Front

What is the definition of acceleration in terms of position functions?

Back

Acceleration is the second derivative of the position function with respect to time, indicating how the velocity of an object changes over time.

Front

How do you find the volume of a solid of revolution using the disk method?

Back

The volume V is given by the integral V = π∫[a,b] (f(x))^2 dx, where f(x) is the function being revolved around the x-axis.

Front

What is the formula for the average value of a function f(x) on the interval [a, b]?

Back

The average value is given by \( \frac{1}{b-a} \int_a^b f(x) \, dx \).

Front

What is the Fundamental Theorem of Calculus?

Back

It states that if F is an antiderivative of f on [a, b], then \( \int_a^b f(x) \, dx = F(b) - F(a) \).

Front

How do you determine if a function is differentiable at a point?

Back

A function is differentiable at a point if the derivative exists at that point, which requires that the function is continuous and the limit of the difference quotient exists.

Front

What is the relationship between the first and second derivatives of a function?

Back

The first derivative indicates the rate of change (slope) of the function, while the second derivative indicates the rate of change of the first derivative (concavity).

Front

What is the significance of critical points in calculus?

Back

Critical points are where the first derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection.

Front

How do you find the area between two curves y=f(x) and y=g(x)?

Back

The area A is given by A = ∫[a,b] (f(x) - g(x)) dx, where f(x) is the upper curve and g(x) is the lower curve.

Front

What is the definition of a limit in calculus?

Back

A limit is the value that a function approaches as the input approaches a certain point.

Front

What is the purpose of using L'Hôpital's Rule?

Back

L'Hôpital's Rule is used to evaluate limits of indeterminate forms like 0/0 or ∞/∞ by taking the derivative of the numerator and denominator.