Every lesson in the Calculus I slide course, in full text: 63 decks, 5376 slides.
Limits: The Graphical and Numerical IdeaTwo problems force calculus into existence: finding the slope of a tangent line, and finding instantaneous velocity. This deck starts there, then gives the informal definition of a limit and shows how to estimate limits from tables and read them off graphs. It covers one-sided limits and the two-sided existence test, removable holes, the three ways a limit can fail to exist, and infinite limits and vertical asymptotes, before closing with an honest first look at epsilon-delta. It targets the classic traps: assuming the limit is the function value, declaring that a limit exists when the one-sided limits disagree, calling an infinite limit an existing limit, and trusting a table on an oscillating function.
Limit Laws and Computing Limits AlgebraicallyThis deck covers the limit laws and the conditions attached to each of them, then works out when direct substitution is legal and what to do when it is not. It handles the zero-over-zero indeterminate form with factoring and cancelling, conjugates, and complex fractions, takes piecewise limits at the seam, and finishes with the Squeeze Theorem and the classic sine-over-x limit. It targets the traps of cancelling a factor and forgetting the hole, treating zero over zero as automatically zero or one, using the quotient law when the bottom limit is zero, and multiplying by a conjugate on only part of the fraction.
Continuity and the Intermediate Value TheoremThis deck gives the three-part definition of continuity at a point and uses it to classify removable, jump, and infinite discontinuities. It then covers continuity on intervals and for the standard function families, composites and passing a limit inside, and choosing parameters that make a piecewise function continuous, before closing with the Intermediate Value Theorem and bisection. It targets the students who check the limit but forget that the function must be defined, who think cancelling a factor erases the hole, and who use the IVT without checking continuity or read it as guaranteeing exactly one root.
Limits at Infinity, End Behavior, and AsymptotesThis deck is about the end behavior of functions. It explains what a limit at infinity means, then covers the divide-by-the-highest-power technique for rational functions and the three degree cases, horizontal and slant asymptotes, radicals in the negative direction, and how exponentials outrun polynomials. It targets the sign error that appears when you pull a variable out of a square root, the myth that a graph never crosses its horizontal asymptote, and the confusion between vertical-asymptote limits and limits at infinity.
The Derivative: Definition, Meaning, and DifferentiabilityThis deck builds the derivative from average rates of change and secant slopes up to the limit definition. It then computes derivatives from that definition for linear, quadratic, rational, and square-root functions, reads the derivative off a graph and in real units, and settles when differentiability fails. It targets the dropped minus sign in the difference quotient, cancelling h before it is a factor, the belief that continuous means differentiable, and confusing a function value with a slope.
Power, Constant, Sum, and Difference RulesThis deck covers the shortcut rules that replace the limit definition: the constant, power, constant-multiple, and sum and difference rules, plus the natural exponential. It also covers rewriting radicals and reciprocals as powers before differentiating, higher-order derivatives, tangent lines, and horizontal tangents. It targets the classic errors: using the power rule on a constant base, forgetting to rewrite, assuming the derivative distributes over a product, and slipping a sign on a negative exponent.
The Product and Quotient RulesThis deck explains why the derivative of a product is not the product of the derivatives, then gives the product rule with its expanding-rectangle picture and the quotient rule, paying close attention to the order of the terms in its numerator. It covers when to simplify instead of grinding, products of three factors, and combining the rules, and finishes with rate-of-change applications such as marginal revenue and average cost. It targets the product-of-derivatives error, the reversed numerator, the unsquared denominator, and reaching for the quotient rule when plain division is easier.
The Chain RuleThis deck is about differentiating a function built inside another function. It covers spotting the outer and inner pieces, both notations for the chain rule, and the generalized power rule, then works through trigonometric, exponential, and radical outer functions, double and triple compositions, chains inside the product and quotient rules, and chain values read from a table, closing with a preview of applied rates. It targets the classic errors of dropping the inner derivative, differentiating the inside in place, and misreading which piece is the inner function.
