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Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = sin² x
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.7.54
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→1⁻ (1-x) tan πx/2
Guida verificata passo dopo passo1
First, identify the form of the limit as x approaches 1 from the left. Substitute x = 1 into the expression (1-x) tan(πx/2) to check if it results in an indeterminate form like 0/0 or ∞/∞.
Notice that as x approaches 1 from the left, (1-x) approaches 0 and tan(πx/2) approaches tan(π/2), which is undefined. However, from the left, tan(πx/2) approaches negative infinity, creating an indeterminate form of 0 * (-∞).
To apply l'Hôpital's Rule, rewrite the expression as a fraction. Consider the limit of (1-x) / (cot(πx/2)), since cotangent is the reciprocal of tangent.
Now, check if the rewritten expression (1-x) / (cot(πx/2)) results in a 0/0 form as x approaches 1 from the left. If it does, l'Hôpital's Rule can be applied.
Apply l'Hôpital's Rule by differentiating the numerator and the denominator separately. Differentiate (1-x) to get -1, and differentiate cot(πx/2) using the chain rule to get -π/2 * csc²(πx/2). Then, evaluate the limit of the new expression as x approaches 1 from the left.
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Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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One-Sided Limits
l'Hôpital's Rule
l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This process can be repeated if the result remains indeterminate.
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5:50Power Rules
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. In the context of limits, these functions can exhibit specific behaviors as their arguments approach certain values, which can lead to indeterminate forms. Understanding the properties and limits of these functions is essential for evaluating limits involving trigonometric expressions.
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Percorso guidato
6:04Introduction to Trigonometric Functions
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