Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ x³ (1/x - sin 1/x)
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 4.6.59
Approximating changes
Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone of fixed height h = 6m when its radius decreases from r = 10 m to r = 9.9 m (S = πr√(r² + h²).
Verified step by step guidance1
Identify the formula for the lateral surface area of a right circular cone, which is given by \( S = \pi r \sqrt{r^2 + h^2} \). Here, \( h = 6 \) meters is constant.
To approximate the change in the lateral surface area, use the concept of differentials. The differential \( dS \) can be expressed as \( dS = \frac{\partial S}{\partial r} \cdot dr \), where \( dr \) is the change in radius.
Calculate the partial derivative of \( S \) with respect to \( r \). This involves using the product rule and chain rule: \( \frac{\partial S}{\partial r} = \pi \sqrt{r^2 + h^2} + \pi r \cdot \frac{r}{\sqrt{r^2 + h^2}} \).
Simplify the expression for \( \frac{\partial S}{\partial r} \) to get \( \frac{\partial S}{\partial r} = \pi \left( \sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}} \right) \).
Substitute \( r = 10 \) m and \( dr = -0.1 \) m into the expression for \( dS \) to approximate the change in the lateral surface area: \( dS = \frac{\partial S}{\partial r} \bigg|_{r=10} \cdot (-0.1) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Calculus
Differential calculus focuses on the concept of the derivative, which measures how a function changes as its input changes. In this context, it helps us approximate the change in the lateral surface area of the cone as the radius changes slightly. By using derivatives, we can find the rate of change of the surface area with respect to the radius.
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Fundamental Theorem of Calculus Part 1
Surface Area of a Cone
The lateral surface area of a right circular cone is given by the formula S = πr√(r² + h²), where r is the radius and h is the height. Understanding this formula is crucial for calculating the surface area before and after the change in radius. It highlights the relationship between the radius and the surface area, which is essential for applying calculus concepts.
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Linear Approximation
Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. In this problem, we can use the derivative of the surface area function to approximate the change in surface area when the radius decreases slightly from 10 m to 9.9 m. This technique simplifies complex calculations by providing a close estimate.
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Related Practice
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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
Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x³ - 6x² + 9x