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
- 8. Definite Integrals3h 25m
4. Applications of Derivatives
Linearization
Problem 4.6.12
Textbook Question
Suppose f is differentiable on (-∞,∞), f(5.99) = 7, and f(6) = 7.002. Use linear approximation to estimate the value of f'(6).
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1
Identify the points given in the problem: f(5.99) = 7 and f(6) = 7.002.
Recall the formula for linear approximation: f(x) ≈ f(a) + f'(a)(x - a), where a is a point close to x.
In this case, let a = 6 and x = 5.99. Substitute the known values into the linear approximation formula.
Rearrange the linear approximation formula to solve for f'(6): f'(6) = (f(5.99) - f(6)) / (5.99 - 6).
Calculate the difference in function values and the difference in x-values to find the estimate for f'(6).
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