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Calculus I

A first course in differential and integral calculus of a single variable. Students discover how to describe change through limits and derivatives, connect derivatives to curve-sketching and optimization, reverse the derivative with definite and indefinite integrals, and apply integrals to compute areas, volumes, and average values. The course culminates with the Fundamental Theorem of Calculus, revealing differentiation and integration as inverse operations.

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Functions & Limits

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Intuitive Limits and Limit Laws appears earlier in the syllabus and supports Continuity, Asymptotes, and the Precise Limit.Continuity, Asymptotes, and the Precise Limit appears earlier in the syllabus and supports Chain Rule and Higher Derivatives.Chain Rule and Higher Derivatives appears earlier in the syllabus and supports Exponential, Logarithmic, and Inverse-Function Derivatives.Exponential, Logarithmic, and Inverse-Function Derivatives appears earlier in the syllabus and supports Extrema, Mean Value Theorem, and Curve Sketching.Extrema, Mean Value Theorem, and Curve Sketching appears earlier in the syllabus and supports Applied Optimization Problems.Applied Optimization Problems appears earlier in the syllabus and supports Antiderivatives and Area via Riemann Sums.Antiderivatives and Area via Riemann Sums appears earlier in the syllabus and supports Area Between Curves and Average Value.Area Between Curves and Average Value appears earlier in the syllabus and supports Volumes by Cylindrical Shells and Arc Length.Volumes by Cylindrical Shells and Arc Length appears earlier in the syllabus and supports More Applications: Work, Center of Mass, and Improper Integrals.prerequisite

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Intuitive Limits and Limit Laws -> Continuity, Asymptotes, and the Precise Limit

Intuitive Limits and Limit Laws appears earlier in the syllabus and supports Continuity, Asymptotes, and the Precise Limit.

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Intuitive Limits and Limit Laws

Introduce limits numerically, graphically, and with simple algebra to capture the idea of "approaching.

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Intuitive Limits and Limit Laws: the core idea

Introduce limits numerically, graphically, and with simple algebra to capture the idea of "approaching. The key thing to notice is: Limit as ‘predicted y-value’ when x gets arbitrarily close (but ≠) to a. A useful example is Frame rate vs smooth motion in video games: limit of (position(t+h)–position(t))/h as h→0 gives instantaneous velocity. Do not treat this as a vocabulary item; the point is to use it to reason about a new situation.

Where would Intuitive Limits and Limit Laws show up in an everyday decision or news headline?

Look for the hidden relationship in the example: Frame rate vs smooth motion in video games: limit of (position(t+h)–position(t))/h as h→0 gives instantaneous velocity.

Intuitive Limits and Limit Laws: the core idea