Domanda del libro di testo
Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x)=3e^x+10 / e^x
342
views
Ch. 2 - Limits
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 2, Problema 79
A right circular cylinder with a height of 10 cm and a surface area of S cm2 has a radius given by r(S)=1/2(√100+2S/π −10).
Find lim S→0^+ r(S) and interpret your result.
Guida verificata passo dopo passo1
Identify the function for the radius: \( r(S) = \frac{1}{2}(\sqrt{100 + \frac{2S}{\pi}} - 10) \).
Understand that you need to find \( \lim_{S \to 0^+} r(S) \).
Substitute \( S = 0 \) into the function: \( r(0) = \frac{1}{2}(\sqrt{100 + \frac{2 \cdot 0}{\pi}} - 10) \).
Simplify the expression: \( r(0) = \frac{1}{2}(\sqrt{100} - 10) \).
Calculate the limit by evaluating the simplified expression.
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Limits
In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is essential for understanding continuity, derivatives, and integrals. In this context, we are interested in the limit of the radius function r(S) as S approaches 0 from the positive side, which helps us determine the behavior of the radius when the surface area is minimal.
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05:50
One-Sided Limits
Surface Area of a Cylinder
The surface area of a right circular cylinder is calculated using the formula S = 2πr(h + r), where r is the radius and h is the height. This concept is crucial for understanding how the radius r(S) is derived from the surface area S, and it provides context for the relationship between the dimensions of the cylinder and its surface area.
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09:07Example 1: Minimizing Surface Area
Function Interpretation
Interpreting the result of a limit involves understanding what the limit signifies in the context of the problem. In this case, finding lim S→0^+ r(S) will reveal the radius of the cylinder as the surface area approaches zero, which can provide insights into the geometric implications of a cylinder with minimal surface area, such as its shape and dimensions.
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04:39Derivative of the Natural Logarithmic Function Example 7
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