To determine if a function is differentiable at a point, consider the following steps:

- Check continuity at the point. If the function is not continuous, it cannot be differentiable.
- Calculate the derivative of the function.
- Evaluate the limit of the derivative as the point of interest is approached from both the left and right.
- If the left and right limits are equal, the function is differentiable at that point.

**Understanding Differentiability: The Basics**

- Define differentiability and explain its importance.

**Understanding Differentiability: The Key to Unlocking a Function’s Secrets**

In the world of mathematics, functions are like intricate puzzles, revealing their hidden secrets through the power of differentiation. Differentiability, a crucial concept in calculus, provides a window into a function’s behavior, offering insights into its smoothness, rate of change, and even its shape.

**Defining Differentiability and Its Significance**

Imagine yourself driving along a winding road. The smoothness of the road can be described by the concept of differentiability. A differentiable function behaves in a *continuous* manner, with no sudden jumps or breaks. This smoothness allows us to analyze the function’s rate of change more precisely.

Differentiability becomes especially valuable when studying the behavior of functions at specific points. Unlike *continuous* functions that may have “flat spots” (points where the derivative is zero), differentiable functions exhibit a well-defined slope at every point in their domain. This slope, known as the *derivative*, provides a measure of how quickly the function changes as the input changes.

By understanding differentiability, we gain a deeper understanding of how functions behave, which is essential in various fields, including physics, engineering, and economics. From predicting the trajectory of a projectile to modeling the growth of a population, differentiability plays a pivotal role in unlocking the secrets of our mathematical world.

## Continuity: The Foundation of Differentiability

In the realm of calculus, **differentiability** reigns supreme, enabling us to delve into the intricate nature of functions and uncover their hidden patterns. Yet, this profound concept rests upon a fundamental pillar: **continuity**.

**Continuity** is the cornerstone of differentiability, ensuring that a function flows smoothly without abrupt jumps or discontinuities. It embodies the idea that as we move infinitesimally close to a point on the function’s graph, the corresponding function value remains within a finite distance.

**Implications of Continuity**

Continuity holds profound implications for the existence and behavior of derivatives.

**Positive Implications:**Continuity ensures that the**limit of the derivative**as the point of evaluation approaches the point of discontinuity exists. This limit provides valuable insights into the function’s behavior at that specific point.**Negative Implications:**Conversely, the absence of continuity spells trouble for differentiability. If a function is discontinuous at a point, it will generally fail to have a**well-defined derivative**at that point.

Understanding the interplay between continuity and differentiability is crucial for navig

## Limit of the Derivative: Unveiling a Function’s Behavior

In the realm of calculus, the limit of the derivative plays a pivotal role in deciphering the intricate behavior of functions. It’s like a window that grants us a glimpse into the function’s characteristics at a precise point.

The **limit of the derivative** at a particular value of *x* represents the *instantaneous rate of change* of the function at that point. It measures the **slope** of the function’s **tangent line** at that instant. By examining the limit of the derivative, we can gain insights into whether the function is **increasing**, **decreasing**, or remains **constant**.

To find the limit of the derivative, we use the concept of **limits**. Given a function *f(x)*, the limit of the derivative at *x* = *a* is defined as:

```
lim (h → 0) [f(a + h) - f(a)] / h
```

This limit represents the slope of the secant line passing through the points *(a, f(a))* and *(a + h, f(a + h))* as *h* approaches zero. As *h* gets infinitely small, the secant line approaches the tangent line, and the slope of the secant line converges to the instantaneous rate of change at *x* = *a*.

Understanding the limit of the derivative empowers us to analyze functions more deeply. By determining whether the derivative is positive, negative, or zero at a particular point, we can identify **local extrema**, **intervals of increase or decrease**, and other salient features of the function’s graph.

This knowledge is invaluable in fields such as **optimization**, where we seek to find maximum or minimum values of functions, and in **modeling real-world phenomena**, where understanding the rate of change of a function over time is crucial. By delving into the limit of the derivative, we unlock a powerful tool that unveils the hidden intricacies of mathematical functions.

**Rolle’s Theorem: Proving Existence and Extrema**

- State and explain Rolle’s Theorem.
- Discuss its applications in proving the existence of derivatives and identifying extreme values.

**Rolle’s Theorem: Unveiling the Secrets of Derivatives and Extremes**

In the captivating world of calculus, we embark on a journey to explore the fascinating theorem named after the brilliant mathematician Michel Rolle. Rolle’s Theorem is a powerful tool that provides groundbreaking insights into the **existence of derivatives** and the **identification of extreme values** for functions that reside in the realm of continuity and differentiability.

Rolle’s Theorem unfolds its magic upon functions that satisfy two fundamental conditions: **continuity** over a closed interval and **differentiability** over the corresponding open interval. Under these constraints, the theorem proclaims that there exists at least one point within the interval where the function’s derivative is **zero**. This revelation opens up a plethora of applications that we will delve into shortly.

**Unveiling the Existence of Derivatives**

Rolle’s Theorem serves as a valuable tool in proving the **existence of derivatives** for functions that comply with the aforementioned conditions. By constructing an auxiliary function, we can leverage the zero derivative property to establish the existence of a derivative for the original function. This approach has far-reaching implications in understanding the behavior of functions and their derivatives.

