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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2:07 minutes

Problem 2.1.2a

Textbook Question

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

g(x)=x²−2x

a. [1, 3]

Verified step by step guidance

Identify the function given: \( g(x) = x^2 - 2x \).

Recall the formula for the average rate of change of a function \( g(x) \) over an interval \([a, b]\): \( \frac{g(b) - g(a)}{b - a} \).

Substitute the interval \([1, 3]\) into the formula, where \( a = 1 \) and \( b = 3 \).

Calculate \( g(1) \) by substituting \( x = 1 \) into the function: \( g(1) = 1^2 - 2 \times 1 \).

Calculate \( g(3) \) by substituting \( x = 3 \) into the function: \( g(3) = 3^2 - 2 \times 3 \).

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Key Concepts

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

Average Rate of Change

The average rate of change of a function over an interval [a, b] is defined as the change in the function's value divided by the change in the input value. Mathematically, it is expressed as (f(b) - f(a)) / (b - a). This concept is crucial for understanding how a function behaves over a specific range and is often used to analyze the function's growth or decline.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For the function g(x) = x² - 2x, evaluating it at points 1 and 3 means calculating g(1) and g(3). This step is essential for finding the values needed to compute the average rate of change over the specified interval.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. The function g(x) = x² - 2x is a quadratic function where a = 1, b = -2, and c = 0. Understanding the properties of quadratic functions, such as their parabolas' shape and vertex, is important for analyzing their behavior over intervals.

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Textbook Question

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 1/x, x = -2