Introduction to Calculus – Notes (Transcript-Based)

Limits

Course context: Limits form foundational concept for derivative and continuity; topics include one-sided limits, limits at infinity, and infinite limits.
Definition of limit (conceptual): A function f(x) has limit L as x approaches a if f(x) can be made arbitrarily close to L by taking x sufficiently close to a (x ≠ a).
One-sided limits:
- Right-hand limit: lim_{x→a^+} f(x)
- Left-hand limit: lim_{x→a^-} f(x)
Basic limit examples (as given in slides):
- lim_{x→2} (x + 3) = 5
- lim_{x→6} x^2 = 36
- lim_{t→-2} t^4 = 16
Limits via graphs: If a function’s graph approaches a y-value as x approaches a from either side, this y-value is the limit; there can be a mismatch between limit existence and function value f(a).
Infinite limits and limits at infinity:
- Infinite limits occur when f(x) grows without bound as x approaches a finite value (or ±∞ as x approaches finite a).
- Limits at infinity describe the behavior of f(x) as x grows without bound or tends to -∞/∞; typical examples include rational and polynomial functions.
Examples from lecture visuals:
- lim_{x→∞} 4 (x − 5)^3
- lim_{x→−∞} 4 − x
Limits of rational functions at infinity: compare degrees of numerator and denominator to determine end behavior.
Common techniques referenced in slides:
- Direct substitution when no indeterminate form arises.
- Algebraic manipulation to remove indeterminate forms.
- Recognizing standard limit forms and using limits of polynomials/sums/products.

Continuity and Differentiability

Continuity at a: f is continuous at a if lim_{x→a} f(x) = f(a).
Differentiability at a: f is differentiable at a if f'(a) exists; derivative is defined as
$f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$
Tangent line intuition: A tangent line at x = a touches the curve y = f(x) at the point (a, f(a)) with slope f'(a).
Relationship: If f'(a) exists, then f is continuous at a. (Differentiability implies continuity; continuity does not imply differentiability.)
Examples in class: compute derivatives using the definition; tangent line equation at a point:
$y - f(a) = f'(a)(x - a)$
Common notational concept:
- A function f with derivative f' exists at a if the limit defining f'(a) exists.

Derivatives and Differentiation Rules

Derivative definition (as above).
Basic derivative rules (summary from lecture slides):
- Power rule: $\frac{d}{dx} x^n = n x^{n-1}$
- Constant multiple rule: $\frac{d}{dx} [c f(x)] = c f'(x)$
- Sum rule: $\frac{d}{dx} [u(x) + v(x)] = u'(x) + v'(x)$
- 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: if y = f(u) and u = g(x), then
  $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = f'(u) \cdot g'(x)$
Derivatives of elementary families (as per slides):
- Exponential: $\frac{d}{dx} e^x = e^x, \quad \frac{d}{dx} a^x = a^x \ln a$
- Logarithmic: $\frac{d}{dx} \ln x = \frac{1}{x}$
- Trigonometric: $\frac{d}{dx} \sin x = \cos x, \quad \frac{d}{dx} \cos x = -\sin x, \quad \frac{d}{dx} \tan x = \sec^2 x$
- Inverse trig/hyperbolic derivatives: listed in course reference pages; standard forms: e.g., $\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$ , etc.
Derivative of higher-order forms are covered (e.g., chain rule applied multiple times, implicit differentiation, etc.).
Examples touched on differentiation applying rules:
- Derivative of simple polynomials: e.g., if y = 3x + 5, then dy/dx = 3.
- Derivative of a composite expression using chain rule.
- Tangent line to y = 2x^2 + 2x + 3 at x = 1: find f'(1) then equation: y = f'(1)(x - 1) + f(1).

Finding Derivatives and Tangent Lines (by Definition and by Rules)

By definition vs. by rules: use the limit definition for a direct computation or apply standard rules for efficiency.
Tangent line example:
- If f(x) = 2x^2 + 2x + 3, tangent line at x = 1 has slope f'(1) and passes through (1, f(1)).
Summary:
- Differentiability implies continuity.
- If f is differentiable at a point, the derivative gives the slope of the tangent line to the graph at that point.

Rules of Differentiation (Summary from Slides)

Summary list includes: Power, Product, Quotient, Chain, and the derivatives of elementary functions (exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, etc.).
Example problems illustrate applying these rules to compute derivatives of polynomials, rational functions, and composed functions.

