Video Lectures for Single-Variable Calculus from the University of Houston
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Limits and Continuity
- Limits and Graphs (11 minutes)
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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)
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Using "limit laws" to compute limits.
- Trigonometric Limits (17 minutes)
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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) { browser }
Definition of continuity at a point. Continuity of polynomials,
rational functions, and trigonometric functions. Left and right
continuity. Continuity on an interval.
Derivatives
- The Derivative (18.5 minutes)
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Slope of the tangent line; definition of the derivative.
Differentiability and nondifferentiability at a point.
- Calculation of Derivatives (25 minutes)
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The power, product, reciprocal, and quotient rules for calculating derivatives.
- Derivatives of Trigonometric Functions (11 minutes)
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The derivatives of sin, cos, tan, cot, sec, csc.
- Leibniz Notation and the Chain Rule (20 minutes)
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Liebniz notation for the derivative. The chain rule.
- Rates of Change and Related Rates (20 minutes)
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The derivative as rate of change. Related rates problems.
- Implicit Differentiation (17.5 minutes)
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Implicit differentiation. The power rule for rational powers.
• Extras for “Early Transcendentals”
ET1. e^{.x}
and ln x (25 minutes) {
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ET2. Inverse Trig Functions (19.5 minutes)
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ET3. Hyperbolic and Inverse Hyperbolic Functions
Applications of Derivatives
- Rectilinear Motion (22 minutes)
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Velocity and acceleration. Acceleration due to gravity. Bounce.
- Higher-Order Derivatives (20 minutes)
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Higher-order derivatives. Concavity. Local approximation
by linear, quadratic, and cubic polynomials.
- The Mean-Value Theorem and Related Results (26 minutes) {
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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) {
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Critical numbers of a function. The first derivative test for local extrema.
- Concavity and the Second Derivative Test (20 minutes) {
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Concavity and the second derivative. The second derivative test for local extrema.
- Limits at ±∞ and Horizontal Asymptotes (20 minutes) {
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Limits at ±∞ and horizontal asymptotes. Calculation of limits at ±∞.
- Curve Sketching (30 minutes) {
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Graphing y = f.(x) using
the first and second derivatives, infinite limits, and limits at ±∞.
- Extreme Values on Intervals (19 minutes) {
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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) {
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- Newtonʼs Method (17.5 minutes) {
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Anti-Derivatives and Integration
- The Area Under a Curve (28 minutes) {
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Approximation of areas with sums of rectangle areas. Right-endpoint,
left-endpoint, and midpoint approximations; upper and lower sums.
- The Integral (28 minutes) {
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Definition of the integral. Signed area. Geometric evaluation
and symmetries. Interval additivity property.
- The Fundamental Theorem of Calculus (26 minutes) {
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Average value theorem. The function Φ(x) =
∫_{a}^{x}
f.(s) ds. The
fundamental theorem of calculus.
- Antidifferentiation and Indeﬁnite Integrals (29 minutes) {
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Indefinite integrals. The power rule for antidifferentiation.
- Change of Variables (Substitution) (21 minutes) {
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Differentials. Using basic “u-substitutions”
to find indefinite integrals and compute definite integrals.
- The Natural Logarithm (19 minutes) {
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The natural log function defined as
∫_{1}^{x}
1/t. dt.
- The Exponential Function (21 minutes) {
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The inverse of the natural logarithm.
- The Inverse Trigonometric Functions (25 minutes) {
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Inverse sine, cosine, tangent, cotangent, secant, and cosecant.
Derivatives and companion indefinite integration formulas.
Applications of Integrals
- Areas Between Curves (19 minutes) {
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- Volumes I (10 minutes) {
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Solids with specified cross-sections.
- Volumes II (10 minutes) {
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Solids of revolution.
- Volumes III (12 minutes) {
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The cylindrical shell method.
- The Centroid of a Planar Region (21 minutes) {
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Calculation of moments and centroids.
- Arc Length and Surface Area (15 minutes) {
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Length of an arc y =
f.(x),
a ≤ x ≤ b. Area of a surface of revolution.
- Polar Coordinates and Graphs (36 minutes) {
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Polar vs. rectangular coordinates; polar graphs; slope of
the tangent line to a polar curve.
- Areas and Lengths Using Polar Coordinates (18 minutes) {
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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.
Advanced Integration Techniques
- Integration by Parts (21 minutes) {
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Integration by parts. Derivation of reduction formulas.
- Integration of Powers and Products of Sine and Cosine (18 minutes) {
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∫.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) {
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∫.sec^{m}x tan^{n}x.dx and ∫.csc^{m}x cot^{n}x.dx
- Trigonometric Substitutions (21 minutes) {
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Sine, tangent, and secant substitutions.
- Partial Fraction Expansions (26 minutes) {
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Partial fraction expansions. Integration of general rational functions.
- Numerical Integration (26 minutes) {
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Trapezoid Rule and Simpsonʼs Rule. Error estimates.
- Improper Integrals (28 minutes) {
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Integrals over unbounded intervals. Integrals over bounded
intervals of functions that are unbounded near an endpoint.
Comparison test for convergence/divergence.
- Indeterminate Forms and LʼHôpitalʼs Rule (22 minutes) {
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Indeterminate forms 0∕0, ∞∕∞, 0∙∞
1^{∞}, 0^{0}, ∞^{0},
and ∞ − ∞. LʼHôpitalʼs rule.
Sequences and Series
- Sequences I (30 minutes) {
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Sequences; the graph of a sequence; the limit of a sequence;
the squeeze theorem. Some special sequences and their limits.
- Sequences II (27 minutes) {
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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) {
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Sequences of partial sums. Geometric series and the
harmonic series.
- The Integral Test (14 minutes) {
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The integral test for convergence of series with positive
terms; p-series. Remainder estimation.
- Comparison Tests (19 minutes) {
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Comparison and limit-comparison tests. The ratio and root tests.
- Alternating Series and Absolute Convergence (25 minutes) {
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Convergence theorem for alternating series. Estimation
of the remainder. Absolute versus conditional convergence.
- Power Series (27 minutes) {
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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^{−1}x.
- Taylor and Maclaurin Series (27 minutes) {
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Maclaurin series. Expansions of e^{.x}, sin x, and cos x,
and related series. Taylor series expansions about x_{0}.
- Taylorʼs Theorem (28 minutes) {
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Taylor polynomials and the remainder term. Convergence of
Taylor series to f.(x).
Created by Selwyn
Hollis.
©2008, University of Houston