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Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?
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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 87
Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.
lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)
Guida verificata passo dopo passo1
First, let's evaluate the limit using algebraic manipulation. Start by substituting x = 0 directly into the expression to check if it results in an indeterminate form. Substitute x = 0 into the expression (e^(2x) + 4e^x - 5) / (e^(2x) - 1).
Notice that substituting x = 0 gives us (1 + 4*1 - 5) / (1 - 1), which is 0/0, an indeterminate form. This suggests that we can use algebraic techniques or l'Hôpital's Rule to evaluate the limit.
For the algebraic method, consider expanding e^(2x) and e^x using their Taylor series expansions around x = 0: e^(2x) ≈ 1 + 2x + 2x^2/2 and e^x ≈ 1 + x + x^2/2. Substitute these approximations into the original expression.
Simplify the expression using the Taylor series expansions: (1 + 2x + 2x^2/2 + 4(1 + x + x^2/2) - 5) / (1 + 2x + 2x^2/2 - 1). Simplify the numerator and the denominator separately.
Now, let's use l'Hôpital's Rule. Since the limit is in the form 0/0, differentiate the numerator and the denominator with respect to x. The derivative of the numerator e^(2x) + 4e^x - 5 is 2e^(2x) + 4e^x, and the derivative of the denominator e^(2x) - 1 is 2e^(2x). Evaluate the limit of the new expression as x approaches 0.
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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 essential for determining continuity, derivatives, and integrals.
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One-Sided Limits
L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It 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 rule simplifies the process of finding limits in complex functions.
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5:50Power Rules
Exponential Functions
Exponential functions, such as e^x, are functions where a constant base is raised to a variable exponent. They are characterized by their rapid growth and unique properties, such as the fact that the derivative of e^x is e^x itself. Understanding exponential functions is crucial for evaluating limits involving expressions like e²ˣ and eˣ, as they often appear in calculus problems.
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6:13Exponential Functions
Pratica correlata
Domanda del libro di testo
More limits Evaluate the following limits.
lim_x→1 (x ln x - x + 1) / (xln²x)
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Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = 2x⁴ + 8x³ + 12x² - x - 2
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Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x² ln x; x³
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Domanda del libro di testo
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x²⁰ ; 1.0001ˣ
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Domanda del libro di testo
{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.
b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.
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