Applications of Derivatives: Rate of Change, Curve Sketching, and Economics

Rate of change ideas:
- Instantaneous rate of change is the derivative.
- Economics application: marginal concepts are derivatives of cost/revenue/consumption functions.
Economics-specific concepts:
- Marginal cost (MC):
  $MC(q) = \frac{dC}{dq}$
- Elasticity of demand: if y = f(x), then
  $\text{elasticity} = \frac{f'(x)}{f(x)} \times 100\%$
- Marginal revenue (MR): for revenue R(q) with demand p = p(q),
- Total revenue: $R(q) = p(q) \cdot q$
- Marginal revenue: $MR(q) = \frac{dR}{dq} = p(q) + q \frac{dp}{dq}$
Example framework problems include:
- Given C(x) or R(q) with a demand equation p = f(q), compute MC, MR and discuss optimization implications.
Typical optimization problems include minimizing costs (or maximizing revenue) under constraints, often using derivative tests (first/second derivative test).

Derivatives of Special Functions and Implicit Differentiation

Logarithmic differentiation:
- Useful for products, quotients, and variable exponents; differentiate by taking logs and differentiating implicitly.
Implicit differentiation:
- Used when y is defined implicitly by a relation between x and y. Steps:
  1) Differentiate both sides with respect to x.
  2) Solve for dy/dx.
- Example: differentiate y + y^3 − x = 7 to find dy/dx.
Higher-order derivatives:
- Second derivative d^2y/dx^2 describes concavity and inflection points.
- Example: For f(x) = 6x^3 − 12x^2 + 6x − 2, compute all higher-order derivatives.

Chain Rule, Logarithmic and Exponential Differentiation, and Applications

Chain rule illustrated with multiple compositions (e.g., y = f(u(x)) and u = g(x)).
Logarithmic differentiation examples including differentiating functions involving ln and exponentials.
Derivatives of inverse trig and hyperbolic functions are covered in advanced sections; standard derivatives can be consulted in reference sheets.

Derivatives: Curve Sketching, Extrema, and Optimization

Curve sketching involves:
- Finding critical points: where f'(x) = 0 or undefined.
- First-derivative test to classify relative extrema by sign changes of f'(x).
- Second-derivative test: if f''(a) < 0, local maximum; if f''(a) > 0, local minimum.
Key notes:
- Relative extrema vs absolute extrema; endpoints matter on closed intervals.
- A candidate for an inflection point requires f'' to be zero or undefined and continuity of f at that point.
Examples touched in class include:
- Finding where a function is increasing/decreasing and where relative extrema occur using sign charts.
- Testing concavity and identifying inflection points via f''(x).
- Curve sketching exercises such as y = (x − 1)^3 + 1 and y = x^2, and other polynomial forms.
Absolute extrema on a closed interval:
- Steps:
  1) Find critical values inside (a, b).
  2) Evaluate f at endpoints a and b and at critical points in (a, b).
  3) Maximum and minimum are the greatest/least of these values.
Applications: optimization problems such as fence-cost minimization and revenue optimization; typical approach uses derivative tests and constraint setup.

Integrals and Antiderivatives

Indefinite integral basics:
- Definition: F'(x) = f(x) and ∫ f(x) dx = F(x) + C.
Basic integration formulas (as summarized in the text):
- Power rule in reverse: ∫ x^n dx = x^{n+1}/(n+1) + C, for n ≠ −1
- ∫ dx/x = ln|x| + C
- ∫ e^x dx = e^x + C
- ∫ a^x dx = a^x/ln a + C, for a > 0, a ≠ 1
- ∫ 1/(x) dx = ln|x| + C
Techniques of integration:
- Substitution (u-substitution): ∫ f(g(x)) g'(x) dx = ∫ f(u) du
- Integration by parts: ∫ u dv = uv − ∫ v du
- Partial fractions: decomposition of a rational function into simpler fractions when integrating.
- Improper integrals: integrals with infinite limits or integrands with infinite discontinuities; two standard types are Type 1 (Infinite intervals) and Type 2 (Discontinuous integrands).
Applications of definite integrals:
- Fundamental Theorem of Calculus:
  $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$
- Area under curves and between curves:
- Area between curves requires intersection points and partitioning into regions where one function is on top of another.
Example applications from the slides include:
- Area calculations between curves such as y = x^2, y = x, and y = 4 − x^2, etc., with various intersection points.
- Finding the area of regions bounded by curves and axes.

Definite Integrals and Applications in Economy

Some applied problems in the slides relate to consumer surplus and producer surplus (detailing graphs with supply and demand).
- Consumer surplus and producer surplus can be computed from the market equilibrium price and the corresponding supply/demand curves.
Example workflow:
- Given demand p = f(q) and supply p = g(q), determine market equilibrium where f(q) = g(q).
- Compute areas under curves to obtain surpluses.
Economics connections emphasize the role of calculus in marginal concepts and optimization under market conditions.

