Textbook Question
49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.
ƒ(x) = 3x⁴ + 4x³ - 12x²
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 4.7.31
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_u→ π/4 (tan u - cot u) / (u - π/4)
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Identify the form of the limit as u approaches π/4. Substitute u = π/4 into the expression (tan u - cot u) / (u - π/4) to check if it results in an indeterminate form like 0/0.
Since substituting u = π/4 results in an indeterminate form 0/0, l'Hôpital's Rule is applicable. According to l'Hôpital's Rule, if the limit of f(u)/g(u) as u approaches a value results in 0/0 or ∞/∞, then the limit can be evaluated as the limit of f'(u)/g'(u).
Differentiate the numerator and the denominator separately. The numerator is tan(u) - cot(u). Differentiate tan(u) to get sec^2(u) and differentiate cot(u) to get -csc^2(u).
Differentiate the denominator, which is simply u - π/4. The derivative of u is 1, and the derivative of a constant π/4 is 0.
Apply l'Hôpital's Rule by taking the limit of the new expression: lim_{u→π/4} (sec^2(u) + csc^2(u)) / 1. Evaluate this limit by substituting u = π/4 into the expression.
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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 concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the continuity and the value that a function approaches, even if it is not explicitly defined at that point. Evaluating limits is crucial for determining the behavior of functions near points of interest, especially in cases of indeterminate forms.
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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 ∞/∞. 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 simplifies the process of finding limits and is particularly useful in calculus when direct substitution fails.
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5:50Power Rules
Trigonometric Functions
Trigonometric functions, such as tangent (tan) and cotangent (cot), are essential in calculus for analyzing periodic behavior and angles. The tangent function is defined as the ratio of the sine and cosine functions, while cotangent is the reciprocal of tangent. Understanding their properties and how they behave near specific angles, like π/4, is crucial for evaluating limits involving these functions.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x ln x - 2x + 3 on (0,∞)
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Textbook Question
Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.
f'(x) > 0, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f"(0) = 0
Textbook Question
{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.
Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?
Textbook Question
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = 2x³ - 3x² + 12
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0⁺ x²ˣ