Calculus I · Chapter 3
Chapter 3: The Derivative
Limits · One-sided & infinite limits · Continuity & the IVT · Rates of change · Definition of the derivative · Graphical differentiation
This is the chapter where calculus truly begins. The central problem: a straight line has one slope everywhere, but a curve bends, its steepness changes from point to point. How do we measure the slope at a single instant? The answer is the limit, and the slope it produces is the derivative, the most important object in the entire course.
Section 3.1
Limits
The Intuitive Idea
Definition: Limit (informal)
We write
$$\lim_{x \to a} f(x) = L$$
if $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to $a$ from both sides, without necessarily equaling $a$. The limit is about where the function is heading, not whether it arrives. The value $f(a)$ may differ from $L$, or may not even exist.
One-Sided Limits and the Existence Rule
$\displaystyle\lim_{x\to a^-}f(x)$ approaches from the left; $\displaystyle\lim_{x\to a^+}f(x)$ from the right. The two-sided limit exists only if both agree:
$$\lim_{x\to a}f(x) = L \iff \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) = L.$$
If the two sides disagree, the limit does not exist.
Approaching $x = a$ from both sides. Drag the slider to move the test point toward $a$. The left (orange) and right (blue) values both close in on the same height $L$, even though the function has a hole exactly at $a$. The limit ignores the hole.
Limit Laws & Techniques
Limit Laws
If $\lim_{x\to a}f(x)$ and $\lim_{x\to a}g(x)$ both exist, then limits distribute over sums, differences, products, quotients (denominator $\neq 0$), powers, and roots. In particular, for any polynomial or rational function $f$ that is defined at $a$:
$$\lim_{x\to a} f(x) = f(a) \quad (\text{direct substitution}).$$
⚠ The $\tfrac{0}{0}$ Indeterminate Form
If substitution gives $\tfrac{0}{0}$, the limit is not automatically undefined, it's indeterminate. You must do algebra first: factor and cancel, rationalize, or simplify, then substitute. A $\tfrac{0}{0}$ form is the signal that a hole is hiding a finite limit.
📘 Example 3.1: Factoring Through a 0/0 Form
Evaluate $\displaystyle\lim_{x\to 3}\frac{x^2 - 9}{x - 3}$.
Substitution gives 0/0, factor instead
$$\frac{x^2 - 9}{x - 3} = \frac{(x-3)(x+3)}{x-3} = x + 3 \quad (x \neq 3).$$
Now substitute: $\lim_{x\to 3}(x + 3) = 6$.
$\displaystyle\lim_{x\to 3}\frac{x^2-9}{x-3} = 6$
📘 Example 3.2: Rationalizing
Evaluate $\displaystyle\lim_{x\to 0}\frac{\sqrt{x+4}-2}{x}$.
Multiply by the conjugate
$$\frac{\sqrt{x+4}-2}{x}\cdot\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2} = \frac{(x+4)-4}{x(\sqrt{x+4}+2)} = \frac{x}{x(\sqrt{x+4}+2)} = \frac{1}{\sqrt{x+4}+2}.$$
Substitute $x = 0$: $\dfrac{1}{\sqrt{4}+2} = \dfrac{1}{4}$.
$\displaystyle\lim_{x\to 0}\frac{\sqrt{x+4}-2}{x} = \frac{1}{4}$
Infinite Limits & Limits at Infinity
Two Different Ideas
- Infinite limit, $\lim_{x\to a}f(x) = \pm\infty$: the output grows without bound near a finite $a$. This signals a vertical asymptote at $x = a$.
- Limit at infinity, $\lim_{x\to\infty}f(x) = L$: the output settles toward $L$ as $x$ grows. This signals a horizontal asymptote $y = L$.
Rational Limits at Infinity: Divide by the Highest Power
For $\lim_{x\to\infty}\dfrac{p(x)}{q(x)}$, divide numerator and denominator by the highest power of $x$ in the denominator. Using $\lim_{x\to\infty}\tfrac{1}{x^n} = 0$, the answer is the ratio of leading coefficients (equal degrees), $0$ (smaller numerator), or $\pm\infty$ (larger numerator).
📘 Example 3.3: Limit at Infinity
Evaluate $\displaystyle\lim_{x\to\infty}\frac{3x^2 - 5x + 1}{2x^2 + 7}$.
Divide every term by $x^2$
$$\frac{3 - \tfrac{5}{x} + \tfrac{1}{x^2}}{2 + \tfrac{7}{x^2}} \;\xrightarrow{x\to\infty}\; \frac{3 - 0 + 0}{2 + 0} = \frac{3}{2}.$$
$\displaystyle\lim_{x\to\infty}\frac{3x^2-5x+1}{2x^2+7} = \frac{3}{2}$, a horizontal asymptote
Section 3.2
Continuity
Continuity at a Point
Definition: Continuous at a Point
$f$ is continuous at $x = a$ if all three conditions hold:
- $f(a)$ is defined,
- $\displaystyle\lim_{x\to a}f(x)$ exists, and
- $\displaystyle\lim_{x\to a}f(x) = f(a)$.
Three ways continuity fails. Click to cycle: a removable discontinuity (hole), a jump (one-sided limits disagree), and an infinite discontinuity (vertical asymptote). The green curve at the end is continuous, no breaks.
