Domanda del libro di testo
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = tan¹⁰x / (5x+3)⁶
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Ch. 3 - Derivatives
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 3, Problema 3.2.76
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
Graph the following curves and determine the location of any vertical tangent lines.
a. x²+y² = 9
Guida verificata passo dopo passo1
First, recognize that the given equation x² + y² = 9 represents a circle centered at the origin with a radius of 3.
To find vertical tangent lines, we need to express y as a function of x. Rearrange the equation to solve for y: y = ±√(9 - x²).
Differentiate y with respect to x to find the derivative y'. For y = √(9 - x²), use the chain rule: y' = (1/2)(9 - x²)^(-1/2) * (-2x). Simplify to get y' = -x / √(9 - x²).
Consider the limit of |y'| as x approaches the endpoints of the domain of y. The domain of y is -3 ≤ x ≤ 3. Check the behavior of y' as x approaches ±3.
As x approaches ±3, the denominator √(9 - x²) approaches 0, causing |y'| to approach infinity. This indicates vertical tangent lines at the points where x = ±3.
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Vertical Tangent Lines
A vertical tangent line occurs at a point on a curve where the slope of the tangent approaches infinity. This typically indicates that the function is increasing or decreasing without bound at that point. In calculus, this is identified when the derivative of the function becomes undefined or approaches infinity, suggesting a steep incline or decline.
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Percorso guidato
05:13
Slopes of Tangent Lines
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation, rather than explicitly as y = f(x). This method involves differentiating both sides of the equation with respect to x and solving for dy/dx. It is particularly useful for curves defined by equations like x² + y² = 9, where y cannot be easily isolated.
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05:14Finding The Implicit Derivative
Endpoints and One-Sided Derivatives
Endpoints of a domain refer to the boundaries of the interval in which a function is defined. At these points, the behavior of the function may differ from that in the interior of the interval. One-sided derivatives are used to analyze the slope of the function from one direction (left or right) at these endpoints, which is crucial for determining the existence of vertical tangents in such cases.
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05:50One-Sided Limits
Pratica correlata
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27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
√x⁴+y² = 5x+2y³
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Domanda del libro di testo
15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)
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Domanda del libro di testo
If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.
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Domanda del libro di testo
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(3x + 1)⁴
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Domanda del libro di testo
A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?
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