Infinite Series and Geometric Sequences

Geometric sequence definition:
- c1 = a, and c{k+1} = ck r (common ratio r, with first term a).
- The ratio r = c{k+1} / ck for all k, with c_k ≠ 0.
Finite sums of geometric sequences:
- Sn = c1 + c2 + … + cn = a (1 − r^n) / (1 − r), for r ≠ 1.
Infinite geometric series (|r| < 1):
- Sum S∞ = a / (1 − r).
Perpetuity and present value example:
- If a perpetuity pays R each year starting now, the present value is
  $PV = R \left( 1 + r^{-1} + r^{-2} + \cdots \right) = \frac{R}{1 - \frac{1}{1+i}} = \frac{R(1+i)}{i}$
  where i is the interest rate per period.
- Example from slides: with R = 100{,}000 and i = 0.02, PV ≈ $5{,}100{,}000$.
Sum of infinite geometric sequences is meaningful only when |r| < 1; otherwise the sum diverges.
Series of practical problems include: perpetuities, present value of streams, and convergence behavior depending on r.

Applications of Series: Area, Consumption, and Revenue Models

The material includes extensive problem sets on:
- Area under curves using definite integrals and series methods.
- Revenue maximization and cost minimization under various demand/price relationships.
- Elasticity and marginal analyses extended to economic models.

Partial Fractions and Rational Functions (Integration Technique)

Goal: decompose a rational function into simpler fractions that can be integrated term-by-term.
Case I: Denominator Q(x) is a product of distinct linear factors.
- Decomposition form:
 $\frac{R(x)}{Q(x)} = \frac{A1}{a1 x + b1} + \frac{A2}{a2 x + b2} + \cdots + \frac{Ak}{ak x + b_k}$
Case II: Repeated linear factors or irreducible quadratics; the decomposition form is adjusted accordingly (details provided in lecture notes).
Examples in the slides illustrate setting up the decomposition forms for various rational functions and solving for constants A_i.
Applications: evaluating definite integrals of rational functions, and advancing to more sophisticated integration techniques.

Fundamental Theorem of Calculus and Accumulation

The definite integral is connected to antiderivatives via:
$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$
where F is any antiderivative of f (i.e., F'(x) = f(x)).
This links area, accumulation, and inverse differentiation.

Area and Region Problems

Techniques to find area between curves:
- Identify intersection points to determine region boundaries.
- Decide whether vertical or horizontal slicing simplifies the integral.
- Compute definite integrals to obtain the area; account for regions where the graph lies below the x-axis.
Examples include areas between curves such as y = 6 − x − x^2 and the x-axis, y = x^2 + x + 2 with x-axis and vertical lines, and more.

Putting It All Together: Study Notes and Practice Focus

Mastery goals:
- Compute limits (including one-sided and limits at infinity).
- Determine continuity and differentiability and understand their relationship.
- Differentiate using rules; apply to curve sketching and optimization.
- Solve applied problems in economics (marginal concepts, elasticity, optimization).
- Implement implicit and logarithmic differentiation when appropriate.
- Use integration techniques (substitution, integration by parts, partial fractions) to evaluate common integrals and to solve area problems.
- Understand improper integrals and convergence criteria for infinite intervals.
- Work with sequences and series, especially geometric sequences and the sum of infinite geometric series; apply these to perpetuity-type problems.
- Apply the Fundamental Theorem of Calculus to relate differentiation and integration.
Academic integrity note: The course explicitly prohibits cheating on exams; violations will be disciplined under KMITL’s academic integrity policy and penalties may apply.

Quick Reference: Key Formulas (LaTeX)

Derivative definitions
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Tangent line
$y - f(a) = f'(a)(x - a)$
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
$\frac{dy}{dx} = f'(u) \cdot g'(x)\quad \text{where } y = f(u(x)), u = g(x)$
Power Rule
$\frac{d}{dx} x^n = n x^{n-1}$
Exponential/Logarithm derivatives
$\frac{d}{dx} e^x = e^x, \quad \frac{d}{dx} a^x = a^x \ln a, \quad \frac{d}{dx} \ln x = \frac{1}{x}$
Fundamental Theorem of Calculus
$\int_{a}^{b} f(x)\,dx = F(b) - F(a)$
Area between curves (generic)
- Identify intersection points, choose appropriate axis orientation, compute definite integrals of top minus bottom function.
Geometric series (finite and infinite)
$Sn = a \frac{1 - r^n}{1 - r},\quad (r\neq 1)$ S\infty = \frac{a}{1 - r},\quad |r| < 1
Present value of a perpetuity (standard form)
$PV = \frac{R}{1 - (1/(1+i))} = \frac{R(1+i)}{i}$

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