Table of contents

0. Functions7h 52m
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 Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Linearization

Calculus

4. Applications of Derivatives

Linearization

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1:55 minutes

Problem 4.6.8

Textbook Question

Use linear approximation to estimate f (5.1) given that f(5) = 10 and f'(5) = -2.

Verified step by step guidance

Identify the function value and derivative at the point of interest: f(5) = 10 and f'(5) = -2.

Recall the formula for linear approximation: L(x) = f(a) + f'(a)(x - a), where a is the point of tangency.

Substitute the given values into the linear approximation formula: L(x) = 10 + (-2)(x - 5).

Set x = 5.1 in the linear approximation formula to estimate f(5.1): L(5.1) = 10 + (-2)(5.1 - 5).

Simplify the expression to find the estimated value of f(5.1) using the linear approximation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is based on the idea that if a function is differentiable, its behavior can be closely approximated by a linear function in the vicinity of that point. The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x - a), where 'a' is the point of tangency.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the function at that point. In the context of linear approximation, the derivative at a specific point is used to determine the slope of the tangent line, which is essential for estimating function values nearby.

Function Evaluation

Function evaluation involves calculating the output of a function for a given input. In this context, we are interested in estimating f(5.1) using known values of f(5) and f'(5). Understanding how to evaluate functions and apply the linear approximation formula is crucial for making accurate estimates based on the information provided.

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