Derivatives of Trig, Exponential, and Log FunctionsThis deck assembles the full transcendental toolkit. It derives sine and cosine from the two special limits, rebuilds the other four trigonometric functions with the quotient rule, and covers the natural exponential that is its own derivative, general bases with their natural-log factor, the natural and general logarithm, and the inverse trigonometric derivatives - then combines all of it with the chain, product, and quotient rules. It targets the sign error on cosine and the co-functions, the degrees-versus-radians error, the missing natural-log factor on a general exponential, and the confusion between the logarithm and the exponential rules.
Implicit and Logarithmic DifferentiationThis deck shows how to differentiate a curve you cannot solve for y, and how to tame ugly products, quotients, and variable exponents by taking a logarithm first. It targets the four classic errors: differentiating a y term as if it were a constant, skipping the product rule on an x-times-y term, failing to collect every dy/dx term before dividing, and forcing the power rule onto a variable base with a variable exponent.
Related RatesThis deck handles two or more quantities that change in time and are linked by an equation. It gives the master procedure and then works the seven classic problems: the ladder, the ripple, the balloon, the cone tank, the separating vehicles, the streetlight shadow, and the angle of elevation. It targets the four errors that cost the most points - substituting the instant's numbers before differentiating, dropping the chain-rule rate factor, using a relation that holds only at one instant, and getting the sign or the units of a rate wrong.
Linear Approximation, Differentials, and Newton's MethodThis deck covers local linearity and the linearization of a function at a point, then differentials and propagated measurement error, using concavity to decide whether an estimate comes out too high or too low, and Newton's Method together with its failure modes. It targets the classic errors: centering at a point whose value you do not know, confusing the differential with the true change, getting the over-or-under direction backwards, and iterating Newton's Method with the previous function value instead of the previous x-value.
Indeterminate Forms and L'Hopital's RuleThis deck explains what makes a limit form indeterminate rather than merely undefined, lists the seven indeterminate forms, and applies L'Hopital's Rule with its hypotheses checked every single time. It targets the four classic errors: applying the rule to a determinate form, using the quotient rule instead of differentiating the top and the bottom separately, skipping the re-check before a second application, and forgetting to exponentiate at the end of a log-then-limit problem.
Extreme Values, Rolle's Theorem, and the MVTThis deck distinguishes absolute from relative extrema, then covers the Extreme Value Theorem and its hypotheses, critical points (including those where the derivative is undefined), Fermat's theorem, and the closed-interval method. It goes on to Rolle's Theorem, the Mean Value Theorem and its consequences, and the First Derivative Test. It targets the classic traps: missing critical points where the derivative does not exist, skipping the endpoints, applying the EVT or the MVT on an open interval or across a discontinuity, and assuming that every critical point is a maximum or a minimum.
Concavity, Inflection Points, and Curve SketchingThis deck explains what the second derivative measures and distinguishes concave up from concave down. It covers inflection points and the sign change they require, the Second Derivative Test and its inconclusive case, and first- and second-derivative sign charts, then works the full curve-sketching checklist end to end on a polynomial, a rational function, and an exponential. It targets the classic errors: calling every zero of the second derivative an inflection point, confusing decreasing with concave down, treating an inconclusive test as proof that there is no extremum, and reading features of a function off the graph of its derivative.
Applied OptimizationThis deck walks through the full applied-optimization workflow: name the objective and the constraint, reduce to one variable, state the realistic domain, find the critical points, and justify the maximum or minimum. The worked classics include fences, a cut-corner box, a minimum-metal can, a poster with margins, closest points, revenue and profit, and a least-cost pipeline. It targets the four errors that cost the most points: differentiating the constraint, leaving two variables in, ignoring the physical domain, and never justifying the answer.
