In this unit, you’ll learn about the essential basics of calculus. Limits and continuity
are the backgrounds for all of AP Calculus so it's crucial to understand these concepts.
This unit makes up about 10-12% of the AP Calculus AB Exam or 4-7% of the AP
Calculus BC Exam.
Calculus is a branch of mathematics that deals with the study of change and motion.
One of the main questions that calculus aims to answer is "Can change occur at an
instant?"
This question refers to the idea that change is not always a gradual process, but can
occur suddenly and at a specific moment in time. To better understand this concept, let's
consider the example of a moving arrow.
Imagine an arrow that is moving across a screen. At a particular moment in time, the
arrow's position on the screen changes. If we were to take a series of snapshots of the
arrow's position at different points in time, we would see that the arrow is moving and its
position is changing over time. However, when we look at a single snapshot, it appears
that the arrow's position has changed instantaneously.
This is an example of how change can occur at an instant. Even though the arrow's
motion is a continuous process, at a specific moment in time, its position changes
abruptly. To further understand this question, we first need to understand the concept of
a limit.
1.1 Introducing Calculus: Can Change Occur at an
Instant?
There are two different types of rates in calculus: average rate of change (AROC)and
instantaneous rate of change (IROC). The average rate of change is the slope of the
, secant line between two points of a function. The formula for the average rate of
change is:
Image courtesy of Medium.
Recall from your algebra class that this is simply the formula for the slope of a straight
line. It is important to note that if the denominator of this formula evaluates to zero, the
average rate of change will be undefined since division by zero is not possible.
1.2 Defining Limits and Using Limit Notation
A limit is a value that a function approaches as the input (or independent variable) gets
closer to a certain value. Limits are used to understand the instantaneous rate of
change. For example, as x approaches 2, the value of the function f(x) = x^2
approaches 4. This means that the limit of f(x) as x approaches 2 is 4.
are the backgrounds for all of AP Calculus so it's crucial to understand these concepts.
This unit makes up about 10-12% of the AP Calculus AB Exam or 4-7% of the AP
Calculus BC Exam.
Calculus is a branch of mathematics that deals with the study of change and motion.
One of the main questions that calculus aims to answer is "Can change occur at an
instant?"
This question refers to the idea that change is not always a gradual process, but can
occur suddenly and at a specific moment in time. To better understand this concept, let's
consider the example of a moving arrow.
Imagine an arrow that is moving across a screen. At a particular moment in time, the
arrow's position on the screen changes. If we were to take a series of snapshots of the
arrow's position at different points in time, we would see that the arrow is moving and its
position is changing over time. However, when we look at a single snapshot, it appears
that the arrow's position has changed instantaneously.
This is an example of how change can occur at an instant. Even though the arrow's
motion is a continuous process, at a specific moment in time, its position changes
abruptly. To further understand this question, we first need to understand the concept of
a limit.
1.1 Introducing Calculus: Can Change Occur at an
Instant?
There are two different types of rates in calculus: average rate of change (AROC)and
instantaneous rate of change (IROC). The average rate of change is the slope of the
, secant line between two points of a function. The formula for the average rate of
change is:
Image courtesy of Medium.
Recall from your algebra class that this is simply the formula for the slope of a straight
line. It is important to note that if the denominator of this formula evaluates to zero, the
average rate of change will be undefined since division by zero is not possible.
1.2 Defining Limits and Using Limit Notation
A limit is a value that a function approaches as the input (or independent variable) gets
closer to a certain value. Limits are used to understand the instantaneous rate of
change. For example, as x approaches 2, the value of the function f(x) = x^2
approaches 4. This means that the limit of f(x) as x approaches 2 is 4.
