Question Bank: Limits and Derivatives

What is the primary focus of calculus?

In the context of limits, what does it mean for a function to approach a value?

What does the derivative represent in a physical context?

Which expression best represents the average velocity of an object over an interval?

If the position function of a falling object is given by s(t) = 4.9t², what is the average velocity between t=1s and t=2s?

At what time is the body dropped from a cliff covering a distance described by s = 4.9t^2?

What happens to the average velocity as the time interval shrinks around a specific point?

What does the derivative represent in a function?

The process of finding the derivative of a function is known as what?

If the limit of a function f(x) as x approaches a is L, what does this indicate?

In the context of calculus, what is a limit?

Which of the following functions has a constant derivative?

Why is the average velocity between two points in time important for understanding derivatives?

What is the derivative of the function f(x) = x at any point?

What is the instantaneous velocity of a body at t=2 seconds if its distance function is s(t) = 4.9t²?

In calculating limits, which method can be used when direct substitution gives an indeterminate form?

If a car's position is given by s(t) = 5t^2, what does the derivative s'(t) signify?

How can limits be understood intuitively?

In which of the following scenarios is the concept of a derivative NOT applicable?

What is the limit of (2x + 3) as x approaches 1?

The term 'tangent line' in the context of derivatives refers to:

Which of the following statements about derivatives is true?

Which calculation method helps derive a function’s instantaneous change?

What happens to the average velocity as the time interval approaches zero?

Considering the function s(t) = 4.9t^2, as t approaches zero, what happens to the velocity?

For the function y = x^2, what is the derivative at x = 3?

For a distance function describing motion, what does a negative derivative indicate?

Which of these expressions is not a valid limit expression?

Why does the average velocity approach the derivative as the interval decreases?

If the derivative of a function is known to be positive, which of the following is true about the function?

What is the value of the limit lim(x→2) (x² - 4)/(x - 2)?

What is the limit lim(x→0) (sin(x)/x)?

If f(x) = x³, what is f'(1) using the definition of the derivative?

What is the limit lim(x→3) (2x + 1)?

What is the limit lim(x→∞) (1/x)?

Determine the limit lim(x→-1) (x² + 2x + 1)/(x + 1).

Evaluate the limit lim(x→0) (tan(x)/x).

What is the limit of the function lim(x→-2) (x² + 4)?

What does the limit lim(x→1) (1/x) evaluate to?

Find the limit lim(x→4) (√x - 2)/(x - 4).

If f(x) = x² for x ≠ 2 and f(2) = 0, what is lim(x→2) f(x)?

Evaluate the limit lim(x→1) (x³ - 1)/(x - 1).

Find the limit lim(x→∞) (x² - 5)/(3x² + 2).

What is the limit lim(x→1) (e^x - e)/(x - 1)?

Determine the limit lim(x→0) (cos(x) - 1)/x².

What is the limit of lim(x→0) (x³)/(sin(x))?

What is the limit lim(x→3) (x² - 9)/(x - 3)?

What is the limit of sin(x)/x as x approaches 0?

Which theorem can be used to establish that lim (x→0) sin(x)/x = 1?

Evaluate lim (x→0) (1 - cos(x))/x².

What is the behavior of tan(x) as x approaches π/2?

Using the Sandwich Theorem, which of the following is true about sin(x) when 0 < x < π/2?

What is the limit of cos(x)/x as x approaches 0?

Which limit is true? lim (x→0) (tan(x)/x) = ?

What is the limit of (sin(2x)/x) as x approaches 0?

Find the limit lim (x→0) (sin(3x)/x).

If f(x) = sin(x), what is lim (x→π) f(x)?

Evaluate lim (x→0) (x - sin(x)) / x³.

Using L'Hospital's Rule, find lim (x→0) (1 - cos(x)) / x².

What can be concluded about sin(x)/tan(x) as x approaches 0?

Evaluate lim (x→π/2) (sin(x))/(1 - cos(x)).

Which of the following statements regarding the limit lim (x→0) (tan(x) - sin(x)) is correct?

What is the derivative of the function f(x) = 3x^2?

If f(x) = x^3 - 4x + 1, what is f'(2)?

What is the derivative of the constant function f(x) = 5?

Determine the derivative of f(x) = sin(x).

What is f'(1) for the function f(x) = x^2 - 2x + 3?

If f(x) = ln(x), what is the derivative f'(x)?

Find the derivative of f(x) = e^x.

Calculate the limit: lim (x -> 2) (x^2 - 4)/(x - 2).

For which value of x is f(x) = x^2 - 4x a maximum?

What does f'(x) represent graphically?

The derivative of f(x) = 1/x is what?

Evaluate f'(x) if f(x) = 2x^3 + 3x^2 - x + 7.

If f(x) = x^4 - 5x^2, what is the critical point?

The second derivative f''(x) indicates what about f(x)?

For the function f(x) = x^2 + 1, what happens at x = 0?