Single Variable Calculus | Project Ulim

Single Variable Calculus

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Unit 1: Limits and continuity

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Defining limits and using limit notation Estimating limit values from graphs Estimating limit values from tables Quiz 1 Determining limits using algebraic properties of limits: limit properties Determining limits using algebraic properties of limits: direct substitution Quiz 2 Determining limits using algebraic manipulation Selecting procedures for determining limits Determining limits using the squeeze theorem Quiz 3 Exploring types of discontinuities

Defining continuity at a point Confirming continuity over an interval Removing discontinuities Quiz 4 Connecting infinite limits and vertical asymptotes Connecting limits at infinity and horizontal asymptotes Quiz 5 Working with the intermediate value theorem Quiz 6 Optional videos More Practice Unit Test

Unit 2: Differentiation: definition and basic derivative rules

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Defining average and instantaneous rates of change at a point Defining the derivative of a function and using derivative notation Estimating derivatives of a function at a point Connecting differentiability and continuity: determining when derivatives do and do not exist More on tangent lines Quiz 1 Applying the power rule Derivative rules: constant, sum, difference, and constant multiple: introduction Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule Quiz 2

Derivatives of cos(x), sin(x), e, and ln(x)The product rule The quotient rule Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions Quiz 3 Optional videos Quiz 4 Quiz 5 Quiz 6 Unit Test

Unit 3: Differentiation: composite, implicit, and inverse functions

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The chain rule: introduction The chain rule: further practice Quiz 1 Implicit differentiation Differentiating inverse functions Differentiating inverse trigonometric functions Differentiating hyperbolic trigonometric functions Selecting procedures for calculating derivatives: strategy Selecting procedures for calculating derivatives: multiple rules Quiz 2

Calculating higher-order derivatives Further practice connecting derivatives and limits An interesting limit involving e and an application to APR Quiz 3 Optional videos Quiz 4 Quiz 5 Quiz 6 Unit Test

Unit 4: Contextual applications of differentiation

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Interpreting the meaning of the derivative in context Straight-line motion: connecting position, velocity, and acceleration Rates of change in other applied contexts (non-motion problems)Quiz 1 Quiz 2 Introduction to related rates Solving related rates problems Quiz 3

Approximating values of a function using local linearity and linearization Quadratic approximations and products of approximations Newton's method Quiz 4 Using L’Hôpital’s rule for finding limits of indeterminate forms Quiz 5 Optional videos Unit Test

Exam 1

Answer questions from recent units.

Unit 5: Applying derivatives to analyze functions

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Using the mean value theorem Quiz 1 Extreme value theorem, global versus local extrema, and critical points Determining intervals on which a function is increasing or decreasing More consequences of the mean value theorem Using the first derivative test to find relative (local) extrema Using the candidates test to find absolute (global) extrema Quiz 2 Quiz 3 Determining concavity of intervals and finding points of inflection: graphical Determining concavity of intervals and finding points of inflection: algebraic

Using the second derivative test to find extrema Sketching curves of functions and their derivatives Quiz 4 Connecting a function, its first derivative, and its second derivative Quiz 5 Solving optimization problems Exploring behaviors of implicit relations Calculator-active practice

Unit 6: Integration and accumulation of change

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Exploring accumulations of change Approximating areas with Riemann sums Riemann sums, summation notation, and definite integral notation Quiz 1 Quiz 2 The fundamental theorem of calculus and accumulation functions Interpreting the behavior of accumulation functions involving area Applying properties of definite integrals Quiz 3 The fundamental theorem of calculus and definite integrals Quiz 4 Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals Quiz 5 Integrating using substitution Change of variables Differential notation

Unit 7: Differential equations

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Modeling situations with differential equations Verifying solutions for differential equations Sketching slope fields Reasoning using slope fields Approximating solutions using Euler’s method Quiz 1 Finding general solutions using separation of variables

Finding particular solutions using initial conditions and separation of variables Exponential models with differential equations Logistic models with differential equations More on separable differential equations Quiz 2 Quiz 3 Unit Test

Exam 2

Answer questions from recent units.

Unit 8: Applications of integration

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Finding the average value of a function on an interval Connecting position, velocity, and acceleration functions using integrals Using accumulation functions and definite integrals in applied contexts Quiz 1 Finding the area between curves expressed as functions of x Finding the area between curves expressed as functions of y Finding the area between curves that intersect at more than two points Quiz 2 Quiz 3 Volumes with cross sections: squares and rectangles Volumes with cross sections: triangles and semicircles Harder volumes with cross sections Volumes with disc method: revolving around x- or y-axis Volumes with disc method: revolving around other axes

Volume with washer method: revolving around x- or y- axis Volume with washer method: revolving around other axes Quiz 4 Using integration to find the weighted average Finding probability using integration

Exam 3

Answer questions from recent units.

Unit 9: Parametric equations, polar coordinates, and vector-valued functions

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Defining and differentiating parametric equations Second derivatives of parametric equations Finding arc lengths of curves given by parametric equations Surface area from parametric equations Quiz 1 Quiz 2 Defining and differentiating vector-valued functions Solving motion problems using parametric and vector-valued functions

Quiz 3 Defining polar coordinates and differentiating in polar form Finding the area of a polar region or the area bounded by a single polar curve Finding the area of the region bounded by two polar curves Calculator-active practice Quiz 4 Unit Test

Exam 4

Answer questions from recent units.

Unit 10: Infinite sequences and series

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Defining convergent and divergent infinite series Working with geometric series Quiz 1 The nth-term test for divergence Integral test for convergence Harmonic series and p-series Comparison tests for convergence Alternating series test for convergence Ratio test for convergence Determining absolute or conditional convergence

Quiz 2 Alternating series error bound Finding Taylor polynomial approximations of functions Lagrange error bound Radius and interval of convergence of power series Finding Taylor or Maclaurin series for a function Representing functions as power series Quiz 3 Optional videos Unit Test

Final Exam

Answer questions from the entire course.

Integrating functions using long division and completing the square

Integrating using trigonometric identities

Estimating definite integrals

Using integration by parts

Reduction formulas

Integrating using linear partial fractions

Integrating using partial fractions (cont.)

Trigonometric substitution

Evaluating improper integrals

Optional videos

The arc length of a smooth, planar curve and distance traveled

Finding surface area using integration

Calculator-active practice