LimitsAndDerivatives.Rmd

# Limits and Derivatives

## The Tangent and Velocity Problems 

**Motivation: Why Limits Appear in Calculus**

Two of the most fundamental problems in calculus are:

1. **The Tangent Problem**  
   → How do we find the slope of a curve at a single point?

2. **The Velocity Problem**  
   → How do we define the velocity of an object at an exact instant in time?

Both problems involve trying to measure something at **a single point**, where no interval exists — and both naturally lead to the concept of a **limit**.

---

### The Tangent Problem


The word *tangent* comes from the Latin **tangens**, meaning *touching*. Informally, a tangent line is a line that:

- touches a curve at a point  
- has the **same direction** as the curve at that point  

But “touching” alone is not mathematically precise:

- A line can touch a curve but not match its direction  
- A line can intersect a curve more than once and still behave like a tangent  

So we need a **precise definition**.


<div class="definition">
A **secant line** is a line that intersects a curve at **two or more points**.

If: 

- \( P \) is a fixed point on the curve  
- \( Q \) is another point on the curve  

Then the slope of the secant line \( PQ \) is:
\[
m_{PQ} = \frac{f(x) - f(a)}{x - a}.
\]
</div>


> **Key Idea:** A tangent line is the **limit of secant lines** as the second point approaches the first point.

Symbolically:
\[
\text{slope of tangent at } x=a
=
\lim_{x \to a} \frac{f(x)-f(a)}{x-a}.
\]

This is the foundational idea that leads directly to **derivatives**.


::: {.example}
Tangent to a Curve

**Problem**

Find the equation of the tangent line to the parabola  
\[
y = x^2
\]
at the point \( P(1,1) \).

**Solution:**

Step 1: Use a nearby point \( Q(x, x^2) \)

Slope of secant line:
\[
m_{PQ} = \frac{x^2 - 1}{x - 1}
\]


Step 2: Take the limit
\[
m = \lim_{x \to 1} \frac{x^2 - 1}{x - 1}
\]
Factor:
\[
= \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1}
= \lim_{x \to 1} (x+1) = 2
\]

Step 3: Use point-slope form
\[
y - 1 = 2(x - 1)
\]
:::

Geometrically, as point \( Q \) moves closer to point \( P \):

- secant lines rotate  
- slopes stabilize  
- the secant line approaches a single limiting line  

That limiting line **is the tangent line**.

---

### Tangents from Experimental Data

Not all functions come from formulas — many come from **measured data**.

In this case:

- The graph is approximated from data points  
- Secant slopes are computed between nearby points  
- The tangent slope is estimated using nearby secants  
- The tangent slope represents an **instantaneous rate of change**  


::: {.example}
Tangent from a Table of Values

**Problem**

A function has the following values near \( x=1 \):

| \(x\) | \(f(x)\) |
|------|--------|
| 0.9  | 1.81 |
| 0.99 | 1.9801 |
| 1.01 | 2.0201 |
| 1.1  | 2.21 |

Estimate the slope of the tangent line at \( x=1 \).


**Solution**

Compute secant slopes:

Left side:
\[
m \approx \frac{1.9801 - 2}{0.99 - 1} = \frac{-0.0199}{-0.01} = 1.99
\]

Right side:
\[
m \approx \frac{2.0201 - 2}{1.01 - 1} = \frac{0.0201}{0.01} = 2.01
\]

Estimated tangent slope:
\[
m \approx 2
\]
:::



### The Velocity Problem

<div class="definition">
Average Velocity

Over a time interval \( [t_1, t_2] \):

\[
\text{Average velocity} =
\frac{\text{change in position}}{\text{change in time}}
=
\frac{s(t_2)-s(t_1)}{t_2-t_1}
\]
</div>

Velocity at a **single instant** cannot be computed directly because no time interval exists.

So we define it as a **limit**:

\[
v(t) = \lim_{h \to 0} \frac{s(t+h)-s(t)}{h}
\]



::: {.example}
Average vs Instantaneous Velocity

**Problem**

A particle moves according to:
\[
s(t) = t^2
\]

1. Find the average velocity on \( [2, 4] \)  
2. Estimate the instantaneous velocity at \( t=2 \)



**Solution**

**1) Average velocity**

\[
v_{avg} = \frac{s(4)-s(2)}{4-2}
= \frac{16 - 4}{2} = 6
\]


**2) Instantaneous velocity**

\[
v(2) = \lim_{h \to 0} \frac{s(2+h)-s(2)}{h}
= \lim_{h \to 0} \frac{(2+h)^2 - 4}{h}
\]

\[
= \lim_{h \to 0} \frac{4 + 4h + h^2 - 4}{h}
= \lim_{h \to 0} \frac{4h + h^2}{h}
\]

\[
= \lim_{h \to 0} (4 + h) = 4
\]

**Final Answers:**
\[
\boxed{v_{avg} = 6}, \qquad \boxed{v(2) = 4}
\]
:::

---

### Connection Between Tangents and Velocity

These are actually the **same mathematical idea**:

| Interpretation | Meaning |
|------|--------|
| Tangent slope | Instantaneous rate of change |
| Velocity | Instantaneous rate of change of position |
| Secant slope | Average rate of change |
| Limit of secants | Derivative |
| Tangent slope | Derivative |
| Velocity | Derivative |


<div class="definition">
**Fundamental Unifying Formula**

\[
\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}
\]

This represents:

- slope of tangent line  
- instantaneous velocity  
- instantaneous rate of change  
- derivative  
</div>



::: {.example}
Interpreting a Tangent Physically

**Problem**

The position of a vehicle is given by:
\[
s(t) = 3t^2 + 2t
\]

Find the slope of the tangent line at \( t=1 \) and interpret its meaning.

