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Calculus 1

8 units370 videos37 hours

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Unit 1: Limits and continuity

Limits intro

Estimating limits from graphs

Estimating limits from tables

Formal definition of limits (epsilon-delta)

Properties of limits

Limits by direct substitution

Limits using algebraic manipulation

Strategy in finding limits

Squeeze theorem

Types of discontinuities

Continuity at a point

Continuity over an interval

Removing discontinuities

Infinite limits

Limits at infinity

Intermediate value theorem

Unit 2: Derivatives: definition and basic rules

Average vs. instantaneous rate of change

Secant lines

Derivative definition

Estimating derivatives

Differentiability

Power rule

Derivative rules: constant, sum, difference, and constant multiple

Combining the power rule with other derivative rules

Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)

Product rule

Quotient rule

Derivatives of tan(x), cot(x), sec(x), and csc(x)

Proof videos

Unit 3: Derivatives: chain rule and other advanced topics

Chain rule

More chain rule practice

Implicit differentiation

Implicit differentiation (advanced examples)

Differentiating inverse functions

Derivatives of inverse trigonometric functions

Strategy in differentiating functions

Differentiation using multiple rules

Second derivatives

Disguised derivatives

Logarithmic differentiation

Proof videos

Unit 4: Applications of derivatives

Meaning of the derivative in context

Straight-line motion

Non-motion applications of derivatives

Introduction to related rates

Solving related rates problems

Approximation with local linearity

L’Hôpital’s rule

L’Hôpital’s rule: composite exponential functions

Unit 5: Analyzing functions

Mean value theorem

Extreme value theorem and critical points

Intervals on which a function is increasing or decreasing

Relative (local) extrema

Absolute (global) extrema

Concavity and inflection points intro

Analyzing concavity and inflection points

Second derivative test

Sketching curves

Connecting f, f', and f''

Solving optimization problems

Analyzing implicit relations

Unit 6: Integrals

Accumulations of change introduction

Approximation with Riemann sums

Summation notation review

Riemann sums in summation notation

Defining integrals with Riemann sums

Fundamental theorem of calculus and accumulation functions

Interpreting the behavior of accumulation functions

Properties of definite integrals

Fundamental theorem of calculus and definite integrals

Reverse power rule

Indefinite integrals of common functions

Definite integrals of common functions

Integrating with u-substitution

Integrating using long division and completing the square

Integrating using trigonometric identities

Proof videos

Unit 7: Differential equations

Differential equations introduction

Verifying solutions for differential equations

Sketching slope fields

Reasoning using slope fields

Separation of variables

Particular solutions to differential equations

Exponential models

Unit 8: Applications of integrals

Average value of a function

Straight-line motion

Non-motion applications of integrals

Area: vertical area between curves

Area: horizontal area between curves

Volume: squares and rectangles cross sections

Volume: triangles and semicircles cross sections

Volume: disc method (revolving around x- and y-axes)

Volume: disc method (revolving around other axes)

Volume: washer method (revolving around x- and y-axes)

Volume: washer method (revolving around other axes)

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