Derivatives
What is the derivative of a constant function?
Zero
What is the integral of a constant function?
Constant value multiplied by x
What is the limit of the function f(x) = 3x + 2 as x approaches 2?
8
What does the derivative of a function represent geometrically?
The slope of the tangent line to the curve at a given point
What does the definite integral of a function represent geometrically?
The area between the curve and the x-axis over a given interval
What is the derivative of the function f(x) = x^3 + 2x^2 - 4x + 1?
f'(x) = 3x^2 + 4x - 4
What is the integral of the function f(x) = 3x^2 + 2x - 4?
F(x) = x^3 + x^2 - 4x + C
What is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2?
4
What is the equation for finding the slope of the tangent line to a curve at a given point?
y - f(a) = f'(a)(x - a)
What is the area under the curve of a non-negative function?
The definite integral of the non-negative function over the interval
What is the derivative of the natural logarithm function, ln(x)?
1/x
What is the integral of the function f(x) = sin(x) + cos(x)?
F(x) = -cos(x) + sin(x) + C
What is the limit of the function f(x) = sin(x) / x as x approaches 0?
What is the first derivative test used for?
Determining the intervals where a function is increasing or decreasing
What is the equation for finding the average value of a function over a closed interval?
(1/b - a) * ∫[a, b] f(x) dx
What is the derivative of the function f(x) = 4e^x - 3cos(x)?
f'(x) = 4e^x + 3sin(x)
What is the integral of the function f(x) = 5/x?
F(x) = 5ln|x| + C
What is the limit of the function f(x) = (1 - cos(x)) / x as x approaches 0?
0
What is the second derivative test used for?
Determining the concavity (concave up or concave down) of a function
What is the fundamental theorem of calculus?
The relationship between integration and differentiation, stating that integration "undoes" differentiation
What is the derivative of the function f(x) = √(3x^2 + 2x + 1)?
f'(x) = (3x + 1) / √(3x^2 + 2x + 1)
What is the integral of the function f(x) = e^x / (1 + e^x)?
F(x) = ln|1 + e^x| + C
What is the limit of the function f(x) = (e^x - 1) / x as x approaches 0?
1
What are the critical points of a function?
The points where the derivative is equal to zero or is undefined
What are the methods for finding the volume of a solid of revolution?
Disk/washer and shell methods