---

**Solution**

\[
v(1) = \lim_{h \to 0} \frac{s(1+h)-s(1)}{h}
\]

\[
s(1) = 5
\]

\[
s(1+h) = 3(1+h)^2 + 2(1+h) = 3(1 + 2h + h^2) + 2 + 2h
\]
\[
= 3 + 6h + 3h^2 + 2 + 2h = 5 + 8h + 3h^2
\]

\[
v(1) = \lim_{h \to 0} \frac{(5 + 8h + 3h^2) - 5}{h}
= \lim_{h \to 0} \frac{8h + 3h^2}{h}
\]

\[
= \lim_{h \to 0} (8 + 3h) = 8
\]


**Interpretation:**  
At \( t=1 \), the vehicle’s instantaneous velocity is **8 units per second**.
:::

---

### Putting It All Together


> **Instantaneous change is the limit of average change.**

This single idea connects:

- tangents → slopes  
- velocity → motion  
- acceleration → changing velocity  
- growth → population dynamics  
- decay → radioactive processes  
- learning curves → AI training  
- finance → marginal change  
- economics → marginal cost  
- physics → kinematics  
- data science → gradients  
- machine learning → optimization  


These problems lead directly to:
\[
\boxed{\text{Derivative} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}}
\]
Which will become the central object of differential calculus.



### Conceptual Takeaways

- Tangents are **limits of secant lines**  
- Velocity is the **limit of average velocities**  
- Both problems are solved using **limits**  
- Instantaneous change requires limiting processes  
- Slopes represent **rates of change**  
- Physical motion and geometry share the same mathematics  
- Calculus begins when we formalize “approaching”  
- The derivative unifies geometry and physics  
- Data-based tangents are meaningful even without formulas  
- Limits are not just abstract — they model real processes  


#### Skills You Should Be Able To Do


- Explain what a tangent line represents  
- Explain what instantaneous velocity means  
- Distinguish secant vs tangent lines  
- Compute slopes of secant lines  
- Interpret slopes as rates of change  
- Use difference quotients  
- Take limits of difference quotients  
- Connect physical motion to slope  
- Understand limits as approximation processes  
- Explain why single-point measurements require limits  
- Interpret data-based tangent estimates  
- Use point-slope form  
- Build tangent line equations  
- Compute average velocity  
- Approximate instantaneous velocity  


#### Problems

---






















## The Limit of a Function 



In the previous section, limits appeared naturally when studying:

- tangent lines  
- instantaneous velocity  

Now we study **limits themselves** as a fundamental concept in calculus, independent of geometry and motion. Limits describe how a function behaves **near a point**, not necessarily at the point.

---

### Intuitive Idea of a Limit


<div class="definition">
Let \( f(x) \) be defined near \( a \).  We write:
\[
\lim_{x \to a} f(x) = L
\]
and say: 
“The limit of \( f(x) \) as \( x \) approaches \( a \) is \( L \)”  
if the values of \( f(x) \) can be made **arbitrarily close to \( L \)** by taking \( x \) sufficiently close to \( a \), **but not equal to \( a \)**.
</div>



<div class="note">
**Limits depend on behavior near a point, not the value at the point.**

- \( f(a) \) may be undefined  
- \( f(a) \) may exist but differ from the limit  
- the limit may exist even if the function is not defined at \( a \)
</div>

---

### Numerical Approach to Limits

We estimate limits using **tables of values**:

- choose values close to \( a \)  
- approach from both sides  
- observe stabilization  


::: {.example}
Numerical Limit

**Problem**

Estimate:
\[
\lim_{x \to 2} (x^2 - 2x + 2)
\]


**Solution**

Evaluate near \( x=2 \):

| \(x\) | \(f(x)\) |
|------|--------|
| 1.9 | 3.71 |
| 1.99 | 3.9701 |
| 2.01 | 4.0301 |
| 2.1 | 4.31 |

Values approach **4**.
:::


Graphically:

- as \( x \to a \)  
- the **y-values approach \( L \)**  

even if:
- the point is missing  
- the point is a hole  
- the point has a different value  


::: {.example}
Limit Exists but Function Value Is Different

**Problem**

Let:
\[
f(x)=
\begin{cases}
\frac{x-1}{x^2-1}, & x\neq 1 \\
2, & x=1
\end{cases}
\]
Find:
\[
\lim_{x \to 1} f(x)
\]


**Solution**

Factor:
\[
\frac{x-1}{(x-1)(x+1)}=\frac{1}{x+1}, \quad x\neq 1
\]

\[
\lim_{x \to 1} f(x) = \frac{1}{2}
\]

Even though \( f(1)=2 \):
\[
\lim_{x\to 1} f(x)=\frac{1}{2}.
\]
:::

---

### One-Sided Limits

Sometimes behavior differs depending on direction.


<div class="definition">

Left-hand limit:  The values of $x$ get closer to $a$ from the left of $a$ 
\[
\lim_{x \to a^-} f(x)
\]

Right-hand limit:  The values of $x$ get closer to $a$ from the right of $a$ 
\[
\lim_{x \to a^+} f(x)
\]
</div>



<div class="theorem">
**Fundamental Rule**

\[
\lim_{x \to a} f(x) = L
\quad \text{iff} \quad
\lim_{x \to a^-} f(x)=L \;\text{and}\; \lim_{x \to a^+} f(x)=L.
\]
</div>



::: {.example}
One-Sided Limits

**Problem**

Let:
\[
f(x)=
\begin{cases}
0, & x<0 \\
1, & x>0
\end{cases}
\]

Find:
\[
\lim_{x\to 0^-} f(x), \quad
\lim_{x\to 0^+} f(x), \quad
\lim_{x\to 0} f(x)
\]


**Solution**

\[
\lim_{x\to 0^-} f(x)=0
\]
\[
\lim_{x\to 0^+} f(x)=1
\]

Since they are not equal:

\[
\lim_{x\to 0} f(x)\ \text{does not exist}
\]
:::

---

### Infinite Limits

Sometimes values grow without bound.
\[
\lim_{x\to a} f(x)=\infty
\quad \text{or} \quad
\lim_{x\to a} f(x)=-\infty.
\]
These **do not mean the limit exists** — they describe divergence behavior.


::: {.example}
Infinite Limit

**Problem**

Evaluate:
\[
\lim_{x \to 0} \frac{1}{x^2}
\]



**Solution**

As \( x\to 0 \), \( x^2 \to 0^+ \), so:

\[
\frac{1}{x^2} \to \infty.
\]
Therefore,
\[
\lim_{x\to 0} \frac{1}{x^2}=\infty.
\]
:::


---

### Vertical Asymptotes


<div class="definition">
A line \( x=a \) is a **vertical asymptote** if:
\[
\lim_{x\to a} f(x)=\infty \quad \text{or} \quad -\infty
\]
(or one-sided versions)
</div>


::: {.example}
Vertical Asymptote

**Problem**

Find the vertical asymptote of:
\[
f(x)=\frac{2x}{x-3}
\]


**Solution**

Denominator \(=0\) at \( x=3 \).

