Department of Mathematics, University of Houston: Video Calculus

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.

Calculation of Limits (17 minutes)

Using "limit laws" to compute limits.

Trigonometric Limits (17 minutes)

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.

Calculation of Derivatives (25 minutes)

The power, product, reciprocal, and quotient rules for calculating derivatives.

Derivatives of Trigonometric Functions (11 minutes)

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

Leibniz Notation and the Chain Rule (20 minutes)

Liebniz notation for the derivative. The chain rule.

Rates of Change and Related Rates (20 minutes)

The derivative as rate of change. Related rates problems.

Implicit Differentiation (17.5 minutes)

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 (20 minutes)

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

The Mean-Value Theorem and Related Results (26 minutes)

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

Critical Numbers and the First Derivative Test (17 minutes)

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

Concavity and the Second Derivative Test (20 minutes)

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

Limits at ±∞ and Horizontal Asymptotes (20 minutes)

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 $±∞$.

Extreme Values on Intervals (19 minutes)

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

Applied Optimization Problems (22 minutes)

Newton's Method (17.5 minutes)

The Area Under a Curve (28 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.

The Fundamental Theorem of Calculus (26 minutes)

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

Antidifferentiation and Indefinite Integrals (29 minutes)

Indefinite integrals. The power rule for antidifferentiation.

Change of Variables (Substitution) (21 minutes)

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

Areas Between Curves (19 minutes)

Volumes I (10 minutes)

Solids with specified cross-sections.

Volumes II (10 minutes)

Solids of revolution.

Volumes III (12 minutes)

The cylindrical shell method.

The Centroid of a Planar Region (21 minutes)

Calculation of moments and centroids.

The Natural Logarithm (19 minutes)

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

The Exponential Function (21 minutes)

The inverse of the natural logarithm.

The Inverse Trigonometric Functions (25 minutes)

Inverse sine, cosine, tangent, cotangent, secant, and cosecant. Derivatives and companion indefinite integration formulas.

Integration by Parts (21 minutes)

Integration by parts. Derivation of reduction formulas.

Integration of Powers and Products of Sine and Cosine (18 minutes)

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

Integration of Powers and Products of Secant and Tangent, Cosecant and Cotangent (23 minutes)

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

Trigonometric Substitutions (21 minutes)

Sine, tangent, and secant substitutions.

Partial Fraction Expansions (26 minutes)

Partial fraction expansions. Integration of general rational functions.

Numerical Integration (26 minutes)

Trapezoid Rule and Simpson's Rule. Error estimates.

Arc Length and Surface Area (15 minutes)

Length of an arc $y = f(x)$, $a ≤ x ≤ b$. Area of a surface of revolution.

Polar Coordinates and Graphs (36 minutes)

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

Areas and Lengths Using Polar Coordinates (18 minutes)

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 and L'Hôpital's Rule (22 minutes)

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.

Alternating Series and Absolute Convergence (25 minutes)

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 $tan^{−1x}$.

Taylor and Maclaurin Series (27 minutes)

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 $f(x)$.