Textbook Question
Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.
v(t) = 2√t; s(0) = 1
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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 4.7.72
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π - 2x) tan x
Verified step by step guidance1
First, identify the form of the limit as x approaches π/2 from the left. Substitute x = π/2 into the expression (π - 2x) tan x to check if it results in an indeterminate form like 0/0 or ∞/∞.
Substitute x = π/2 into the expression: (π - 2x) becomes 0, and tan x approaches ∞ as x approaches π/2 from the left. This results in the indeterminate form 0 * ∞, which can be rewritten as 0/0 by expressing tan x as sin x / cos x.
Rewrite the expression as (π - 2x) / (1/tan x) = (π - 2x) / (cos x/sin x) = (π - 2x) sin x / cos x. This is now in the form 0/0, which allows us to apply l'Hôpital's Rule.
Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, then the limit is the same as the limit of f'(x)/g'(x) as x approaches that point. Differentiate the numerator and the denominator separately.
Differentiate the numerator (π - 2x) to get -2, and differentiate the denominator sin x / cos x using the quotient rule. The derivative of sin x is cos x, and the derivative of cos x is -sin x. Apply these derivatives to find the new limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is essential for determining the continuity and differentiability of functions.
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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, are fundamental in calculus, particularly when dealing with limits involving angles. The behavior of these functions near specific points, like π/2, can lead to unique limit scenarios, such as vertical asymptotes. Understanding their properties and graphs is crucial for evaluating limits that involve trigonometric expressions.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Shipping crates A square-based, box-shaped shipping crate is designed to have a volume of 16 ft³. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?
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Textbook Question
{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.
x³ + y³ = 3xy (Folium of Descartes)
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Textbook Question
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) = sin⁻¹ x
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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) = sin x + x - 1; x₀ = 0.5
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Textbook Question
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = 3x⁵ - 25x³ + 60x on [-2,3]
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