Velocity from position The graph of s = f ( t ) s = f(t) represents the position of an object moving along a line at time t ≥ 0 t≥0 . <IMAGE> a. Assume the velocity of the object is 0 when t = 0 t=0 . For what other values of t is the velocity of the object zero?

Welcome back, everyone. The position of a moving particle is shown below in the graph F of T. At what times velocity is 0, and we're given 4 answer choices A says 0 and D, B, A, B, and C, C, A, B, C, and D, and D says time equals 0. So essentially what we want to do is consider this position function. We have to understand that F of T represents position, right? We are given F of T on the vertical axis, on the horizontal axis, we're given time. And since this is a position function, we have to recall that the first derivative of the position function represents the velocity function, right? And essentially the velocity function is simply going to be equal to the slope. Off The position function, right? Because if we're taking the first derivative, it represents the slope of the tangent line to the curve. So if we want to identify. The velocity of 0, we want to make sure that velocity is equal to 0, and it follows that the first derivative. Must be equal to 0 and this means that the slope. Of the tangent line at a specific point must be equal to 0, and we're looking at a horizontal tangent line. Which is basically parallel to the time axis, and we can see that if we start with the origin and if we go up, these would be the points. That 3% a local minima and maxima. So for example, one of the points would be time equals A. If we draw a tangent line passing through that local maximum, we're going to see that this tangent line is horizontal. Also, we can identify the local minimum at point B. So the next point is time equals B. There's another local maximum at point C, right? We once again have the horizontal tangent line. So the next point is time equals C, and when we go down, that's it, right? Essentially, we have the intercepts and zeros. Well, these do not count, right? Because they do not represent local minima or maxima. So what we can do is simply conclude that the final answer to this problem corresponds to the answer choice B. Thank you for watching.

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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

0. Functions

Introduction to Functions

Calculus

0. Functions

Introduction to Functions

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Problem 41a

Textbook Question

Velocity from position The graph of $s = f (t)$ represents the position of an object moving along a line at time $t is greater than or equal to 0$ . <IMAGE>
a. Assume the velocity of the object is 0 when $t equals 0$ . For what other values of t is the velocity of the object zero?

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Step 1: Understand that the velocity of an object is the derivative of its position function with respect to time, i.e., v(t) = f'(t).

Step 2: Since the velocity is zero at t = 0, we need to find other values of t where the derivative f'(t) = 0.

Step 3: Analyze the graph of s = f(t) to identify points where the tangent to the curve is horizontal, as these correspond to points where f'(t) = 0.

Step 4: Look for critical points on the graph where the slope of the tangent line is zero, indicating potential values of t where the velocity is zero.

Step 5: Verify these points by considering the behavior of the graph around these critical points to ensure they are indeed points where the velocity is zero.

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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, denoted as s = f(t), describes the location of an object along a line at any given time t. Understanding this function is crucial because it provides the basis for analyzing the object's motion. The graph of this function visually represents how the position changes over time, allowing us to infer other properties like velocity and acceleration.

Velocity

Velocity is defined as the rate of change of position with respect to time, mathematically expressed as v(t) = f'(t), where f'(t) is the derivative of the position function. It indicates how fast and in what direction the object is moving. When the velocity is zero, it signifies that the object is momentarily at rest, which is essential for determining points of interest in motion analysis.

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. In the context of velocity, finding critical points helps identify when the object's velocity is zero, which corresponds to moments when the object stops or changes direction. Analyzing these points is vital for understanding the overall motion of the object.

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