AP Exam: Rates of Change, Tangent Lines, and Derivatives

What is the slope of a tangent line?

The limit of secant lines that pass the given point and another point nearby. It is equal to the limit as h approaches 0 of (f(a + h) - f(a))/h

What is the slope of the curve y = f(x) at the point (a, f(a))?

The limit as h approaches 0 of (f(a + h) -f(a))/h (it is equal to the slope of the tangent line of the curve at that point)

What is the instantaneous rate of change of the function f(x) with respect to x?

The limit as h approaches 0 of (f(x + h) - f(x))/h

What is sensitivity?

Sensitivity escribes how small changes in the input variable cause changes in the output variable. Sensitivity = the limit as h approaches 0 of (f(x + h) -f(x))/h

What is the derivative of f(x)?

The slope of the tangent line of y = f(x) at x. It is the same as the limit as h approaches 0 of (f(x + h) - f(x))/h

How many derivatives are there at a point a?

There are two- the right hand derivative and the left hand derivative, both of which can be expressed as (f(a + h) -f(a))/h

Does differentiability imply continuity? Does continuity imply differentiability?

Differentiability does imply continuity, but continuity does not imply differentiability.

What are marginal cost and marginal revenue?

Marginal cost is the derivative of a cost function and marginal revenue is the derivative of a revenue function

What is the derivative of tanx?

sec^2 x

what is the derivative of cot x?

\-csc^2 x

What is the derivative of sec x?

sec x (tan x)

What is the derivative of csc x?

\-csc x (cot x)