Right side:
\[
x\to 3^+ \Rightarrow f(x)\to \infty
\]

Left side:
\[
x\to 3^- \Rightarrow f(x)\to -\infty
\]

\[
x=3 \text{ is a vertical asymptote}
\]
:::


Some functions never settle to a value.


::: {.example}
Limit Does Not Exist (Oscillation)

**Problem**

\[
\lim_{x\to 0} \sin\left(\frac{\pi}{x}\right)
\]



**Solution**

As \( x\to 0 \):

- function oscillates infinitely between \( -1 \) and \( 1 \)
- does not approach a single value

\[
\text{Limit does not exist}
\]
:::


---

### Putting It All Together



**A limit describes behavior near a point — not at the point.**  Limits are about **tendency**, **approach**, **direction**, and **local structure**, not exact values.



#### Conceptual Takeaways

- Limits describe **local behavior**, not point values   
- A function value and a limit are independent concepts  
- Limits are about **approach**, not arrival  
- Tables and graphs are approximations, not proofs  
- Numerical methods can mislead (rounding errors, cancellation)  
- One-sided limits detect directional behavior  
- Limits fail if:  
  - oscillation occurs  
  - sides disagree  
  - divergence occurs  
- Infinite limits describe **unbounded growth**, not numbers  
- Vertical asymptotes are geometric expressions of infinite limits  
- Limits unify algebra, geometry, physics, and data analysis  
- Limits formalize the idea of “getting close”  



#### Skills You Should Be Able To Do


- Distinguish limit value from function value  
- Estimate limits numerically  
- Estimate and interpret limits graphically 
- Use correct limit notation  
- Recognize non-existent limits  
- Explain one-sided behavior/ limits  
- Identify infinite limits and vertical asymptotes  
- Understand asymptotic behavior  
- Recognize oscillation and analyze oscillating functions  
- Interpret discontinuities  
- Analyze rational functions near singularities  
- Distinguish removable vs non-removable discontinuities  
- Interpret divergence  
- Detect numerical instability  

#### Problems

---

















## Calculating Limits Using the Limit Laws 


In earlier sections, limits were often **estimated numerically or graphically**. While useful for intuition, these methods can be unreliable.  This section introduces **Limit Laws**, which provide **algebraic rules** for computing limits exactly and systematically.

These laws allow us to break complicated expressions into simpler pieces and compute limits using known limits.

---

### Limit Laws

Assume that  
\[
\lim_{x \to a} f(x) \quad \text{and} \quad \lim_{x \to a} g(x)
\]
both exist, and that \( c \) is a constant.


<div class="definition">
**Algebraic Limit Laws**

1. **Sum Law**
\[
\lim_{x \to a}[f(x)+g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
\]

2. **Difference Law**
\[
\lim_{x \to a}[f(x)-g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)
\]

3. **Constant Multiple Law**
\[
\lim_{x \to a}[c f(x)] = c \cdot \lim_{x \to a} f(x)
\]

4. **Product Law**
\[
\lim_{x \to a}[f(x)g(x)] = \lim_{x \to a} f(x)\cdot \lim_{x \to a} g(x)
\]

5. **Quotient Law**
\[
\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, 
\quad \text{if } \lim_{x \to a} g(x) \neq 0
\]
</div>


<div class="definition">
**Special Limits**

1.  **Constants and Identity**
\[
\lim_{x \to a} c = c, \qquad \lim_{x \to a} x = a
\]

2.  **Powers and Roots**
\[
\lim_{x \to a} x^n = a^n, \qquad 
\lim_{x \to a} \sqrt[n]{x} = \sqrt[n]{a}
\]

3.  **Power Law**
\[
\lim_{x \to a} [f(x)]^n = \left(\lim_{x \to a} f(x)\right)^n
\]
</div>


<div class="definition">
**Direct Substitution Property:**  If \( f(x) \) is a **polynomial** or a **rational function** and \( a \) is in its domain, then:
\[
\lim_{x \to a} f(x) = f(a)
\]
</div>

> For continuous algebraic functions, limits can be found by direct substitution.


::: {.example}
Polynomial Limit

Evaluate the limit:
\[
\lim_{x \to 5} (2x^2 - 3x + 4)
\]

**Solution**

\[
= 2(5^2) - 3(5) + 4
\]
\[
= 50 - 15 + 4
\]
\[
= 39
\]
:::

::: {.example}
Rational Function

Evaluate the limit:
\[
\lim_{x \to -2} \frac{x^3 + 2x^2 - 1}{5 - 3x}
\]

**Solution**

\[
= \frac{(-2)^3 + 2(-2)^2 - 1}{5 - 3(-2)}
\]
\[
= \frac{-8 + 8 - 1}{5 + 6}
\]
\[
= \frac{-1}{11}
\]
:::

::: {.example}
Indeterminate Form \( \frac{0}{0} \)

Evaluate the limit:
\[
\lim_{x \to 1} \frac{x^2 - 1}{x - 1}
\]


**Solution**

**Step 1: Factor**
\[
= \frac{(x-1)(x+1)}{x-1}
\]

**Step 2: Cancel**
\[
= x+1
\]

**Step 3: Take limit**
\[
\lim_{x \to 1} (x+1) = 2
\]
:::

::: {.example}
Difference of Squares Structure

Evaluate the limit:
\[
\lim_{h \to 0} \frac{(3+h)^2 - 9}{h}
\]


**Solution**

**1.  Expand numerator**
\[
= \frac{9 + 6h + h^2 - 9}{h}
\]
\[
= \frac{6h + h^2}{h}
\]

**2.  Simplify**
\[
= 6 + h
\]

**3.  Limit**
\[
\lim_{h \to 0} (6+h) = 6
\]
:::

::: {.example}
Rationalization

Evaluate the limit:
\[
\lim_{t \to 0} \frac{\sqrt{t+9} - 3}{t}
\]


