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

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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2:18 minutes

Problem 5a

Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

a. $[0, 2]$

Verified step by step guidance

Identify the formula for average velocity over an interval [a, b], which is given by the change in position divided by the change in time: $ v_{avg} = \frac{s(b) - s(a)}{b - a} $.

Determine the values of $ s(a) $ and $ s(b) $ from the table for the interval [0, 2]. These represent the position of the object at times t = 0 and t = 2, respectively.

Substitute the values of $ s(0) $ and $ s(2) $ into the average velocity formula: $ v_{avg} = \frac{s(2) - s(0)}{2 - 0} $.

Simplify the expression by calculating the difference $ s(2) - s(0) $ to find the change in position.

Divide the change in position by the change in time (which is 2 seconds) to find the average velocity over the interval [0, 2].

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Key Concepts

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

Average Velocity

Average velocity is defined as the change in position divided by the time interval over which that change occurs. Mathematically, it is expressed as (s(t2) - s(t1)) / (t2 - t1), where s(t) represents the position function. This concept is crucial for understanding how an object's position changes over time and is fundamental in analyzing motion.

Position Function

The position function s(t) describes the location of an object at any given time t. It provides a mathematical representation of the object's trajectory along a line. Understanding this function is essential for calculating average velocity, as it allows us to determine the specific positions at the start and end of the time interval.

Time Interval

A time interval is the duration over which motion is observed, typically denoted as [t1, t2]. In this context, it is the period during which the average velocity is calculated. Recognizing the significance of the time interval is important for accurately determining changes in position and understanding the object's motion dynamics.

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Textbook Question

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\left(0\right)$

Textbook Question

Let $f\left(x\right)=\frac{x^2-4}{x-2}$ . <IMAGE>

Calculate $f\left(x\right)$ for each value of $x$ in the following table.

Textbook Question

Let $f\left(x\right)=\frac{x^2-4}{x-2}$ . <IMAGE>

Make a conjecture about the value of ${\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}}$ .

Textbook Question

The function $s(t)$ represents the position of an object at time t moving along a line. Suppose $s(2)=136$ and $s(3)=156$ . Find the average velocity of the object over the interval of time $[2, 3]$ .

Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

c. $[0, 1]$

Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

a. s(t)=−16t^2+80t+60 at t=3

Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

c. s(t)=40 sin 2t at t=0

Textbook Question

Tangent lines with zero slope

a. Graph the function f(x)=x^2−4x+3.

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