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4 min read•february 15, 2024
Welcome back to AP Calculus with Fiveable! This topic focuses on determining when a derivative exists and relating it to the concept of continuity. Let's combine our skills in determining continuity over an interval with the knowledge of derivatives and rates of change to continue building our derivative skills. 🧱
Most of the curves you encounter in AP Calculus will be both continuous and differentiable. But we can run into problems if it turns out that it doesn't fit both conditions! Brush up on the rules for continuity with this guide: Confirming Continuity over an Interval.
For a curve to be differentiable, it must have a derivative at every point in its domain. This means that if you zoom in very close, the graph will be smooth and look like a straight line. This line does not have to be only horizontal! ↗️
Take a look at this graph of , and notice that as we zoom into the point , the curve looks like a line.
For a curve to be differentiable at a single point, the right-hand limit of the derivative must equal the left-hand derivative, as well as the derivative at that point.
If a curve is differentiable, it must also be continuous. However, if a curve is continuous, it does not have to be differentiable! A curve is not differentiable if: it is discontinuous, has a sharp shift in the rate of change, or contains a vertical tangent.
Let’s check out these scenarios and observe why the curve is not differentiable at these points.
For most discontinuous graphs, we can see that the derivatives approaching the point from either side will not be equal.
For example, looks like the following in the graph below. Since the denominator cannot equal 0, . The domain of this graph is
We also see that , while . Hence, we can conclude that this graph is not continuous, and therefore not differentiable at .
We can apply the same logic to a graph with a jump discontinuity, such as the following graph. This graph is also discontinuous because while .
For a removable discontinuity, we have to take it a step further. Take a look at the graph of below. We can see that there is a removable discontinuity at .
We can determine, by hand or with a calculator, that the derivative is equal to at all points except for . But what about at ? The calculator will tell us that the derivative is undefined, or does not exist. Since but , we determine that the function is not differentiable at .
Therefore, if the curve is discontinuous at a point, it cannot be differentiable.
Take a look at the graph of below.
What would the derivative at be equal to? If we think about the slope of a line tangent to at , the slope would be because the change in values would be a number divided by the change in values: . Therefore, the derivative at does not exist, and makes the curve not differentiable at .
We can determine that a curve will not be differentiable at a corner or a cusp because the derivative will not approach the same value on both sides of the point. Additionally, there may be a vertical tangent! Let's observe the graphs below.
This is the graph of . We can see that this graph would have a vertical tangent line as we approach , so it cannot be differentiable.
Let’s look at another graph! This is the graph of . This function has a corner, and is not diffferentiable because while , and cannot be equal.
Nice work! Now you can identify when the curve is differentiable at a point. 🙌
Let’s work on a few questions and make sure we have the concept down!
Identify the number of points where the piecewise function is not differentiable.
The correct answer here is ! Let’s list them out: ⬇️
That was excellent work. Let’s try to determine differentiability with an algebraic method.
The following free-response question (FRQ) is from the 2003 AP Calculus AB examination administered by College Board. All credit to College Board.
The function can be represented by the following piecewise function:
Given that is continuous at , is the function differentiable?
To solve this question, we need to check if the derivative approaches the same value from both sides of the function. Let’s get started!
To check the derivative on the left-hand side, we need to take the derivative of the equation representing the function when .
Therefore, . Nice work!
To check the derivative on the right-hand side, we need to take the derivative of the equation representing the function when .
Therefore, . Almost there…
To check the derivative at the point, we need to take the derivative of the equation representing the function when , which is the same as when .
Therefore, .
Now we check that all of the limits equal each other, and find that the function is differentiable at ! Check the graph below to confirm our work.
As we can see from the graph, the function is smooth at , and is therefore differentiable. Amazing work! 👏
Great job! 🚀👩🚀
You're mastering these concepts, and with practice, you'll navigate derivatives and continuity with confidence. Determining when derivatives exist and ensuring continuity is a crucial skill in AP Calculus. As you encounter questions on the exam, remember to check for domain restrictions and assess piecewise continuity.
Continuous Function
: A continuous function is one that has no breaks, holes, or jumps in its graph. It can be drawn without lifting the pen from the paper.Differentiability
: Differentiability refers to the property of a function where it has a derivative at every point in its domain. In other words, the function is smooth and has a well-defined slope at each point.Differentiable Function
: A differentiable function is a type of function that has derivatives at every point within its domain. This means that the slope or rate of change can be determined at any point on the graph.Function
: A relationship between two sets where each input (domain) value corresponds to exactly one output (range) value.Limit
: The limit of a function is the value that the function approaches as the input approaches a certain value or infinity. It represents the behavior of the function near a specific point.Limit definition of the derivative
: The limit definition of the derivative is a mathematical expression used to find the instantaneous rate of change (slope) of a curve at a specific point. It involves taking the limit as the change in x approaches zero.Piecewise function
: A piecewise function is a mathematical function that is defined by different formulas or rules for different intervals of the input variable. It consists of multiple "pieces" or sections, each with its own formula.Point of discontinuity
: A point of discontinuity occurs when there is a break or jump in the graph of a function at a specific input value. The function fails to be continuous at this point.Step function
: A step function is a piecewise-defined function that jumps from one constant value to another at specific intervals. It resembles a series of steps on a staircase.4 min read•february 15, 2024
Welcome back to AP Calculus with Fiveable! This topic focuses on determining when a derivative exists and relating it to the concept of continuity. Let's combine our skills in determining continuity over an interval with the knowledge of derivatives and rates of change to continue building our derivative skills. 🧱
Most of the curves you encounter in AP Calculus will be both continuous and differentiable. But we can run into problems if it turns out that it doesn't fit both conditions! Brush up on the rules for continuity with this guide: Confirming Continuity over an Interval.