**Solution**

**1.  Multiply by conjugate:**
\[
\frac{\sqrt{t+9} - 3}{t} \cdot \frac{\sqrt{t+9} + 3}{\sqrt{t+9} + 3}
\]

\[
= \frac{(t+9)-9}{t(\sqrt{t+9}+3)}
\]

**2.  Cancel \( t \):**
\[
= \frac{1}{\sqrt{t+9}+3}
\]

**3.  Take limit:**
\[
= \frac{1}{3+3} = \frac{1}{6}
\]
:::


::: {.example}
Absolute Value

Evaluate the limit:
\[
\lim_{x \to 0} |x|
\]


**Solution**

**1.  Right-hand:**
\[
\lim_{x \to 0^+} x = 0
\]

**2.  Left-hand:**
\[
\lim_{x \to 0^-} (-x) = 0
\]

Therefore:
\[
\lim_{x \to 0} |x| = 0
\]
:::



::: {.example}
Nonexistent Limit

Evaluate the limit:
\[
\lim_{x \to 0} \frac{|x|}{x}
\]


**Solution**

**1.  Right-hand:**
\[
\lim_{x \to 0^+} \frac{x}{x} = 1
\]

**2.  Left-hand:**
\[
\lim_{x \to 0^-} \frac{-x}{x} = -1
\]

Since they differ:
\[
\text{The limit does not exist.}
\]
:::


### The Squeeze Theorem (Sandwich Theorem)

The **Squeeze Theorem** (also called the **Sandwich Theorem**) is a method for evaluating limits of functions that are **trapped between two simpler functions** whose limits are known and equal.


<div class="definition">
Theorem (Squeeze Theorem)

Suppose that for all \( x \) near \( a \) (except possibly at \( a \)),
\[
g(x) \le f(x) \le h(x)
\]
and
\[
\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L.
\]
Then,
\[
\lim_{x \to a} f(x) = L.
\]
</div>


If a function is **always between two other functions**, and both of those functions approach the **same value**, then the trapped function has **no choice** but to approach that same value.

> The outer functions “squeeze” the middle function into a single limit value.

Visually, think of three curves:

- Top curve: \( h(x) \)   
- Middle curve: \( f(x) \)   
- Bottom curve: \( g(x) \) 

As \( x \to a \), both \( g(x) \) and \( h(x) \) approach the same height \( L \).  Since \( f(x) \) is stuck between them, it must also approach \( L \).


::: {.example}
Oscillating Function

Evaluate the limit:
\[
\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right)
\]


**Solution**

**Step 1: Bound the sine function**
\[
-1 \le \sin\left(\frac{1}{x}\right) \le 1
\]

**Step 2: Multiply by \( x^2 \)**
\[
- x^2 \le x^2 \sin\left(\frac{1}{x}\right) \le x^2
\]

**Step 3: Take limits**
\[
\lim_{x \to 0} (-x^2) = 0, \qquad \lim_{x \to 0} x^2 = 0
\]

**Conclusion (Squeeze Theorem):**
\[
\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0
\]
:::


::: {.example}
Absolute Value Oscillation

Evaluate the limit:
\[
\lim_{x \to 0} x \cos\left(\frac{1}{x}\right)
\]


**Solution**

**Bounding:**
\[
-1 \le \cos\left(\frac{1}{x}\right) \le 1
\]

Multiply by \( x \):
\[
- |x| \le x\cos\left(\frac{1}{x}\right) \le |x|
\]

Limits:
\[
\lim_{x \to 0} (-|x|) = 0, \qquad \lim_{x \to 0} |x| = 0
\]

**Conclusion:**
\[
\lim_{x \to 0} x\cos\left(\frac{1}{x}\right) = 0
\]
:::


::: {.example}
Fractional Squeeze

Evaluate the limit: 

\[
\lim_{x \to 0} \frac{x^2}{\sqrt{x^2 + 1}}
\]


**Solution**

\[
\sqrt{x^2 + 1} \ge 1
\]

So:
\[
0 \le \frac{x^2}{\sqrt{x^2+1}} \le x^2
\]

Taking limits:
\[
\lim_{x \to 0} 0 = 0, \qquad \lim_{x \to 0} x^2 = 0
\]

Therefore,
\[
\lim_{x \to 0} \frac{x^2}{\sqrt{x^2 + 1}} = 0
\]
:::


###  Putting It All Together

#### Conceptual Takeaways

- Limits behave **algebraically** like numbers.  
- Complicated limits can be broken into **simpler components**.  
- Not all limits can be evaluated by substitution.  
- Indeterminate forms like \( \frac{0}{0} \) require algebraic simplification.  
- The value of a function at a point **does not affect the limit**.  
- Some limits require:  
  - One-sided analysis  
  - Algebraic restructuring  
  - Bounding arguments (Squeeze Theorem)  

#### Skills Checklist

You should now be able to:

- Apply all **five limit laws**  
- Use **direct substitution**  
- Factor and simplify expressions  
- Rationalize expressions  
- Identify **indeterminate forms**  
- Compute **one-sided limits**  
- Detect **nonexistent limits**  
- Apply the **Squeeze Theorem**  
- Distinguish between function value and limit value  

#### Problems






















## The Precise Definition of a Limit 





















## Continuity 

In earlier sections, we observed that many limits can be found simply by **direct substitution**:
\[
\lim_{x \to a} f(x) = f(a).
\]
Functions with this property are called **continuous** at \( a \).  Intuitively, continuity means **no sudden jumps, breaks, or holes** in the graph.  A continuous process changes **gradually**, without interruption.


<div class="definition">
A function \( f \) is **continuous at a number \( a \)** if:
\[
\lim_{x \to a} f(x) = f(a).
\]
</div>


This definition requires **three conditions**:

1. \( f(a) \) is defined  
2. \( \lim_{x \to a} f(x) \) exists  
3. \( \lim_{x \to a} f(x) = f(a) \)  

If any of these fail, the function is **discontinuous at \( a \)**.


Intuitively, this means that:

- Small changes in \( x \) produce small changes in \( f(x) \)  
- No gaps, jumps, or holes  
- Graph can be drawn **without lifting your pen**  

---

### Types of Discontinuities

**1. Removable Discontinuity (Hole)**
Occurs when:

- Limit exists  
- Function value is missing or incorrect  

Example:
\[
f(x) = \frac{x^2-4}{x-2}
\]

Factor:
\[
f(x) = \frac{(x-2)(x+2)}{x-2} = x+2 \quad (x \ne 2)
\]

Limit exists, but \( f(2) \) undefined → removable discontinuity.


