Textbook Question
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x² - 4x + 2
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 4.7.38
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)
Verified step by step guidance1
Identify the form of the limit as x approaches infinity. The expression is (e^(3x)) / (3e^(3x) + 5). As x approaches infinity, both the numerator and the denominator approach infinity, which is an indeterminate form of type ∞/∞.
Since the limit is in the indeterminate form ∞/∞, we can apply l'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value is in the form 0/0 or ∞/∞, then it can be evaluated as the limit of f'(x)/g'(x), provided this new limit exists.
Differentiate the numerator and the denominator separately. The derivative of the numerator, e^(3x), with respect to x is 3e^(3x). The derivative of the denominator, 3e^(3x) + 5, with respect to x is 9e^(3x).
Apply l'Hôpital's Rule by taking the limit of the new fraction formed by the derivatives: lim_(x→∞) (3e^(3x)) / (9e^(3x)).
Simplify the expression by canceling out e^(3x) from the numerator and the denominator, resulting in lim_(x→∞) 3/9. Simplify this fraction to find the limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches infinity helps determine the behavior of the function at extreme values.
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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 these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits in complex expressions.
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Guided course
5:50Power Rules
Exponential Functions
Exponential functions, such as e^x, are functions where a constant base is raised to a variable exponent. They grow rapidly as x increases, which is crucial when evaluating limits at infinity. Understanding their growth behavior helps in determining the dominant terms in expressions, especially when comparing them to polynomial or constant terms.
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Related Practice
Textbook Question
{Use of Tech} Absolute maxima and minima
a. Find the critical points of f on the given interval.
b. Determine the absolute extreme values of f on the given interval.
c. Use a graphing utility to confirm your conclusions.
f(x) = 2ᶻ sin x on [-2,6]
Textbook Question
{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = cos⁻¹ x - x; x₀ = 0.75
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Textbook Question
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>
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Textbook Question
Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.
f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (√x(2x⁶ - 4³√)dx