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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1:55 minutes

Problem 3i

Textbook Question

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

i. f(0)=1

Verified step by step guidance

Examine the graph of the function y = f(x) at the point where x = 0. This involves identifying the y-coordinate of the point on the graph that corresponds to x = 0.

Determine the value of f(0) by observing the y-coordinate of the point on the graph where x = 0. This value is the output of the function when the input is 0.

Compare the observed value of f(0) with the given statement f(0) = 1. Check if the y-coordinate at x = 0 matches the value 1.

If the y-coordinate at x = 0 is equal to 1, then the statement f(0) = 1 is true. Otherwise, the statement is false.

Conclude whether the statement is true or false based on the comparison made in the previous step. This involves confirming the accuracy of the statement by referencing the graph.

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Key Concepts

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

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. For example, if f(x) is a function, then f(0) means we are looking for the value of the function when x equals 0. Understanding how to read and interpret function values from a graph is crucial for answering questions about specific points on the graph.

Graph Interpretation

Graph interpretation is the ability to analyze and extract information from a visual representation of a function. This includes identifying key points, such as intercepts and local maxima or minima, as well as understanding the overall shape and behavior of the graph. In this context, it is essential to determine the value of f(0) by locating the point where the graph intersects the y-axis.

True/False Statements

True/false statements require critical thinking to assess the validity of claims based on given information. In calculus, this often involves verifying whether a specific condition holds true for a function, such as checking if f(0) equals a certain value. This process typically involves both analytical skills and visual confirmation from the graph.

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Average Rates of Change

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

g(t)=2+cos t

b. [0,π]

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.

P(θ)=θ³ − 4θ² + 5θ; [1,2]

Textbook Question