**2. Jump Discontinuity**
Left and right limits exist but are **not equal**.

Example: Greatest integer function \( f(x) = \lfloor x \rfloor \)  
Discontinuous at every integer.

**3. Infinite Discontinuity**
Function grows without bound.

Example:
\[
f(x) = \frac{1}{x}
\]
Discontinuous at \( x=0 \).

---

### One-Sided Continuity


<div class="definition">
A function is right-continuous at \( a \) if:
\[
\lim_{x \to a^+} f(x) = f(a)
\]

A function is left-continuous at \( a \) if:
\[
\lim_{x \to a^-} f(x) = f(a).
\]
</div>


<div class="definition">
A function is **continuous on an interval** if it is continuous at **every point** in the interval (with one-sided continuity at endpoints if needed).
</div>


<div class="theorem">
If \( f \) and \( g \) are continuous at \( a \), then the following are also continuous at \( a \):

- \( f+g \)  
- \( f-g \)  
- \( cf \)    
- \( fg \)  
- \( \frac{f}{g} \), if \( g(a) \ne 0 \)  
</div>

This leads to the following group of continuous functions.



<div class="theorem">
The following are continuous **everywhere on their domains**:

- Polynomials  
- Rational functions  
- Root functions  
- Trigonometric functions  
- Inverse trigonometric functions  
- Exponential functions  
- Logarithmic functions  
</div>

---

### Composite Functions

If \( t \) is continuous at \( a \) and \( f \) is continuous at \( t(a) \), then the composite function
\[
(f \circ t)(x) = f(t(x))
\]
is continuous at \( a \).


<div class="theorem">
A continuous function of a continuous function is continuous.
</div>


If a function is continuous at \( a \), then:
\[
\lim_{x \to a} f(x) = f(a).
\]
This allows **direct substitution** for limits.


::: {.example}
Direct Substitution

Evaluate the following limit:
\[
\lim_{x \to 2} (3x^2 - 5x + 1).
\]

**Solution**

Polynomial → continuous

\[
\lim_{x \to 2} (3x^2 - 5x + 1) = 3(2)^2 - 5(2) + 1 = 12 - 10 + 1 = 3.
\]
:::

::: {.example}
Rational Function

Evaluate the following limit:
\[
\lim_{x \to 1} \frac{x^2+1}{x+2}.
\]

**Solution**

Denominator ≠ 0 → continuous

\[
\lim_{x \to 1} \frac{x^2+1}{x+2} = \frac{1^2+1}{1+2} = \frac{2}{3}.
\]
:::

::: {.example}
Composite Function

Where is $h$ continuous?
\[
h(x) = \sin(x^2).
\]


**Solution**

- \( x^2 \) is continuous  
- \( \sin x \) is continuous  
→ \( \sin(x^2) \) is continuous everywhere
:::

::: {.example}
Detecting Discontinuity

Where is $f$ continuous?
\[
f(x) = \ln(x) + \frac{1}{x-1}.
\]


**Solution**


- \( \ln(x) \): domain \( x>0 \)  
- \( \frac{1}{x-1} \): undefined at \( x=1 \)

So discontinuous at \( x=1 \), continuous elsewhere on domain.
:::

---

### Intermediate Value Theorem (IVT)


<div class="definition">  **Intermediate Value Theorem (IVT)**:
If \( f \) is continuous on \( [a,b] \) and \( N \) is any number between \( f(a) \) and \( f(b) \), then there exists \( c \in (a,b) \) such that:
\[
f(c) = N.
\]
</div>

**Meaning:** A continuous function takes **every intermediate value**.


::: {.example}
Root Existence

Show that the equation
\[
f(x) = x^3 - 4x - 1 = 0
\]
has a root in \( (1,2) \).


**Solution:**
Compute:
\[
f(1) = -4 < 0, \qquad f(2) = 8 - 8 - 1 = -1 < 0
\]

Try interval:
\[
f(1.5) = 3.375 - 6 - 1 = -3.625 < 0
\]
\[
f(2) > 0
\]

Sign change → by IVT, at least one root exists in the interval.
:::

---

### Putting It All Together

#### Conceptual Takeaways

- Continuity links **limits and function values**  
- Discontinuity comes from:  
  - Undefined values  
  - Broken limits  
  - Mismatched limits and values  
- Most common functions are continuous on their domains  
- Algebra preserves continuity  
- Composition preserves continuity  
- Continuity guarantees **predictable behavior**  
- Continuous functions cannot “skip” values (IVT)  

---

### Skills You Should Be Able To Do

You should now be able to:

- Define continuity formally  
- Identify discontinuities algebraically and graphically  
- Classify types of discontinuities  
- Use continuity to evaluate limits  
- Apply direct substitution correctly  
- Analyze domains of functions  
- Determine continuity of composite functions  
- Use algebra of continuous functions  
- Apply the Intermediate Value Theorem  
- Prove existence of roots  
- Connect graphs, limits, and function values  
- Reason about function behavior without computation  

---

### Problems
























## Limits at Infinity; Horizontal Asymptotes


In earlier sections, we studied **infinite limits** where \( x \to a \) and the function values grow without bound.  In this section, we instead study what happens when:
\[
x \to \infty \quad \text{or} \quad x \to -\infty.
\]
That is, we analyze the **end behavior** of functions.


We investigate:

- What happens to \( f(x) \) when \( x \) becomes **very large**?  
- What happens to \( f(x) \) when \( x \) becomes **very negative**?  

