Calculus 1, part 2 of 2: Derivatives with applications

A total of 56 hours of lectures

This is an academic level course for university and college engineering. Due to its size, it is divided into two parts. This page describes the second of those two parts.

Level - Intermediate

You need to be familiar with the contents of the four Precalculus courses and Calculus 1, part 1 (Limits and continuity) to comfortably follow along in this course.

Curriculum

Make sure that you check with your professor what parts of the course you will need for your exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

Calculus 1, part 2 of 2

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Get the outline

A detailed list of all the lectures in part 2 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.

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Course Objectives & Outcomes for part 2

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How to solve problems concerning derivatives of real-valued functions of 1 variable (illustrated with 330 solved problems) and why these methods work.
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Write equations of tangent lines to graphs of functions.
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Prove, apply, and illustrate the formulas for computing derivatives: the Sum Rule, the Product Rule, the Scaling Rule, the Quotient and Reciprocal Rule.
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Use the Chain Rule in problem solving with related rates.
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Understand the connection between the signs of derivatives and the monotonicity of functions; apply first- and second-derivative tests.
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Determine and classify stationary (critical) points for differentiable functions.
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Main theorems of Differential Calculus: Fermat’s Theorem, Mean Value Theorems (Lagrange, Cauchy), Rolle’s Theorem, and Darboux Property.
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Formulate and prove l’Hospital’s rule; apply it for computing limits of indeterminate forms; algebraical tricks to adapt the rule for various situations.
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Classes of functions: C^0, C^1, … , C^∞; connections between these classes, and examples of their members.
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Logarithmic differentiation: when and how to use it.

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Definition of derivatives of real-valued functions of one real variable, with a geometrical interpretation and many illustrations.
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Derive the formulas for the derivatives of basic elementary functions.
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Prove and apply the Chain Rule; recognise the situations in which this rule should be applied and draw diagrams helping in the computations.
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Use derivatives for solving optimisation problems.

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Understand the connection between the second derivative and the local shape of graphs (convexity, concavity, inflection points).
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Use derivatives as help in plotting real-valued functions of one real variable.
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Formulate, prove, illustrate with examples, apply, and explain the importance of the assumptions in main theorems of Differential Calculus.
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Higher order derivatives; an intro to Taylor / Maclaurin polynomials and their applications for approximations and for limits (more in Calculus 2).
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Implicit differentiation with some illustrations showing horizontal and vertical tangent lines to implicit curves.
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A sneak peek into some future applications of derivatives.