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

5. Graphical Applications of Derivatives

Intro to Extrema

Calculus

5. Graphical Applications of Derivatives

Intro to Extrema

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2:18 minutes

Problem 4.1.71

Textbook Question

Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?

Verified step by step guidance

To find when the object is farthest to the right, we need to determine the maximum value of the position function s(t) = 32t - t^4 over the interval 0 ≤ t ≤ 3.

First, find the derivative of the position function, s'(t), to determine the critical points. The derivative is s'(t) = 32 - 4t^3.

Set the derivative equal to zero to find the critical points: 32 - 4t^3 = 0. Solve for t to find the critical points.

Evaluate the position function s(t) at the critical points and at the endpoints of the interval, t = 0 and t = 3, to determine which gives the maximum value.

Compare the values of s(t) at these points to determine when the object is farthest to the right. The largest value of s(t) will indicate the time when the object is farthest to the right.

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Key Concepts

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

Position Function

The position function s(t) describes the location of an object at any given time t. In this case, s(t) = 32t - t⁴ represents the position of an object moving horizontally, where t is measured in seconds and s in feet. Understanding this function is crucial for determining the object's position over time and analyzing its motion.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. To find when the object is farthest to the right, we need to calculate the derivative of the position function, s'(t), and set it equal to zero. This helps identify potential maximum positions within the given interval.

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function at critical points. By evaluating the second derivative, s''(t), we can ascertain whether a critical point is a local maximum, minimum, or neither. This is essential for confirming that the object is indeed at its farthest right position when the first derivative is zero.

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