AP Calculus Exam Practice - Derivatives Focus I

Front

What does W′(2) > 3 indicate in the context of a water tank filling with water?

Back

Two hours after the tank begins filling with water, the rate at which the depth of the water is rising is increasing at a rate greater than 3 feet per hour per hour.

Front

How do you calculate the average rate of change of a function f(x) on a closed interval [a, b]?

Back

The average rate of change is given by the formula: \( \frac{f(b) - f(a)}{b - a} \).

Front

Given the function f(x) = 3x^2 - x^3, what is the average rate of change on the interval [1, 5]?

Back

The average rate of change is -13.

Front

What is the significance of a tangent line to a curve at a point?

Back

A tangent line represents the instantaneous rate of change of the function at that point.

Front

If a line x + y = k is tangent to the graph of y = x^2 + 3x + 1, how do you find the value of k?

Back

Set the derivative of the curve equal to the slope of the line and solve for k.

Front

What does it mean for a function to be not differentiable at a point?

Back

A function is not differentiable at a point if it has a discontinuity, vertical tangent, or cusp at that point.

Front

What is the derivative of f(x) = ln(e^(2x))?

Back

The derivative f'(x) = 2.

Front

What is the relationship between the first derivative and the slope of a function?

Back

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

Front

What does a vertical tangent indicate about the derivative of a function?

Back

A vertical tangent indicates that the derivative is undefined at that point.

Front

What is the definition of a horizontal tangent?

Back

A horizontal tangent occurs where the derivative of the function is zero, indicating a local maximum or minimum.