Calculus I: Limits, Rates of Change, and the Slope of a Curve – Study Notes

Limits and Continuity

Average Rates of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. It is a foundational concept for understanding derivatives and instantaneous rates of change.

• Definition: For a function f(x) over the interval [a, b], the average rate of change is given by: $\frac{f(b) - f(a)}{b - a}$

• Interpretation: This represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the graph of the function.

• Example: If f(x) = x^2 and the interval is [1, 3], then the average rate of change is: $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$

Slope of a Curve at a Point (Instantaneous Rate of Change)

The slope of a curve at a point is the instantaneous rate of change of the function at that point, also known as the derivative. It is found by taking the limit of the average rate of change as the interval shrinks to a single point.

• Definition: The slope at x = a is: $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

• Interpretation: This is the slope of the tangent line to the curve at the point (a, f(a)).

• Example: For f(x) = x^2 at x = 2: $\lim_{h \to 0} \frac{(2 + h)^2 - 4}{h} = \lim_{h \to 0} \frac{4 + 4h + h^2 - 4}{h} = \lim_{h \to 0} \frac{4h + h^2}{h} = \lim_{h \to 0} (4 + h) = 4$

Limits from Graphs

Evaluating limits from graphs involves analyzing the behavior of a function as the input approaches a particular value, using the visual information provided by the graph.

• Key Steps:

  1. Identify the value that x approaches (say, c).

  2. Observe the left-hand limit (as x approaches c from the left) and the right-hand limit (from the right).

  3. If both limits are equal, that value is the limit of the function at c.

• Notation: $\lim_{x \to c} f(x)$

• Example: If the graph approaches 3 from both sides as x approaches 2, then $\lim_{x \to 2} f(x) = 3$.

Limits Using Tables

Using a table of values is a numerical method to estimate the limit of a function as x approaches a specific value.

• Procedure:

  1. Choose values of x that get closer to the target value from both sides.

  2. Calculate the corresponding f(x) values.

  3. If the f(x) values approach a single number, that is the estimated limit.

• Example: To estimate $\lim_{x \to 1} (x^2 + 2)$, use x values like 0.9, 0.99, 1.01, 1.1, etc. The f(x) values approach 3.

Using Limit Rules (Limit Laws)

Limit laws are algebraic properties that allow the computation of limits for combinations of functions, provided the individual limits exist.

• Sum Law: $\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$

• Product Law: $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

• Quotient Law: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$

• Power Law: $\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$

• Example: $\lim_{x \to 2} (3x^2 + 4) = 3 \cdot (2)^2 + 4 = 12 + 4 = 16$

Limits Involving Exponential and Logarithmic Functions

Limits can also be evaluated for exponential and logarithmic functions using their continuity and properties.

• Exponential Function: $\lim_{x \to a} e^x = e^a$

• Logarithmic Function: $\lim_{x \to a} \ln(x) = \ln(a)$, provided a > 0

• Example: $\lim_{x \to 0^+} \ln(x) = -\infty$

Limits of Average Rates of Change (as h → 0)

Taking the limit of the average rate of change as h approaches 0 leads to the definition of the derivative, representing the instantaneous rate of change.

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

• Interpretation: This process finds the slope of the tangent line at x = a.

• Example: For f(x) = x^3 at x = 1: $\lim_{h \to 0} \frac{(1 + h)^3 - 1}{h} = \lim_{h \to 0} \frac{1 + 3h + 3h^2 + h^3 - 1}{h} = \lim_{h \to 0} (3 + 3h + h^2) = 3$

The Squeeze Theorem

The Squeeze Theorem is a method for finding the limit of a function that is "squeezed" between two other functions with known limits at a point.

• Statement: If $g(x) \leq f(x) \leq h(x)$ for all x near a (except possibly at a), and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

• Example: To find $\lim_{x \to 0} x^2 \sin(1/x)$:

  • Since $-1 \leq \sin(1/x) \leq 1$, then $-x^2 \leq x^2 \sin(1/x) \leq x^2$.

  • Both $\lim_{x \to 0} -x^2 = 0$ and $\lim_{x \to 0} x^2 = 0$.

  • By the Squeeze Theorem, $\lim_{x \to 0} x^2 \sin(1/x) = 0$.

Additional info: These notes are based on the listed test topics and standard Calculus I curriculum. For more detailed examples and proofs, refer to your course textbook or lecture materials.