## Τρίτη 31 Ιανουαρίου 2023

### Contents

Limits and Graphs (11 minutes)
The concept of limit from an intuitive, graphical point of view. Left and right-sided limits. Infinite one-sided limits and vertical asymptotes.
Using "limit laws" to compute limits.

Limits involving sine and cosine. Vertical asymptotes of $tan$, $cot$, $sec$, $csc$. The limit of $sin(x)∕x$ as $x → 0$ and related limits.

Continuity (19.5 minutes)
Definition of continuity at a point. Continuity of polynomials, rational functions, and trigonometric functions. Left and right continuity. Continuity on an interval.

The Derivative (18.5 minutes)
Slope of the tangent line; definition of the derivative. Differentiability and nondifferentiability at a point.
The power, product, reciprocal, and quotient rules for calculating derivatives.

The derivatives of $sin$, $cos$, $tan$, $cot$, $sec$, $csc$.

Liebniz notation for the derivative. The chain rule.

The derivative as rate of change. Related rates problems.

Implicit differentiation. The power rule for rational powers.

• Extras for “Early Transcendentals”
ET1. ex and ln x (25 minutes)
ET2. Inverse Trig Functions (25 minutes)
Rectilinear Motion (22 minutes)
Velocity and acceleration. Acceleration due to gravity. Bounce.

Higher-order derivatives. Concavity. Local approximation by linear, quadratic, and cubic polynomials.

Rolle's theorem and the mean-value theorem. Invervals where a function is increasing/decreasing/constant.

Critical numbers of a function. The first derivative test for local extrema.

Concavity and the second derivative. The second derivative test for local extrema.

Limits at ±∞ and horizontal asymptotes. Calculation of limits at $±∞$.
Curve Sketching (30 minutes)
Graphing $y = f(x)$ using the first and second derivatives, infinite limits, and limits at $±∞$.

Global (absolute) maximum and minimum values on closed intervals. Endpoint (one-sided) derivatives. The second derivative and extrema on open intervals.

Newton's Method (17.5 minutes)

Approximation of areas with sums of rectangle areas. Right-endpoint, left-endpoint, and midpoint approximations; upper and lower sums.

The Integral (28 minutes)
Definition of the integral. Signed area. Geometric evaluation and symmetries. Interval additivity property.

Average value theorem. The function $Φ(x) = ∫^a_x f(s) ds$. The fundamental theorem of calculus.

Indefinite integrals. The power rule for antidifferentiation.

Differentials. Using basic “u-substitutions” to find indefinite integrals and compute definite integrals.

Volumes I (10 minutes)
Solids with specified cross-sections.

Volumes II (10 minutes)
Solids of revolution.
Volumes III (12 minutes)
The cylindrical shell method.

Calculation of moments and centroids.

The natural log function defined as $∫^1_x 1/t dt$.

The inverse of the natural logarithm.

Inverse sine, cosine, tangent, cotangent, secant, and cosecant. Derivatives and companion indefinite integration formulas.
Integration by parts. Derivation of reduction formulas.

$∫cos^m(x)sin^n(x)dx$. Also $∫cos(ax)sin(bx)dx$, etc.

$∫sec^m(x)tan^n(x)dx$ and $∫csc^m(x)cot^n(x).dx$

Sine, tangent, and secant substitutions.

Partial fraction expansions. Integration of general rational functions.

Trapezoid Rule and Simpson's Rule. Error estimates.
Length of an arc $y = f(x)$, $a ≤ x ≤ b$. Area of a surface of revolution.

Polar vs. rectangular coordinates; polar graphs; slope of the tangent line to a polar curve.

Area of a polar region; length of a polar arc.

Parametric Curves
Parametric description of curves in the plane. Slope, arc length, and area.

The Conic Sections
Geometric definitions of parabolas, ellipses, and hyperbolas. Equations in the case of symmmetry about the coordinate axes. Rotation of axes.
Improper Integrals (28 minutes)
Integrals over unbounded intervals. Integrals over bounded intervals of functions that are unbounded near an endpoint. Comparison test for convergence/divergence.

Indeterminate forms $0∕0$, $∞∕∞$, $0∙∞$, $1^∞$, $0^0$, $∞^0$, and $∞ − ∞$. L'Hôpital's rule.

Sequences I (30 minutes)
Sequences; the graph of a sequence; the limit of a sequence; the squeeze theorem. Some special sequences and their limits.

Sequences II (27 minutes)
Precise definition of the limit of a sequence. Monotonicity and boundedness; convergence of bounded, monotonic sequences. Recursively defined sequences, fixed points, and web plots.

Series (22 minutes)
Sequences of partial sums. Geometric series and the harmonic series.

The Integral Test (14 minutes)
The integral test for convergence of series with positive terms; p-series. Remainder estimation.
Comparison Tests (19 minutes)
Comparison and limit-comparison tests. The ratio and root tests.

Convergence theorem for alternating series. Estimation of the remainder. Absolute versus conditional convergence.

Power Series (27 minutes)
Functions defined by power series. Ratio and root tests for absolute convergence. Differentiation and integration. Closed forms for series derived from geometric series. Series expansions of $ln(1+x)$ and $tan^{−1x}$.

Maclaurin series. Expansions of $e^x$, $sinx$, and $cosx$, and related series. Taylor series expansions about x0.

Taylor's Theorem (28 minutes)
Taylor polynomials and the remainder term. Convergence of Taylor series to $f(x)$.