Calculus Finals Review (MC) Fall 2024

Front

What is the derivative of the function f(x) = (x - 1)(x + 2)?

Back

f'(x) = (x - 1)'(x + 2) + (x - 1)(x + 2)' = 1(x + 2) + (x - 1)(1) = 2x + 1. To find f'(0), plug in 0: f'(0) = 1.

Front

How do you find the speed of an object given its distance function s(t) = t^3 - 6t^2 - 4 when acceleration is 0?

Back

First, find the acceleration by taking the second derivative of s(t). Set the acceleration to 0 and solve for t. Then, plug that t value into s'(t) to find the speed.

Front

What does it mean for the slope of the tangent line to be positive?

Back

The slope of the tangent line is positive when the function is increasing at that point. This occurs when the derivative of the function is greater than 0.

Front

What is the limit of a function as x approaches a certain value?

Back

The limit is the value that a function approaches as the input approaches a certain value. For example, lim (x -> 1) of f(x) = 4 means as x gets closer to 1, f(x) approaches 4.

Front

Back

Front

What is the Product Rule in calculus?

Back

The Product Rule states that if you have two functions u(x) and v(x), then the derivative of their product is: (uv)' = u'v + uv'.

Front

Define acceleration in terms of calculus.

Back

Acceleration is the second derivative of the position function with respect to time, or the derivative of velocity.

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

How do you determine if a function is increasing or decreasing?

Back

A function is increasing where its first derivative is positive and decreasing where its first derivative is negative.

Front

What is the significance of a limit approaching infinity?

Back

If a limit approaches infinity, it indicates that the function grows without bound as the input approaches a certain value.