Sometimes:

- \( f(x) \to L \) (approaches a finite number)  
- \( f(x) \to \infty \) or \( f(x) \to -\infty \)  
- \( f(x) \) oscillates and does not approach anything  

---

### Limits at Infinity


<div class="definition">
Let \( f \) be defined for large values of \( x \). Then:
\[
\lim_{x \to \infty} f(x) = L.
\]
means the values of \( f(x) \) can be made arbitrarily close to \( L \) by taking \( x \) sufficiently large.
</div>


Notation:
\[
f(x) \to L \quad \text{as } x \to \infty
\]


<div class="definition">
Let \( f \) be defined for large negative values of \( x \). Then:
\[
\lim_{x \to -\infty} f(x) = L
\]
means the values of \( f(x) \) approach \( L \) as \( x \) becomes large in the negative direction.
</div>


<div class="definition">
The line \( y = L \) is a **horizontal asymptote** of \( y = f(x) \) if:
\[
\lim_{x \to \infty} f(x) = L
\quad \text{or} \quad
\lim_{x \to -\infty} f(x) = L
\]
</div>


A function may have:

- One horizontal asymptote  
- Two horizontal asymptotes  
- No horizontal asymptotes  


::: {.example}
Reciprocal Function

\[
\lim_{x \to \infty} \frac{1}{x} = 0, \qquad
\lim_{x \to -\infty} \frac{1}{x} = 0
\]

So the horizontal asymptote is:

\[
y = 0
\]
:::

::: {.example}
Polynomial Growth

\[
\lim_{x \to \infty} x^3 = \infty, \qquad
\lim_{x \to -\infty} x^3 = -\infty
\]

No horizontal asymptote.
:::

::: {.example}
Exponential Function

\[
\lim_{x \to -\infty} e^x = 0, \qquad
\lim_{x \to \infty} e^x = \infty
\]

Horizontal asymptote:

\[
y = 0 \quad (\text{as } x \to -\infty)
\]
:::

::: {.example}
Trigonometric Oscillation

\[
\lim_{x \to \infty} \sin x \quad \text{does not exist}
\]

The function oscillates forever → no limit → no asymptote.
:::





<div class="note">
**Key Rule**

For a rational function:
\[
f(x) = \frac{P(x)}{Q(x)}
\]

Let:

- \( n = \deg(P) \)  
- \( m = \deg(Q) \)  

**Case 1: \( n < m \)**
\[
\lim_{x \to \pm\infty} f(x) = 0
\]
Horizontal asymptote: \( y = 0 \)

**Case 2: \( n = m \)**
\[
\lim_{x \to \pm\infty} f(x) = \frac{\text{leading coeff of } P}{\text{leading coeff of } Q}
\]
Horizontal asymptote: ratio of leading coefficients

**Case 3: \( n > m \)**
\[
\lim_{x \to \pm\infty} f(x) = \pm\infty
\]
No horizontal asymptote  
(possibly slant/oblique asymptote)
</div>


::: {.example}
Evaluate the following limit:

\[
\lim_{x \to \infty} \frac{3x^2 - 2x - 2}{5x^2 + 4x + 1}
\]


**Solution:**

Divide by \( x^2 \):

\[
\frac{3 - \frac{2}{x} - \frac{2}{x^2}}{5 + \frac{4}{x} + \frac{1}{x^2}}
\]

Take limits:

\[
\frac{3 - 0 - 0}{5 + 0 + 0} = \frac{3}{5}
\]

Horizontal asymptote:

\[
y = \frac{3}{5}
\]
:::


A function may have **two different horizontal asymptotes**:

\[
\lim_{x \to \infty} f(x) = L_1, \qquad
\lim_{x \to -\infty} f(x) = L_2
\]

Example:

\[
\lim_{x \to \infty} \arctan x = \frac{\pi}{2}, \qquad
\lim_{x \to -\infty} \arctan x = -\frac{\pi}{2}
\]

Two horizontal asymptotes:
\[
y = \frac{\pi}{2}, \qquad y = -\frac{\pi}{2}
\]

---

### Infinite Limits at Infinity


<div class="definition">
\[
\lim_{x \to \infty} f(x) = \infty
\]
means the values of \( f(x) \) grow arbitrarily large as \( x \to \infty \).

Similarly:
\[
\lim_{x \to -\infty} f(x) = -\infty
\]

means the values of \( f(x) \) grow arbitrarily large and negative as \( x \to - \infty \).
</div>





::: {.example}
Radical Example

\[
\lim_{x \to \infty} \left(\sqrt{x^2 + 1} - x \right)
\]

**Solution:**

Multiply by conjugate:
\[
= \lim_{x \to \infty} 
\frac{1}{\sqrt{x^2 + 1} + x}
\]

Denominator grows → expression goes to 0

\[
\lim_{x \to \infty} \left(\sqrt{x^2 + 1} - x \right) = 0.
\]
:::


---

### Graph Interpretation

- Horizontal asymptote → graph flattens toward a line  
- Infinite limit → graph grows upward/downward without bound  
- Oscillation → no limit  
- Different ends → different asymptotes  

---

### Putting It All Together

#### Conceptual Takeaways

- Limits at infinity describe **long-term behavior**  
- Horizontal asymptotes describe **end behavior lines**  
- A function does not need to touch its asymptote  
- A function may cross a horizontal asymptote  
- Rational functions have predictable asymptotic behavior  
- Degree comparison determines asymptotes  
- Some functions approach constants (e.g., \( \frac{1}{x}, e^{-x}, \arctan x \))  
- Some grow without bound (polynomials, exponentials)  
- Some never settle (trigonometric oscillation)  
- End behavior helps with **graphing and modeling**  

---

#### Skills You Should Be Able To Do

You should now be able to:

- Interpret limits at infinity conceptually  
- Compute \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \)  
- Identify horizontal asymptotes  
- Classify infinite vs finite limits  
- Analyze rational functions using degree comparison  
- Use algebraic division by powers of \( x \)  
- Apply conjugates for radical limits  
- Recognize oscillating functions  
- Determine end behavior of polynomials  
- Connect limits to graph shape  
- Sketch graphs using asymptotes and intercepts 

####  Problems































## Derivatives and Rates of Change 

The problems of finding the **tangent line** to a curve and finding the **velocity** of a moving object both involve the same special type of limit.  
This type of limit is called a **derivative**.

Derivatives are fundamental because they represent:

- Slopes of curves  
- Velocities of moving objects  
- Rates of change in science, engineering, economics, biology, and social sciences  

---

### Tangent Lines and Limits

Let a curve \( C \) be defined by:
\[
y = f(x)
\]
and let \( P(a, f(a)) \) be a point on the curve.  Choose a nearby point \( Q(x, f(x)) \), where \( x \neq a \).

