AP Calculus - Unit 3

Front

What is the definition of a derivative in calculus?

Back

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. It represents the slope of the tangent line to the function's graph at that point.

Front

How do you find the derivative of a function using the quotient rule?

Back

If you have a function in the form of f(x) = g(x)/h(x), the derivative is given by f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2.

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, rather than explicitly solving for y.

Front

What is the chain rule in calculus?

Back

The chain rule is a formula for computing the derivative of the composition of two or more functions. If y = f(g(x)), then the derivative is dy/dx = f'(g(x)) * g'(x).

Front

What is the product rule for differentiation?

Back

The product rule states that if you have two functions u(x) and v(x), the derivative of their product is given by (uv)' = u'v + uv'.

Front

How do you find the derivative of a polynomial function?

Back

To find the derivative of a polynomial function, apply the power rule: if f(x) = ax^n, then f'(x) = n * ax^(n-1).

Front

What is the significance of the first derivative test?

Back

The first derivative test is used to determine the local maxima and minima of a function by analyzing the sign of the derivative before and after critical points.

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function and to classify critical points as local maxima, minima, or points of inflection based on the sign of the second derivative.

Front

What does it mean for a function to be increasing or decreasing?

Back

A function is increasing on an interval if its derivative is positive throughout that interval, and it is decreasing if its derivative is negative.

Front

What is the relationship between the derivative and the slope of a tangent line?

Back

The derivative of a function at a given point gives the slope of the tangent line to the graph of the function at that point.