Antiderivatives, Riemann Sums, and the FTCThis deck reverses differentiation into antiderivatives, including the constant of integration you must never drop. It then builds area from left, right, and midpoint Riemann sums, defines the definite integral as signed area, and proves out both parts of the Fundamental Theorem. It targets the classic errors: losing the plus C, using the power rule at an exponent of negative one, calling net area total area, dropping the chain factor in Part 1 of the FTC, and reversing the evaluation bar.
u-Substitution and Area Between CurvesThis deck treats substitution as the chain rule run backwards: choosing u, converting dx into du, back-substituting, and changing the limits on a definite integral. It then covers the area between two curves - top minus bottom, and left minus right in y - and net change, distinguishing displacement from total distance. It targets the traps of leaving a stray x in the integrand, losing a constant factor from du, keeping the old x-limits, subtracting in the wrong order, and computing distance without splitting at the sign changes.
1.1 Review of FunctionsThe formal definition of a function, function notation and evaluation, domain and range in interval notation, graphs and the vertical line test, zeros and intercepts, building new functions by arithmetic and by composition, and even and odd symmetry — the vocabulary every later section of calculus is stated in.
1.2 Basic Classes of FunctionsLinear functions and the meaning of slope, polynomials and degree, the roots of a quadratic and what the discriminant predicts, power functions and end behaviour, the algebraic and transcendental families, piecewise-defined functions, and the four transformations of a graph — the classification that lets Chapter 3 give one differentiation rule per family.
1.3 Trigonometric FunctionsRadian measure and why it is the right unit, the six trigonometric functions read off the unit circle, the major angles, the graphs with their periods and asymptotes, amplitude and period of a sinusoid, the Pythagorean and addition identities, and solving trigonometric equations for every solution rather than one.
1.4 Inverse FunctionsOne-to-one functions and the horizontal line test, the definition of an inverse through its two cancellation equations, finding an inverse algebraically by swapping and solving, the reflection of graphs in the line y equals x, restricting a domain to force an inverse to exist, and the inverse trigonometric functions with their conventional ranges.
1.5 Exponential and Logarithmic FunctionsThe form and graph of an exponential function, growth against decay, compound interest and where the number e comes from, the logarithm as the exponential's inverse, the three laws of logarithms and why they are exponent laws read backwards, change of base, solving exponential and logarithmic equations, and the hyperbolic functions.
2.1 A Preview of CalculusThe tangent problem and the recognition of a tangent as the limit of secant lines, instantaneous velocity as the limit of average velocities, the area problem solved by inscribed polygons and rectangles, and the observation that both ancient problems dissolve under the same limiting manoeuvre — the idea Section 2.2 will define.
2.2 The Limit of a FunctionThe intuitive definition of a limit and the crucial fact that it ignores the value at the point, estimating limits from tables and graphs and the ways a table can mislead, the three ways a limit fails to exist, one-sided limits and the theorem connecting them to the two-sided limit, and infinite limits with vertical asymptotes.
2.3 The Limit LawsThe basic limit laws and what they require, direct substitution for polynomials and rational functions, the five algebraic moves for resolving an indeterminate quotient, one-sided laws for piecewise functions, and the Squeeze Theorem with the fundamental trigonometric limits it delivers.
2.4 ContinuityThe three-part definition of continuity at a point and the three ways it fails, the classification of discontinuities as removable, jump or infinite, continuity on an interval with one-sided continuity at endpoints, the algebra of continuous functions and continuity of composites, and the Intermediate Value Theorem as the course's first existence theorem.
2.5 The Precise Definition of a LimitThe epsilon-delta definition of a limit and the challenge-and-response way of reading it, proving limits for linear and quadratic functions with the backwards scratch work and forwards proof, the epsilon-delta forms of one-sided and infinite limits, and using the definition to prove a limit law.
3.1 Defining the DerivativeThe tangent line as the limit of secant lines, the two equivalent forms of the difference quotient, computing a derivative at a point directly from the definition, writing the equation of a tangent line, velocity as an instantaneous rate of change, and estimating a derivative from a table of values.
3.2 The Derivative as a FunctionThe derivative as a function in its own right, sketching the graph of f prime from the graph of f, the Lagrange and Leibniz notations, the theorem that differentiability implies continuity and its useful contrapositive, the three ways a derivative fails, and higher-order derivatives with their interpretation.
