Calculus 1A: Limits, Derivatives, and Rates of Change – Study Notes

Limits and Continuity

Understanding Limits

Limits are fundamental to calculus, describing the behavior of a function as the input approaches a particular value. They are essential for defining derivatives and continuity.

• Limit Definition: The limit of a function f(x) as x approaches a is the value that f(x) gets closer to as x gets closer to a.

• Notation: $\lim_{x \to a} f(x)$

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

• Infinite Limits: If f(x) increases or decreases without bound as x approaches a, the limit is $\infty$ or $-\infty$.

• Does Not Exist (DNE): If the left and right limits are not equal, or the function oscillates infinitely, the limit does not exist.

Continuity and Differentiability

A function is continuous at a point if its limit exists at that point and equals the function's value. Differentiability requires continuity and a well-defined tangent (no sharp corners or cusps).

• Continuous at a: $\lim_{x \to a} f(x) = f(a)$

• Differentiable at a: The derivative $f'(a)$ exists.

• Relationship: Differentiability implies continuity, but continuity does not guarantee differentiability.

Table: Limits, Continuity, and Differentiability

a	$\lim_{x \to a^-} f(x)$	$\lim_{x \to a^+} f(x)$	$\lim_{x \to a} f(x)$	$f(a)$	Continuous at a?	Differentiable at a?
-1	2	2	2	2	y	y
0	0	0	0	0	y	y
2	-2	2	DNE	2	n	n
4	0	4	DNE	0	n	n

y = yes, n = no

Derivatives and the Definition of the Derivative

Limit Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change or the slope of the tangent line at that point. It is defined as:

• Definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• This formula calculates the slope of the secant line as the two points get infinitely close, becoming the tangent line.

Example: Compute the Derivative Using the Limit Definition

• Given $f(t) = t^3 + 5t^2 - t$

• Compute $f'(t)$ using the limit definition:

$\lim_{h \to 0} \frac{(t+h)^3 + 5(t+h)^2 - (t^3 + 5t^2 - t)}{h}$

• Expand and simplify the numerator, cancel $h$, and take the limit as $h \to 0$ to find $f'(t) = 3t^2 + 10t - 1$.

Rates of Change: Average and Instantaneous

Average Rate of Change

The average rate of change of a function over an interval $[a, b]$ is the slope of the secant line connecting $(a, f(a))$ and $(b, f(b))$.

• Formula: $\frac{f(b) - f(a)}{b - a}$

• Example: For $x = 0$ to $x = 1$, if $f(1) = 1$ and $f(0) = 0$, then average rate of change is $\frac{1-0}{1-0} = 1$.

Instantaneous Rate of Change

The instantaneous rate of change at a point is the derivative at that point, representing the slope of the tangent line.

• Example: If the tangent at $x = 2$ has a rise of $-0.8$ and a run of $0.6$, then the instantaneous rate is $\frac{-0.8}{0.6} = -\frac{4}{3}$.

Secant and Tangent Lines

• The secant line connects two points on a curve; its slope is the average rate of change.

• The tangent line touches the curve at one point; its slope is the instantaneous rate of change.

• As the two points of the secant line get closer, the secant line approaches the tangent line.

Graphical Interpretation of Derivatives

Relating $f(x)$ and $f'(x)$ Graphs

• The graph of $f(x)$ shows the function's values; the graph of $f'(x)$ shows the slopes of the tangent lines to $f(x)$ at each point.

• To estimate $f'(a)$ from $f(x)$, draw the tangent at $x = a$ and find its slope.

• To estimate $f(a)$ from $f'(x)$, look for the value of $f(x)$ at $x = a$ on the original function's graph.

Rules for Differentiation

Basic Derivative Rules

• Constant Rule: $\frac{d}{dx} (c) = 0$

• Power Rule: $\frac{d}{dx} (x^n) = n x^{n-1}$

• Sum Rule: $\frac{d}{dx} (f(x) + g(x)) = f'(x) + g'(x)$

• Product Rule: $\frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Examples of Differentiation

• Example 1: $f(t) = 3$ $f'(t) = 0$

• Example 2: $g(t) = e^{-t}$ $g'(t) = -e^{-t}$

• Example 3: $h(t) = \sqrt{t} - t^{1/2}$ $h'(t) = \frac{1}{2} t^{-1/2} - \frac{1}{2} t^{-1/2} = 0$

• Example 4: $f(t) = (t-1)(t^2+4)$ $f'(t) = 3t^2 - t$

• Example 5 (Quotient Rule): $g(x) = \frac{x}{x-3}$ $g'(x) = \frac{(x-3)\cdot 1 - x \cdot 1}{(x-3)^2} = \frac{-3}{(x-3)^2}$

• Example 6 (Chain Rule): $f(t) = e^{1-t}$ $f'(t) = -e^{1-t}$

• Example 7 (Chain Rule): $f(t) = [\sin(t^2)]^5$ $f'(t) = 5[\sin(t^2)]^4 \cdot \cos(t^2) \cdot 2t$

Summary Table: Differentiation Rules and Examples

Function	Rule Used	Derivative
$f(t) = 3$	Constant Rule	$0$
$g(t) = e^{-t}$	Chain Rule	$-e^{-t}$
$h(t) = \sqrt{t} - t^{1/2}$	Power Rule	$0$
$f(t) = (t-1)(t^2+4)$	Product Rule	$3t^2 - t$
$g(x) = \frac{x}{x-3}$	Quotient Rule	$\frac{-3}{(x-3)^2}$
$f(t) = e^{1-t}$	Chain Rule	$-e^{1-t}$
$f(t) = [\sin(t^2)]^5$	Chain Rule	$5[\sin(t^2)]^4 \cdot \cos(t^2) \cdot 2t$

Key Takeaways

• Limits are foundational for defining derivatives and continuity.

• The derivative measures instantaneous rate of change and is defined via a limit.

• Average rate of change is the slope of a secant line; instantaneous rate is the slope of the tangent.

• Continuity is required for differentiability, but not vice versa.

• Mastery of differentiation rules (power, product, quotient, chain) is essential for calculus success.