Fundamental Concepts in Calculus: Derivatives, Limits, and Tangents

Derivatives and Differentiation

Definition and Basic Computation

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a foundational concept in calculus, representing instantaneous rate of change or the slope of the tangent line to the curve at a point.

• Notation: The derivative of a function y = f(x) is denoted as f'(x) or \( \frac{dy}{dx} \).

• Basic Rule: For a function y = f(x), the derivative at x = a is defined as: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Example: If \( y = 2\cos(x) - e^{x^2} - 6 \), then \( \frac{dy}{dx} = -2\sin(x) - 2x e^{x^2} \).

Applications: Tangent and Normal Lines

The tangent line to a curve at a point is a straight line that just touches the curve at that point and has the same slope as the curve there. The normal line is perpendicular to the tangent line at the point of contact.

• Equation of Tangent Line: At point (x0, y0), the tangent line is: $y - y_0 = f'(x_0)(x - x_0)$

• Equation of Normal Line: The normal line has slope \( -1/f'(x_0) \): $y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$

• Example: For y = 7 - 3x2 at x = 1, find the tangent line by computing the derivative and evaluating at x = 1.

Limits and Continuity

Definition and Evaluation

A limit describes the value that a function approaches as the input approaches a certain value. Limits are essential for defining derivatives and understanding function behavior near points of interest.

• Notation: \( \lim_{x \to a} f(x) \)

• Basic Properties:

  • If \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \), then \( \lim_{x \to a} f(x) = L \).

  • If the left and right limits are not equal, the limit does not exist (DNE).

• Example: \( \lim_{x \to 2} \frac{3x - 2}{x - 2} \) may require algebraic simplification or recognizing a discontinuity.

Vertical Asymptotes

A vertical asymptote occurs at values of x where a function grows without bound (approaches infinity or negative infinity). These are typically found where the denominator of a rational function is zero and the numerator is not zero at that point.

• How to Find: Set the denominator equal to zero and solve for x.

• Example: For \( f(x) = \frac{x^2 - 2x - 3}{x^2 - 3x - 4} \), set \( x^2 - 3x - 4 = 0 \) to find vertical asymptotes.

Continuity

A function is continuous at a point if the limit exists at that point and equals the function's value there.

• Three Conditions for Continuity at x = a:

  1. f(a) is defined

  2. \( \lim_{x \to a} f(x) \) exists

  3. \( \lim_{x \to a} f(x) = f(a) \)

• Example: If \( \lim_{x \to 2^-} g(x) = 2 \), \( \lim_{x \to 2^+} g(x) = 2 \), and g(2) = 2, then g(x) is continuous at x = 2.

Special Limits and Indeterminate Forms

Evaluating Limits Involving Square Roots and Rational Functions

Some limits require algebraic manipulation, such as multiplying by the conjugate or factoring, to resolve indeterminate forms like 0/0.

• Example: \( \lim_{x \to 2} \frac{2x - 2}{\sqrt{x + 2} - 2} \) may require multiplying numerator and denominator by the conjugate of the denominator.

Limits Defining the Derivative

The definition of the derivative uses a limit to describe the instantaneous rate of change of a function at a point.

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

• Example: If \( \lim_{h \to 0} \frac{(8 + h)^2 - 64}{h} \), recognize this as the derivative of \( f(x) = x^2 \) at x = 8.

Function Composition and Values

Evaluating Functions and Their Compositions

Function composition involves applying one function to the results of another. This is often tested by providing values of functions at specific points and asking for the value of a composition.

• Example: If f(3) = -2, F(3) = 4, and g(3) = 1/2, then evaluating expressions like f(g(3)) or F(f(3)) requires substituting the given values.

Graphing and Analyzing Functions

Sketching Graphs and Identifying Limits

Understanding the behavior of functions graphically is crucial for interpreting limits, asymptotes, and continuity.

• Example: For \( f(x) = \frac{x^2 - 2x - 3}{x^2 - 3x - 4} \), sketch the graph and find limits as x approaches values where the denominator is zero or as x approaches infinity.

Summary Table: Key Calculus Concepts

Concept	Definition	Example
Derivative	Instantaneous rate of change; slope of tangent	$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Limit	Value a function approaches as input approaches a point	$\lim_{x \to 2} (3x - 2)$
Continuity	No breaks, jumps, or holes at a point	f(x) is continuous at x = a if $\lim_{x \to a} f(x) = f(a)$
Vertical Asymptote	Line x = a where function grows without bound	Denominator zero, numerator nonzero
Tangent Line	Line with same slope as curve at a point	$y - y_0 = f'(x_0)(x - x_0)$

Additional info:

• Some questions require knowledge of the chain rule, product rule, and algebraic manipulation for limits.

• Understanding the graphical interpretation of derivatives and limits is essential for deeper calculus comprehension.