3.3 Differentiation RulesThe constant and power rules, the sum, difference and constant multiple rules, the product rule with the area proof that explains its two terms, the quotient rule and why its order matters, and the discipline of deciding which rule applies outermost when several are needed at once.
3.4 Derivatives as Rates of ChangeThe amount of change formula and the estimate it licenses, motion along a line with velocity, speed and acceleration and the rule for speeding up or slowing down, marginal cost, revenue and profit in economics, population growth rates, and the discipline of attaching correct units to every applied derivative.
3.5 Derivatives of Trigonometric FunctionsThe two fundamental trigonometric limits and why they require radians, the derivatives of sine and cosine proved from them with the addition formula, the other four derivatives by the quotient rule and the co-function pattern, the four-step cycle of higher derivatives, and simple harmonic motion.
3.6 The Chain RuleThe chain rule for compositions and the intuition that rates multiply along a chain, the power form for a function raised to a power, chains of three or more functions, combining the chain rule with the product and quotient rules, the Leibniz form and why it resembles cancellation, and the proof.
3.7 Derivatives of Inverse FunctionsThe inverse function theorem and the reciprocal-slope picture behind it, differentiating a general inverse, the extension of the power rule to rational exponents, and the derivatives of all six inverse trigonometric functions obtained by a right-triangle argument — every one of them algebraic.
3.8 Implicit DifferentiationDifferentiating an equation in x and y without solving for y, the chain-rule factor every y term produces, tangent lines to curves that fail the vertical line test, reading horizontal and vertical tangents off an implicit derivative, and computing a second derivative implicitly.
3.9 Derivatives of Exponential and Logarithmic FunctionsThe natural exponential as the function that is its own derivative and why that singles out e, the natural logarithm's derivative by implicit differentiation, general bases and their stray logarithm factor, logarithmic differentiation for awkward products and powers, and the completed proof of the power rule for every real exponent.
4.1 Related RatesExpressing changing quantities as derivatives with respect to time, finding a geometric equation that relates them, differentiating it implicitly, and only then substituting the instant's values — worked through the sliding ladder, the inflating balloon, the filling cone and two objects approaching at right angles.
4.10 AntiderivativesReversing differentiation: the general antiderivative and why the constant is essential, indefinite integral notation, the reversed power rule and the basic formulas, initial-value problems, and the family of vertically shifted curves sharing one derivative.
4.2 Linear Approximations and DifferentialsThe linearization of a function at a point and its use for approximating values, the differential as the change along the tangent, the direction of the error decided by concavity and its second-order growth with the step, and relative and percentage error in propagated measurements.
4.3 Maxima and MinimaAbsolute and local extrema and the difference between them, the Extreme Value Theorem and its two hypotheses, Fermat's theorem that an interior extremum forces a zero derivative, critical points including those where the derivative fails to exist, and the closed-interval method for finding absolute extrema.
4.4 The Mean Value TheoremRolle's theorem and the Mean Value Theorem, the proof by subtracting the chord, the two hypotheses and counterexamples showing both are needed, and the three corollaries — a zero derivative gives a constant, equal derivatives differ by a constant, and the derivative's sign determines increase or decrease.
4.5 Derivatives and the Shape of a GraphThe first derivative test for increase, decrease and local extrema; concavity as the sign of the second derivative and as the direction the tangents lie; points of inflection where the concavity changes; and the second derivative test with the three functions that show why it can be inconclusive.
4.6 Limits at Infinity and AsymptotesLimits at infinity and horizontal asymptotes, the degree rule for rational functions, oblique asymptotes obtained by polynomial division, the end behaviour of polynomials and of the transcendental families, and the complete curve-sketching procedure combining asymptotes with the derivative tests of Section 4.5.
4.7 Applied Optimization ProblemsTranslating a description into an objective function, using a constraint to reduce it to a single variable, determining the physically meaningful domain, and applying the extreme-value machinery — worked through fencing, an open box, a minimal-surface can, a shortest distance and a revenue problem.
