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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Multiple Choice

Which of the following best describes the remainder estimate for the integral test when determining the convergence of a series $\sum \underset{\infty}{n = 1} a_{n}$ with $a_{n} = f (n)$ , where $f$ is positive, continuous, and decreasing for $x \geq 1$ ?

The remainder

R_{N}

\sum \underset{\infty}{n = N + 1} a_{n}

satisfies

\int_{N + 1}^{\infty} f (x) d x \leq R_{N} \leq \int_{N}^{\infty} f (x) d x

The remainder

R_{N}

\sum \underset{\infty}{n = N + 1} a_{n}

is always greater than

\int_{N + 1}^{\infty} f (x) d x

The remainder

R_{N}

\sum \underset{\infty}{n = N + 1} a_{n}

equals exactly

\int_{N}^{\infty} f (x) d x

The remainder

R_{N}

\sum \underset{\infty}{n = N + 1} a_{n}

is always less than

\int_{N}^{\infty} f (x) d x

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Verified step by step guidance

Step 1: Recall the integral test for convergence of a series. It states that if a series $ \sum_{n=1}^{\infty} a_n $ has terms $ a_n = f(n) $, where $ f(x) $ is positive, continuous, and decreasing for $ x \geq 1 $, then the convergence of the series is determined by the convergence of the improper integral $ \int_{1}^{\infty} f(x) \, dx $.

Step 2: Understand the remainder $ R_N $. The remainder $ R_N $ represents the sum of the terms of the series from $ n = N+1 $ to infinity, i.e., $ R_N = \sum_{n=N+1}^{\infty} a_n $. This is the portion of the series left after summing the first $ N $ terms.

Step 3: Relate the remainder $ R_N $ to the integral. The integral test provides bounds for $ R_N $ using improper integrals. Specifically, $ \int_{N+1}^{\infty} f(x) \, dx \leq R_N \leq \int_{N}^{\infty} f(x) \, dx $. This inequality arises because the integral $ \int_{N+1}^{\infty} f(x) \, dx $ approximates the sum $ \sum_{n=N+1}^{\infty} a_n $ from below, while $ \int_{N}^{\infty} f(x) \, dx $ approximates it from above.

Step 4: Clarify why $ R_N $ is greater than $ \int_{N+1}^{\infty} f(x) \, dx $. The sum $ \sum_{n=N+1}^{\infty} a_n $ includes discrete terms $ a_n $, which are slightly larger than the corresponding values of $ f(x) $ used in the integral $ \int_{N+1}^{\infty} f(x) \, dx $. This is because the integral smooths out the function $ f(x) $ over the interval.

Step 5: Clarify why $ R_N $ is less than $ \int_{N}^{\infty} f(x) \, dx $. The integral $ \int_{N}^{\infty} f(x) \, dx $ starts at $ N $, including the contribution of $ f(N) $, which is not part of $ R_N $. Thus, $ \int_{N}^{\infty} f(x) \, dx $ overestimates $ R_N $.