The slope of the **secant line** \( PQ \) is:
\[
m_{PQ} = \frac{f(x) - f(a)}{x - a}.
\]
As \( Q \to P \) (that is, as \( x \to a \)), the secant line approaches the **tangent line**.


<div class="definition">
The tangent line to \( y = f(x) \) at the point \( (a, f(a)) \) is the line through \( (a, f(a)) \) with slope:
\[
m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a},
\]
if this limit exists.
</div>




<div class="definition">
Let:
\[
h = x - a \Rightarrow x = a + h.
\]
Then the slope becomes:
\[
m = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.
\]
This form is often easier to compute.
</div>


::: {.example}
Tangent Line via Definition

Find the tangent line to \( y = x^2 \) at \( (1,1) \).

**Solution:**

\[
m = \lim_{x \to 1} \frac{x^2 - 1}{x - 1}
\]

Factor:

\[
= \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1}
\]

\[
= \lim_{x \to 1} (x+1) = 2
\]

Slope \( m = 2 \)

Equation (point-slope form):

\[
y - 1 = 2(x - 1)
\]

\[
y = 2x - 1
\]
:::


---

### Derivatives

Because this type of limit appears everywhere, it is given a special name.


<div class="definition">
The **derivative** of \( f \) at \( a \), denoted \( f'(a) \), is:
\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.
\]
Equivalently:
\[
f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.
\]
</div>

Geometrically, we've seen that $f^{\prime}$ represents the slope of a tangent line to the graph of $f$.
\[
f'(a) = \text{slope of the tangent line to } y=f(x) \text{ at } x=a
\]


If \( s = f(t) \) is the **position function** of an object:
\[
\text{Average velocity} = \frac{f(t+h)-f(t)}{h}
\]
Taking the limit:
\[
v(t) = \lim_{h \to 0} \frac{f(t+h)-f(t)}{h}
\]
So, the velocity of an object is:
\[
v(t) = f'(t)
\]



::: {.example}
Velocity from Position

Let:
\[
s(t) = 4.9t^2.
\]
Find the velocity.

**Solution:**

Compute velocity:
\[
v(t) = \lim_{h \to 0} \frac{4.9(t+h)^2 - 4.9t^2}{h}
\]
\[
= \lim_{h \to 0} \frac{4.9(t^2 + 2th + h^2 - t^2)}{h}
\]
\[
= \lim_{h \to 0} \frac{4.9(2th + h^2)}{h}
\]
\[
= \lim_{h \to 0} 4.9(2t + h)
\]
\[
= 9.8t
\]

So:

\[
v(t) = 9.8t.
\]
:::




::: {.example}
Derivative at a Point

Let:
\[
f(x) = x^2 - 8x + 9
\]
Find \( f'(a) \).

**Solution:**
\[
f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}
\]
\[
= \lim_{h \to 0} \frac{(a+h)^2 - 8(a+h) + 9 - (a^2 - 8a + 9)}{h}
\]
\[
= \lim_{h \to 0} \frac{2ah + h^2 - 8h}{h}
\]
\[
= \lim_{h \to 0} (2a + h - 8)
\]
\[
= 2a - 8.
\]

So:
\[
f'(a) = 2a - 8.
\]
:::

---

### Tangent Line Using Derivatives

The equation of the tangent line at \( (a, f(a)) \) is:

\[
y - f(a) = f'(a)(x - a)
\]



::: {.example}
Tangent Line Using Derivative

Let:
\[
f(x) = x^2 - 8x + 9
\]
Find the tangent line to $f$ at \( x=3 \).

**Solution:**

\[
f'(x) = 2x - 8
\Rightarrow f'(3) = -2
\]
At the point $x = 3$,
\[
f(3) = 9 - 24 + 9 = -6
\]
Therefore, the equation is
\[
y + 6 = -2(x - 3)
\]
or
\[
y = -2x.
\]
:::


---

### Rates of Change

If \( y = f(x) \), then:

- Average rate of change:
\[
\frac{\Delta y}{\Delta x} = \frac{f(x_2)-f(x_1)}{x_2-x_1}
\]

- Instantaneous rate of change:
\[
\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = f'(x)
\]


The derivative represents:

- Slope of tangent line  
- Velocity (physics)  
- Marginal cost (economics)  
- Growth rate (biology)  
- Reaction rate (chemistry)  
- Power (physics)  
- Population growth  
- Debt growth  
- Production efficiency  
- Learning curves  


::: {.example}
Real-World Interpretation

Let \( C(x) \) be the cost (in dollars) to produce \( x \) units.

Then:
\[
C'(x) = \text{marginal cost}
\]
Units:
\[
\frac{\text{dollars}}{\text{unit}}
\]

If:
\[
C'(1000) = 9
\]

Then, the cost of producing the next unit at 1000 units is approximately \$9.
:::



---

### Putting It All Together

#### Conceptual Takeaways

- Derivatives arise from limits of secant slopes  
- Tangent lines are limiting positions of secant lines  
- Derivatives unify geometry and motion  
- Derivatives measure **local behavior**  
- Zooming in makes curves look linear  
- Velocity is a derivative  
- Rates of change are derivatives  
- Tangents and rates of change are the same mathematically  
- Derivatives apply across disciplines  
- Slopes encode physical meaning  
- Units of derivatives matter  
- Local linearity is fundamental  
- Calculus connects algebra, geometry, and motion  



#### Skills You Should Be Able To Do

You should now be able to:

- Define a derivative using limits  
- Compute derivatives from first principles  
- Interpret derivatives geometrically  
- Interpret derivatives physically  
- Find tangent line equations  
- Compute slopes of tangent lines  
- Use point-slope form correctly  
- Model velocity using derivatives  
- Interpret units of derivatives  
- Explain rates of change  
- Distinguish average vs instantaneous change  
- Apply derivatives to real-world contexts  
- Connect graphs to derivatives  
- Understand local linearization  
- Use both \( h \to 0 \) and \( x \to a \) definitions  
- Explain the meaning of a derivative in words  
- Interpret derivative values contextually  

---


####  Problems

































## The Derivative as a Function 

Previously, we defined the derivative of a function \( f \) at a fixed number \( a \) by
\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.
\]
Now we allow the point \( a \) to vary and replace it with a variable \( x \). This gives the **derivative as a function**:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
\]

This means:

- The derivative is **not just a number** at one point.  
- The derivative is a **new function** that assigns a slope value to every input \( x \) where the limit exists.  