**Identifying Extreme Values**

Rolle’s Theorem extends its influence beyond the mere existence of derivatives. It also plays a pivotal role in **identifying extreme values** for functions. By combining Rolle’s Theorem with the **First Derivative Test**, we gain the ability to pinpoint intervals of increase, decrease, and extreme values. This knowledge is instrumental in understanding the shape and behavior of functions, both graphically and algebraically.

In conclusion, Rolle’s Theorem stands as a testament to the power of mathematics in revealing the intricate workings of functions. Its applications in proving the existence of derivatives and identifying extreme values have transformed our understanding of the mathematical landscape. As we continue to delve deeper into the world of calculus, we will encounter even more remarkable theorems and concepts that illuminate the hidden secrets of functions and their derivatives.

## Mean Value Theorem: Unveiling the Connection Between Average and Instantaneous Rates

In the realm of calculus, the Mean Value Theorem serves as a bridge between the average rate of change and the instantaneous rate of change of a function. It unveils a deep connection between these two concepts, providing invaluable insights into a function’s behavior.

The theorem states that if a function *f(x)* is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number *c* in (a, b) such that the instantaneous rate of change at *c* is equal to the average rate of change over the entire interval. In other words, there is a point *c* where the tangent line to the graph of *f(x)* is parallel to the secant line connecting the points *(a, f(a))* and *(b, f(b))*.

This remarkable result has profound implications. It tells us that the average rate of change over an interval can be calculated by evaluating the derivative at a single point within that interval. This is particularly useful when dealing with functions that exhibit non-linear behavior, as it allows us to approximate the average rate of change without having to resort to complex integration techniques.

Furthermore, the Mean Value Theorem provides a deeper understanding of the function’s local behavior. By identifying a point *c* where the instantaneous rate of change equals the average rate of change, it helps us pinpoint where the function is increasing or decreasing most rapidly. It also allows us to draw inferences about the function’s concavity and the presence of inflection points.

In essence, the Mean Value Theorem is a powerful tool that connects the global and local properties of a function. It enables us to analyze the overall trend of a function based on its behavior at a single point, providing valuable insights into the function’s characteristics and behavior.

**First Derivative Test: Identifying Local Extrema**

- Introduce the First Derivative Test and its applications.
- Discuss how it can be used to determine intervals of increase and decrease, as well as locate local extrema.

**The First Derivative Test: Unraveling Local Extrema**

In the realm of calculus, understanding the behavior of functions is crucial. Among the key tools that unlock this knowledge is the *First Derivative Test*. This powerful technique allows us to *identify local extrema*, or points where a function reaches its *maximum or minimum value* within a specific interval.

**The Essence of the Test**

The First Derivative Test hinges on the idea that **at a local extremum, the first derivative is either zero or undefined**. This stems from the fact that as a function approaches its peak or trough, its rate of change gradually diminishes. At the exact point of extrema, this rate becomes *zero*, indicating a *change in the function’s direction*.

**Practical Applications**

To apply the test, we first calculate the first derivative of the function in question. Then, we analyze its behavior at different points within the interval of interest:

**If the derivative is positive**, the function is*increasing*, signifying that it’s moving towards a maximum.**If the derivative is negative**, the function is*decreasing*, indicating a movement towards a minimum.**If the derivative is zero**, it could potentially be a local extremum; further analysis at neighboring points is necessary.

**Identifying Extrema**

Using the information gathered from the derivative analysis, we can pinpoint local extrema by examining the sign changes in the first derivative. Here’s how:

**Maximum:**If the derivative changes from positive to negative, we have found a local maximum.**Minimum:**If the derivative changes from negative to positive, we have found a local minimum.

The First Derivative Test is an invaluable tool that empowers us to unravel the behavior of functions and identify their local extrema. By understanding its principles and applications, we gain insights into the shape and characteristics of functions, which is essential for various fields such as optimization, modeling, and engineering.

## Unveiling Concavity and Inflections: The Second Derivative Test

In the realm of calculus, the **Second Derivative Test** emerges as a powerful tool to **dissect the behavior** of functions beyond their **rate of change**. This test shines a light on two crucial aspects of a function’s graphical landscape: **concavity** and **inflection points**.

**Defining Concavity**

Concavity describes the **curvature of a graph**. When a graph **curves upward**, it is said to be **concave up**, while a graph that **curves downward** is **concave down**. The Second Derivative Test provides a clear-cut means of determining concavity.

**Positive or Negative? The Key to Concavity**

The **sign of the second derivative** holds the key to concavity. If the **second derivative is positive**, the graph is **concave up**. Conversely, if the **second derivative is negative**, the graph is **concave down**. This simple rule allows us to **identify the regions** where the function is curving upward or downward.

**Inflection Points: Where Curves Change Direction**

**Inflection points** are special points on a graph where the **concavity changes**. At these points, the graph transitions from concave up to concave down or vice versa. The Second Derivative Test identifies inflection points by locating the points where the **second derivative is zero**. This crucial information enables us to predict the **shape** of the graph and anticipate its behavior.

**A Deeper Dive into Function Behavior**

The Second Derivative Test empowers us to **understand** a function’s behavior in **greater depth**. By examining the sign of the second derivative, we can **determine intervals** of concavity, **locate inflection points**, and **predict the overall shape** of the graph. This knowledge enhances our ability to **visualize** and **interpret** the function’s characteristics, providing insights that would otherwise remain hidden.