📘 Example 3.4: Testing Continuity of a Piecewise Function
Is $f(x) = \begin{cases} x^2 + 1 & x \leq 1 \\ 3x - 1 & x > 1 \end{cases}$ continuous at $x = 1$?
- $f(1) = 1^2 + 1 = 2$, defined.
- Left limit: $\lim_{x\to1^-}(x^2+1) = 2$. Right limit: $\lim_{x\to1^+}(3x-1) = 2$. They agree, so the limit is $2$.
- Limit equals $f(1)$: $2 = 2$. ✓
The Intermediate Value Theorem
Theorem: Intermediate Value Theorem (IVT)
If $f$ is continuous on $[a, b]$ and $N$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c$ in $(a, b)$ with $f(c) = N$. A continuous function cannot skip values. The most common use: if $f(a)$ and $f(b)$ have opposite signs, then $f$ has a root somewhere between $a$ and $b$.
📘 Example 3.5: Locating a Root with the IVT
Show that $f(x) = x^3 + x - 1$ has a root between $0$ and $1$.
Check the signs at the endpoints
$f(0) = -1 < 0$ and $f(1) = 1 > 0$. Since $f$ is continuous (a polynomial) and changes sign, the IVT guarantees a $c$ in $(0,1)$ with $f(c) = 0$.
A root exists in $(0, 1)$
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Section 3.3
Rates of Change
Average vs. Instantaneous Rate of Change
The average rate of change of $f$ over $[a, b]$ is the slope of the secant line:
$$\frac{f(b) - f(a)}{b - a}.$$
The instantaneous rate of change at $x = a$ is the limit as the interval shrinks to a point, the slope of the tangent line:
$$\lim_{h\to 0}\frac{f(a+h) - f(a)}{h}.$$
This limit is the derivative. Velocity is the instantaneous rate of change of position; marginal cost is the instantaneous rate of change of cost.
📘 Example 3.6: Average Rate of Change
A population is $P(t) = 100 + 20t - t^2$ (thousands, $t$ in years). Find the average rate of change from $t = 1$ to $t = 4$.
Slope of the secant
$P(1) = 100 + 20 - 1 = 119$; $P(4) = 100 + 80 - 16 = 164$.
$$\frac{164 - 119}{4 - 1} = \frac{45}{3} = 15 \text{ thousand per year}.$$
Average rate of change $= 15{,}000$ people per year
Section 3.4
Definition of the Derivative
Definition: The Derivative
The derivative of $f$ at $x$ is
$$f'(x) = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h},$$
provided the limit exists. It gives the slope of the tangent line to $y = f(x)$ at the point $(x, f(x))$, and the instantaneous rate of change of $f$. Notations: $f'(x)$, $\dfrac{dy}{dx}$, $\dfrac{d}{dx}f(x)$, $y'$.
The secant becomes the tangent. As $h \to 0$, the second point slides toward the fixed point and the secant line (orange) rotates into the tangent line (blue). The secant slope $\frac{f(x+h)-f(x)}{h}$ converges to the derivative $f'(x)$. This single picture is the definition of the derivative.
secant slope = ,
The Four-Step Process
To compute $f'(x)$ from the definition:
- Form $f(x + h)$.
- Compute the difference $f(x+h) - f(x)$.
- Divide by $h$ to get the difference quotient $\dfrac{f(x+h)-f(x)}{h}$, and simplify so $h$ cancels from the denominator.
- Take the limit as $h \to 0$.
📘 Example 3.7: Derivative from the Definition
Use the definition to find $f'(x)$ for $f(x) = x^2 - 3x$.
- $f(x+h) = (x+h)^2 - 3(x+h) = x^2 + 2xh + h^2 - 3x - 3h$.
- $f(x+h) - f(x) = 2xh + h^2 - 3h = h(2x + h - 3)$.
- $\dfrac{f(x+h)-f(x)}{h} = 2x + h - 3$.
- $f'(x) = \lim_{h\to0}(2x + h - 3) = 2x - 3$.
⚠ Differentiable ⇒ Continuous (not the reverse)
If $f$ is differentiable at $a$, it is continuous at $a$. But continuity does not guarantee differentiability: $f(x) = |x|$ is continuous at $0$ yet has no derivative there, the sharp corner has no single tangent slope. Derivatives also fail at vertical tangents and breaks.
Section 3.5
Graphical Differentiation
Often you have only a graph, not a formula. You can still sketch the derivative by reading slopes off the original curve and plotting them as heights.
Reading $f'$ from the Graph of $f$
| Where $f$ is… | The slope $f'$ is… |
|---|---|
| rising (increasing) | positive, graph of $f'$ above the axis |
| falling (decreasing) | negative, graph of $f'$ below the axis |
| at a peak or valley (horizontal tangent) | zero, graph of $f'$ crosses the axis |
| steeper | $f'$ is farther from the axis |
Top: the function $f$. Bottom: its derivative $f'$. The moving dot reads the tangent slope on $f$ and plots it as a height on $f'$. Notice $f'$ crosses zero exactly where $f$ has a peak or valley.
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Looking Ahead
Computing every derivative from the four-step limit process is slow. Chapter 4 develops shortcut rules, power, product, quotient, chain, that make differentiation fast and mechanical, including the beautifully simple derivatives of $e^x$ and $\ln x$.