Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (2x +1)² dx
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.R.116a
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cos xⁿ) / x²ⁿ
Verified step by step guidance1
Recognize that the limit involves an indeterminate form 0/0 as x approaches 0, which suggests the use of L'Hôpital's Rule or a series expansion.
Consider using the Taylor series expansion for cos(x) around x = 0: cos(x) ≈ 1 - x²/2 + x⁴/24 - ... . For cos(xⁿ), substitute xⁿ into the series: cos(xⁿ) ≈ 1 - (xⁿ)²/2 + (xⁿ)⁴/24 - ... .
Substitute the series expansion into the limit expression: (1 - (1 - (xⁿ)²/2 + ...)) / x²ⁿ = ((xⁿ)²/2 - ...) / x²ⁿ.
Simplify the expression: ((x²ⁿ)/2 - ...) / x²ⁿ = (1/2) - ... . As x approaches 0, higher order terms become negligible.
Conclude that the limit evaluates to 1/2 as x approaches 0, since the dominant term in the expansion is (1/2).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function
In calculus, the limit of a function describes the behavior of that function as its input approaches a certain value. It is a fundamental concept used to define continuity, derivatives, and integrals. Evaluating limits often involves techniques such as substitution, factoring, or applying L'Hôpital's rule when dealing with indeterminate forms.
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Limits of Rational Functions: Denominator = 0
Taylor Series Expansion
The Taylor series expansion is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For the cosine function, the expansion around zero is given by cos(x) = 1 - x²/2! + x⁴/4! - ... This series is useful for approximating functions and simplifying the evaluation of limits, especially when dealing with small values of x.
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L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule is particularly helpful in simplifying complex limit problems, such as the one presented in the question.
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Related Practice
Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
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Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (⁴√x³ + √x⁵) dx
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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3
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Textbook Question
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cosⁿ x) / x²
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Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x⁴ - 50x² on [-1, 5]
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