4.8 L'Hôpital's RuleThe rule for indeterminate quotients of the forms zero over zero and infinity over infinity, repeated application, converting products, differences and exponential forms into quotients, the proof of the growth ranking, and the forms that only appear indeterminate.
4.9 Newton's MethodThe iteration derived from the tangent line, its rapid convergence with correct digits roughly doubling each step, the ways it fails through a vanishing derivative, a cycle or divergence, the importance of the initial guess, and the comparison with bisection.
5.1 Approximating AreasSigma notation and the four summation formulas, approximating the area under a curve with rectangles, the left, right and midpoint rules, upper and lower sums bracketing the answer, and the limit of Riemann sums that defines area exactly.
5.2 The Definite IntegralThe definite integral as the limit of Riemann sums, its notation and the meaning of each part, net signed area against total area, which functions are integrable, the properties inherited from sums, and the average value of a function.
5.3 The Fundamental Theorem of CalculusThe Mean Value Theorem for Integrals, the area function and Part 1 proving its derivative is the integrand, Part 2 evaluating any definite integral from an antiderivative, the chain rule for variable limits, and the net change reading.
5.4 Integration Formulas and the Net Change TheoremThe basic integration formulas applied to definite integrals, rewriting integrands the table does not cover, the Net Change Theorem, displacement against distance travelled, and the symmetry shortcuts for even and odd functions.
5.5 SubstitutionReversing the chain rule: choosing the substitution, computing the differential, rewriting every x in terms of the new variable, back-substituting or changing the limits, adjusting constants but never variables, and recognising when substitution cannot help.
5.6 Integrals Involving Exponential and Logarithmic FunctionsIntegrating the natural exponential and its substitutions, general exponentials with other bases, the reciprocal with its absolute value, the logarithmic form for a quotient whose numerator is the denominator's derivative, and applications to growth and decay.
5.7 Integrals Resulting in Inverse Trigonometric FunctionsThe three inverse trigonometric integration formulas and their general forms with a constant, reaching them by substitution, completing the square for a quadratic denominator, the domains on which each is valid, and telling them apart from the logarithmic forms they resemble.
6.1 Areas between CurvesThe area between two curves as the integral of top minus bottom, why the axis is irrelevant, splitting where the curves cross, integrating with respect to y using horizontal strips, and choosing the variable that avoids splitting.
6.2 Determining Volumes by SlicingVolume as the integral of a cross-sectional area, solids with known cross-sections, the disk method for a region touching the axis, the washer method for one separated from it, revolving about lines other than the coordinate axes, and why the axis decides the variable.
6.3 Volumes of Revolution: Cylindrical ShellsSlicing parallel to the axis of revolution, a shell's volume as circumference times height times thickness, why the method needs no inversion of the boundary, shells about lines other than the coordinate axes, and choosing between shells and washers.
6.4 Arc Length of a Curve and Surface AreaArc length from the Pythagorean theorem on an infinitesimal triangle, the formula in either variable, why so few arc length integrals are elementary, the surface area of a solid of revolution, and the frustum's slant height.
6.5 Physical ApplicationsMass from a varying density, work as the integral of a varying force, Hooke's law and springs, the work of pumping a tank, and hydrostatic force on a vertical plate — all from the same representative-piece discipline.
6.6 Moments and Centers of MassMoments and the centre of mass for point masses and for a rod with varying density, the centroid of a plane region and the factor of one half in the vertical moment, the symmetry principle, and the Theorem of Pappus.
6.7 Integrals, Exponential Functions, and LogarithmsDefining the natural logarithm as an integral, proving its algebraic properties from that definition, defining e and the exponential as its inverse, giving irrational exponents a meaning, and repairing the circularity of the earlier treatment.
6.8 Exponential Growth and DecayThe differential equation whose rate is proportional to the amount, its solution by separating variables, doubling times and half-lives, Newton's law of cooling applied to the temperature difference, and the limits of the model.
6.9 Calculus of the Hyperbolic FunctionsThe hyperbolic functions as the even and odd parts of the exponential, the identity that names them, their derivatives and integrals, the inverse functions with their logarithmic formulas, and the catenary.
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