Domanda del libro di testo
Find an equation of the line tangent to the given curve at a.
y = (x + 5) / (x - 1); a = 3
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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 3.5.74a
Find an equation of the line tangent to the following curves at the given value of x.
y = csc x; x = π/4
Guida verificata passo dopo passo1
First, understand that the equation of a tangent line to a curve at a given point is y = mx + b, where m is the slope of the tangent line and b is the y-intercept.
To find the slope of the tangent line, calculate the derivative of the function y = csc(x). The derivative of csc(x) is -csc(x)cot(x).
Evaluate the derivative at the given point x = π/4. Substitute x = π/4 into the derivative -csc(x)cot(x) to find the slope m.
Next, find the y-coordinate of the point on the curve by substituting x = π/4 into the original function y = csc(x). This gives you the point (π/4, y).
Use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point on the curve and m is the slope, to write the equation of the tangent line.
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Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. This concept is crucial for finding the equation of the tangent line, as it provides the necessary slope to use in the point-slope form of a line.
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Percorso guidato
05:13
Slopes of Tangent Lines
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line at any point on the curve. For the function y = csc x, the derivative can be calculated using differentiation rules, which will be needed to find the slope at x = π/4.
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Percorso guidato
05:44Derivatives
Cosecant Function
The cosecant function, denoted as csc x, is the reciprocal of the sine function, defined as csc x = 1/sin x. Understanding the properties and behavior of the cosecant function is essential for evaluating the function and its derivative at specific points, such as x = π/4, which is necessary for determining the equation of the tangent line.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
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