# Video Calculus Series

The following videos cover concepts from Calculus I and II.

Created by Selwyn Hollis. ©2022 University of Houston Department of Mathematics.

To see additional practice problems and solutions combined with these videos, visit our AP Calculus Help by Topic page.

## 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. -
**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^{−1}*x*.**Taylor and Maclaurin Series**(27 minutes)

Maclaurin series. Expansions of*e*, sin^{.x}*x*, and cos*x*, and related series. Taylor series expansions about*x*_{0}.**Taylor's Theorem**(28 minutes)

Taylor polynomials and the remainder term. Convergence of Taylor series to*f*.(*x*).

**•**Extras for

*“Early Transcendentals”*

ET1.

**(25 minutes)**

*e*and ln^{x}*x*ET2.

**Inverse Trig Functions**(25 minutes)