For a curve to be differentiable, it must have a derivative at every point in its domain. This means that if you zoom in very close, the graph will be smooth and look like a straight line. This line does not have to be only horizontal! ↗️
Take a look at this graph of , and notice that as we zoom into the point , the curve looks like a line.
For a curve to be differentiable at a single point, the right-hand limit of the derivative must equal the left-hand derivative, as well as the derivative at that point.
If a curve is differentiable, it must also be continuous. However, if a curve is continuous, it does not have to be differentiable! A curve is not differentiable if: it is discontinuous, has a sharp shift in the rate of change, or contains a vertical tangent.
Let’s check out these scenarios and observe why the curve is not differentiable at these points.
For most discontinuous graphs, we can see that the derivatives approaching the point from either side will not be equal.
For example, looks like the following in the graph below. Since the denominator cannot equal 0, . The domain of this graph is
We also see that , while . Hence, we can conclude that this graph is not continuous, and therefore not differentiable at .
We can apply the same logic to a graph with a jump discontinuity, such as the following graph. This graph is also discontinuous because while .
For a removable discontinuity, we have to take it a step further. Take a look at the graph of below. We can see that there is a removable discontinuity at .
We can determine, by hand or with a calculator, that the derivative is equal to at all points except for . But what about at ? The calculator will tell us that the derivative is undefined, or does not exist. Since but , we determine that the function is not differentiable at .
Therefore, if the curve is discontinuous at a point, it cannot be differentiable.
Take a look at the graph of below.
What would the derivative at be equal to? If we think about the slope of a line tangent to at , the slope would be because the change in values would be a number divided by the change in values: . Therefore, the derivative at does not exist, and makes the curve not differentiable at .
We can determine that a curve will not be differentiable at a corner or a cusp because the derivative will not approach the same value on both sides of the point. Additionally, there may be a vertical tangent! Let's observe the graphs below.
This is the graph of . We can see that this graph would have a vertical tangent line as we approach , so it cannot be differentiable.
Let’s look at another graph! This is the graph of . This function has a corner, and is not diffferentiable because while , and cannot be equal.
Nice work! Now you can identify when the curve is differentiable at a point. 🙌
Let’s work on a few questions and make sure we have the concept down!
Identify the number of points where the piecewise function is not differentiable.
The correct answer here is ! Let’s list them out: ⬇️
That was excellent work. Let’s try to determine differentiability with an algebraic method.
The following free-response question (FRQ) is from the 2003 AP Calculus AB examination administered by College Board. All credit to College Board.
The function can be represented by the following piecewise function:
Given that is continuous at , is the function differentiable?
To solve this question, we need to check if the derivative approaches the same value from both sides of the function. Let’s get started!
To check the derivative on the left-hand side, we need to take the derivative of the equation representing the function when .
Therefore, . Nice work!
To check the derivative on the right-hand side, we need to take the derivative of the equation representing the function when .
Therefore, . Almost there…
To check the derivative at the point, we need to take the derivative of the equation representing the function when , which is the same as when .
Therefore, .
Now we check that all of the limits equal each other, and find that the function is differentiable at ! Check the graph below to confirm our work.
As we can see from the graph, the function is smooth at , and is therefore differentiable. Amazing work! 👏
Great job! 🚀👩🚀
You're mastering these concepts, and with practice, you'll navigate derivatives and continuity with confidence. Determining when derivatives exist and ensuring continuity is a crucial skill in AP Calculus. As you encounter questions on the exam, remember to check for domain restrictions and assess piecewise continuity.
Continuous Function
: A continuous function is one that has no breaks, holes, or jumps in its graph. It can be drawn without lifting the pen from the paper.Differentiability
: Differentiability refers to the property of a function where it has a derivative at every point in its domain. In other words, the function is smooth and has a well-defined slope at each point.Differentiable Function
: A differentiable function is a type of function that has derivatives at every point within its domain. This means that the slope or rate of change can be determined at any point on the graph.Function
: A relationship between two sets where each input (domain) value corresponds to exactly one output (range) value.Limit
: The limit of a function is the value that the function approaches as the input approaches a certain value or infinity. It represents the behavior of the function near a specific point.Limit definition of the derivative
: The limit definition of the derivative is a mathematical expression used to find the instantaneous rate of change (slope) of a curve at a specific point. It involves taking the limit as the change in x approaches zero.Piecewise function
: A piecewise function is a mathematical function that is defined by different formulas or rules for different intervals of the input variable. It consists of multiple "pieces" or sections, each with its own formula.Point of discontinuity
: A point of discontinuity occurs when there is a break or jump in the graph of a function at a specific input value. The function fails to be continuous at this point.Step function
: A step function is a piecewise-defined function that jumps from one constant value to another at specific intervals. It resembles a series of steps on a staircase.© 2024 Fiveable Inc. All rights reserved.
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