f^-1(y)))
What is the integral of sin^2(x) with respect to x?
(1/2)(x - sin(x)cos(x)) + C
Fill in the blank: A function is ________ at a point if it can be drawn without lifting the pencil.
continuous
What is the derivative of the exponential function a^x?
a^x ln(a)
What is the integral of cos^2(x) with respect to x?
(1/2)(x + sin(x)cos(x)) + C
Derivatives of arctan, arcsin, arccos.
(a/a^2+x^2) for arctan, 1/root(a^2-x^2) for arcsin, -1/root(a^2-x^2) for arccos
Derivatives of arctan, arcsin, arccos.
(a/a^2+x^2) for arctan, 1/root(a^2-x^2) for arcsin, -1/root(a^2-x^2) for arccos