For each value of \( x \) we can interpret \( f'(x) \) as the **slope of the tangent line** to the graph of \( y=f(x) \) at the point \( (x,f(x)) \).  Geometrically, the graph of \( f' \) records the slopes of \( f \).


::: {.example}
Polynomial Function

**Problem:**  
Let \( f(x) = x^3 - x \). Find \( f'(x) \).

**Solution:**

\begin{align*} 
f'(x) &= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\
&= \lim_{h \to 0} \frac{(x+h)^3 - (x+h) - (x^3 - x)}{h} \\
&= \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x - h - x^3 + x}{h} \\
&= \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 - h}{h} \\
&= \lim_{h \to 0} \left( 3x^2 + 3xh + h^2 - 1 \right) \\
&= 3x^2 - 1.
\end{align*}
:::

We can interpret the slope of $f$ in terms of increasing and decreasing:

- Where \( f'(x) > 0 \): function increasing  
- Where \( f'(x) < 0 \): function decreasing  
- Where \( f'(x) = 0 \): horizontal tangent lines  

---


::: {.example}
Square Root Function and Domain of the Derivative

**Problem:**  
Let \( f(x) = \sqrt{x} \). Find \( f'(x) \) and its domain.

**Solution:**

\[
f'(x) = \lim_{h \to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}
\]

Multiply by the conjugate:
\begin{align*}
f'(x) &= \lim_{h \to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}\\
& = \lim_{h \to 0} \frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}{h(\sqrt{x+h}+\sqrt{x})} \\
& = \lim_{h \to 0} \frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})} \\
& = \lim_{h \to 0} \frac{h}{h(\sqrt{x+h}+\sqrt{x})} \\
& = \lim_{h \to 0} \frac{1}{\sqrt{x+h}+\sqrt{x}} \\
& = \frac{1}{2\sqrt{x}}.
\end{align*}


**Domain:**  
Since division by \( \sqrt{x} \) requires \( x>0 \) (because we take roots of positive numbers and we cannot divide by 0),
\[
\text{Domain of } f'(x): (0,\infty)
\]
:::



<div class="definition">
A function \( f \) is **differentiable at \( a \)** if \( f'(a) \) exists.  It is **differentiable on an interval** if the derivative exists at every point in that interval.
</div>


::: {.example}
Absolute Value Function

Let \( f(x) = |x| \).  Determine where $f$ is differentiable.


**Solution:**

Piecewise form:
\[
f(x) =
\begin{cases}
x, & x>0 \\
-x, & x<0
\end{cases}
\]

Derivative:
\[
f'(x) =
\begin{cases}
1, & x>0 \\
-1, & x<0
\end{cases}
\]
At \( x=0 \):

-  Left-hand limit \( \to -1 \)  
-  Right-hand limit \( \to 1 \)  

Since the limits are not equal:
\[
f'(0) \text{ does not exist}.
\]
So \( f(x)=|x| \) is **not differentiable at \( x=0 \)**.
:::


---

### Continuity vs Differentiability


<div class="definition">
If \( f \) is differentiable at \( a \), then \( f \) is continuous at \( a \).
</div>


**Important:**  The converse is false:

> A function can be continuous but not differentiable.


How a Function Can Fail to Be Differentiable?  

A function is **not differentiable** at a point if:

1. **Corner / kink** (e.g. \( |x| \) at \( x=0 \))  
2. **Discontinuity** (jump, hole, asymptote)  
3. **Vertical tangent** (slope becomes infinite)  

---

### Higher Derivatives

If \( f \) is differentiable, then its derivative \( f' \) is also a function and may itself be differentiable.


<div class="note">

- First derivative:  
  \[
  f'(x) = \frac{dy}{dx}
  \]

- Second derivative:  
  \[
  f''(x) = \frac{d^2 y}{dx^2}
  \]

- Third derivative:
  \[
  f'''(x) = \frac{d^3 y}{dx^3}
  \]

- \( n \)-th derivative:
  \[
  f^{(n)}(x) = \frac{d^n y}{dx^n}
  \]
</div>



::: {.example}
Second Derivative

Let \( f(x) = x^3 - x \).  Find the second derivative of $f$.

**Solution:**

\[
f'(x) = 3x^2 - 1
\]
\[
f''(x) = 6x
\]
:::


We can interpret derivatives in the following way:

- \( f'(x) \): slope of the curve  
- \( f''(x) \): rate of change of the slope  
- \( f''(x) > 0 \): slope increasing  
- \( f''(x) < 0 \): slope decreasing  




Physically, if \( s(t) \) is the position of a function:

- Velocity:  
  \[
  v(t) = s'(t)
  \]  
- Acceleration:  
  \[
  a(t) = s''(t)
  \]  
- Jerk:
  \[
  j(t) = s'''(t)
  \]  

Each derivative represents a **rate of change of the previous quantity**.

---

### Putting It All Together

The derivative transforms a function into a **new function** that describes change.  It unifies geometry (slopes), physics (motion), economics (marginal change), and data science (rates, trends, dynamics) into one mathematical concept:

> **The derivative is the mathematics of change itself.**

#### Conceptual Takeaways

- The derivative is a **function**, not just a number.  
- \( f'(x) \) represents the **slope of the tangent line** at every point.  
- The derivative encodes **how the function changes**.  
- Differentiability implies continuity.  
- A function may be continuous but not differentiable.  
- Higher derivatives measure **changes of changes**:  
  - slope → curvature → acceleration → jerk → etc.  




#### Skills You Should Be Able To Do

By the end of this section, you should be able to:

- Compute derivatives using the **limit definition**  
- Interpret \( f'(x) \) geometrically  
- Interpret \( f'(x) \) physically (rate of change)  
- Identify where a function is:  
  - differentiable  
  - not differentiable  
- Explain why differentiability implies continuity  
- Find second and higher derivatives  
- Interpret second derivatives conceptually  
- Connect:  
  - \( f \) → position   
  - \( f' \) → velocity  
  - \( f'' \) → acceleration  
  - \( f''' \) → jerk  





####  Problems