Table of contents

0. Functions7h 55m

Introduction to Functions
18m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 5m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

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 31m

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
52m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

13. Intro to Differential Equations2h 55m

Basics of Differential Equations
48m

Slope Fields
19m

Euler's Method
22m

Separable Differential Equations
1h 25m

14. Sequences & Series5h 36m

Sequences
1h 22m

Review of Factorials
11m

Series
44m

Convergence Tests
3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

14. Sequences & Series

Series

Calculus

14. Sequences & Series

Series

Previous problemNext problem

2:03 minutes

Problem 10.R.95a

Textbook Question

Building a tunnel — first scenario
A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.

a.How far does the crew dig in 10 weeks? 20 weeks? N weeks?

Verified step by step guidance

Identify the type of sequence described in the problem. Since each week the crew digs 0.95 times the distance dug the previous week, this is a geometric sequence where each term is multiplied by a common ratio \(r = 0.95\).

Write the general formula for the distance dug in the \(n^{th}\) week. The first term \(a_1\) is 100 meters, so the distance dug in week \(n\) is given by \(a_n = a_1 \times r^{n-1} = 100 \times 0.95^{n-1}\).

To find the total distance dug after \(N\) weeks, recognize that this is the sum of the first \(N\) terms of a geometric sequence. Use the formula for the sum of the first \(N\) terms: \(S_N = a_1 \times \frac{1 - r^N}{1 - r}\).

Apply the sum formula to calculate the total distance dug after 10 weeks by substituting \(N=10\), \(a_1=100\), and \(r=0.95\) into the formula: \(S_{10} = 100 \times \frac{1 - 0.95^{10}}{1 - 0.95}\).

Similarly, calculate the total distance dug after 20 weeks by substituting \(N=20\) into the sum formula: \(S_{20} = 100 \times \frac{1 - 0.95^{20}}{1 - 0.95}\). For a general number of weeks \(N\), use \(S_N = 100 \times \frac{1 - 0.95^{N}}{1 - 0.95}\).

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Key Concepts

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

Geometric Sequences

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant ratio. In this problem, the distance dug each week decreases by a factor of 0.95, making it a geometric sequence with first term 100 and common ratio 0.95.

Sum of a Finite Geometric Series

The total distance dug over multiple weeks is the sum of the terms in the geometric sequence. The sum of the first n terms of a geometric series can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

Generalization to N Terms

To find the distance dug after N weeks, the formula for the sum of the first N terms of the geometric series is used. This allows calculation of the total distance for any number of weeks, not just specific values like 10 or 20.

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Define the remainder of an infinite series.

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Textbook Question

Building a tunnel — first scenario

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.

b.What is the longest tunnel the crew can build at this rate?

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Textbook Question

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯

89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.

a. Find the base-10 representation of the binary number 0.001̅1.

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Textbook Question

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.

b. Find an upper bound for the remainder Rₙ.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

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Textbook Question

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.

c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.