Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
2. Intro to Derivatives
Tangent Lines and Derivatives
1:50 minutes
Problem 106
Textbook Question
Textbook QuestionUse the definition of the derivative to evaluate the following limits.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of the Derivative
The derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is expressed as f'(a) = lim(h→0) [f(a+h) - f(a)] / h. This concept is fundamental in calculus as it provides a way to determine the instantaneous rate of change of a function.
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Derivatives
Natural Logarithm Properties
The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. Key properties include ln(ab) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). Understanding these properties is essential for simplifying expressions involving logarithms, especially when evaluating limits.
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3:56
Change of Base Property
Limit Evaluation Techniques
Limit evaluation techniques involve methods to find the value that a function approaches as the input approaches a certain point. Common techniques include direct substitution, factoring, and using L'Hôpital's Rule for indeterminate forms. Mastery of these techniques is crucial for solving problems that involve limits, particularly in the context of derivatives.
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One-Sided Limits
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