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

Related Rates

Calculus

4. Applications of Derivatives

Related Rates

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2:17 minutes

Problem 3.6.5

Textbook Question

Suppose w(t) is the weight (in pounds) of a golden retriever puppy t weeks after it is born. Interpret the meaning of w'(15) = 1.75.

Verified step by step guidance

Understand that w(t) represents the weight of the puppy as a function of time, where t is measured in weeks.

Recognize that w'(t) is the derivative of w(t) with respect to t, which represents the rate of change of the puppy's weight with respect to time.

The expression w'(15) = 1.75 indicates that at t = 15 weeks, the rate of change of the puppy's weight is 1.75 pounds per week.

Interpret this to mean that when the puppy is 15 weeks old, its weight is increasing at a rate of 1.75 pounds each week.

This rate of change is an instantaneous rate, meaning it describes how fast the weight is increasing exactly at 15 weeks, not over an interval of time.

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Key Concepts

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

Derivative Interpretation

The derivative of a function at a given point represents the rate of change of that function at that point. In this context, w'(15) indicates how the weight of the puppy is changing at 15 weeks. Specifically, it tells us that the weight is increasing at a rate of 1.75 pounds per week at that time.

Function Notation

Function notation, such as w(t), is used to express the relationship between an independent variable (t, in this case, time in weeks) and a dependent variable (w, the weight of the puppy). Understanding this notation is crucial for interpreting the behavior of the function over time and how changes in t affect w.

Contextual Interpretation

Interpreting the meaning of mathematical results in context is essential. Here, w'(15) = 1.75 means that at 15 weeks, the puppy's weight is not just increasing, but it is doing so at a specific rate, which can inform decisions about its health and growth patterns. This contextual understanding helps relate mathematical concepts to real-world scenarios.