Calculus I Study Notes: Limits, Continuity, and Introduction to Derivatives

Limits and Instantaneous Velocity

Understanding Instantaneous Velocity and Tangent Lines

The concept of instantaneous velocity is foundational in calculus, relating to the rate of change of position with respect to time. It is closely connected to the slope of the tangent line to a function's graph at a specific point.

• Instantaneous Velocity: The velocity of an object at a specific moment, found by taking the derivative of the position function.

• Tangent Line Slope: The slope of the tangent line to the graph of a function at a point is the derivative at that point.

• Parallel Concepts: Both involve finding the limit of the average rate of change as the interval approaches zero.

• Formula: $v_{inst} = \lim_{h \to 0} \frac{s(t+h) - s(t)}{h}$

• Example: For $s(t) = -16t^2 + 128t$, the instantaneous velocity at $t$ is $s'(t)$.

Average Velocity and Position Functions

Calculating Average Velocity Over Intervals

Average velocity over an interval is the change in position divided by the change in time.

• Definition: $v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

• Application: For $s(t) = -16t^2 + 128t$, calculate $v_{avg}$ over intervals [1,4], [1,3], [1,2].

• Example: $v_{avg}$ on [1,4]: $\frac{s(4) - s(1)}{4-1}$

Conjecture About Instantaneous Velocity

• By making a table of average velocities over shrinking intervals, one can conjecture the instantaneous velocity at a point.

• Example: For $s(t) = -16t^2 + 80t + 60$, compute $v_{avg}$ as $t$ approaches 3.

Limits: Numerical, Algebraic, and Graphical Approaches

Evaluating Limits Numerically and Algebraically

Limits describe the behavior of a function as the input approaches a particular value.

• Definition: $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$.

• Techniques: Direct substitution, factoring, rationalizing, and using special limit laws.

• Examples:

  • $\lim_{x \to 4} (2x + 5)$

  • $\lim_{x \to 2} \frac{3x^2 - 8x + 4}{x-2}$

  • $\lim_{x \to 0} \frac{\sin x}{x}$

Evaluating Limits from Graphs

Graphical analysis allows for the estimation of limits by observing the behavior of the function near the point of interest.

• Identify left-hand and right-hand limits.

• Check for jumps, holes, or asymptotes.

• Example: Use the graph to find $\lim_{x \to a} f(x)$ and $f(a)$ for various $a$.

Types of Discontinuity and Continuity

Identifying Discontinuities

A function is discontinuous at points where it is not defined, has a jump, or an infinite limit.

• Removable Discontinuity: A hole in the graph; can be fixed by redefining $f(a)$.

• Non-removable Discontinuity: Jumps or infinite asymptotes; cannot be fixed by redefining $f(a)$.

• Example: $f(x) = \frac{x^2+2x}{x+1}$ has a removable discontinuity at $x = -1$.

Intervals of Continuity

• State intervals where the function is continuous using interval notation.

• Example: If $f(x)$ is discontinuous at $x = 2$, then $(-\infty, 2) \cup (2, \infty)$ is the interval of continuity.

Continuity Checklist

• $f(a)$ is defined.

• $\lim_{x \to a} f(x)$ exists.

• $\lim_{x \to a} f(x) = f(a)$

Intermediate Value Theorem (IVT)

Statement and Application

The IVT guarantees that if a function is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = N$.

• Example: Show $25 - 8x^2 - x^3$ has a solution on $[-2,4]$ by checking $f(-2)$ and $f(4)$ and confirming a sign change.

Introduction to Derivatives

Limit Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change and is defined by a limit.

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

• Application: Find $f'(x)$ for $f(x) = 3x^2 + 2x - 10$ at $x = 1$.

• Tangent Line Equation: $y = f'(a)(x-a) + f(a)$

Notation for Derivatives

• $f'(x)$

• $\frac{df}{dx}$

• $\frac{dy}{dx}$

• $D_x f(x)$

Graphical Interpretation of Derivatives

The graph of $f'(x)$ can be sketched from $f(x)$ by analyzing slopes, intervals of increase/decrease, and points of non-differentiability.

• Where $f'(x) > 0$: $f(x)$ is increasing.

• Where $f'(x) < 0$: $f(x)$ is decreasing.

• Where $f'(x) = 0$: $f(x)$ has a horizontal tangent (possible local max/min).

• Non-differentiable Points: Corners, cusps, or discontinuities in $f(x)$.

Summary Table: Types of Discontinuity

Type	Description	Removable?	Example
Removable	Hole in the graph	Yes	$f(x) = \frac{x^2-1}{x-1}$ at $x=1$
Jump	Sudden change in value	No	Piecewise function with different values at $x=a$
Infinite	Vertical asymptote	No	$f(x) = \frac{1}{x}$ at $x=0$

Summary Table: Limit Laws

Law	Formula	Description
Sum Law	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$	Limit of sum is sum of limits
Product Law	$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$	Limit of product is product of limits
Quotient Law	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ (if denominator ≠ 0)	Limit of quotient is quotient of limits

Additional info:

• Some graphs and tables referenced in the original material are not fully reproduced here, but the main concepts and methods are described.

• Students should practice evaluating limits both algebraically and graphically, and understand the connection between average and instantaneous rates of change.

• Continuity and differentiability are key concepts for further study in calculus.