Domanda del libro di testo
Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (csc x) = -csc x cot x
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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 3.5.7
Find an equation of the line tangent to the curve y = sin x at x = 0.
Guida verificata passo dopo passo1
First, understand that the equation of a tangent line to a curve at a given point is given by the formula: y - y₁ = m(x - x₁), where m is the slope of the tangent line and (x₁, y₁) is the point of tangency.
To find the slope of the tangent line, calculate the derivative of the function y = sin(x). The derivative, y' = cos(x), gives the slope of the tangent line at any point x.
Evaluate the derivative at the point of tangency, x = 0. Substitute x = 0 into y' = cos(x) to find the slope m at x = 0.
Determine the y-coordinate of the point of tangency by substituting x = 0 into the original function y = sin(x). This gives y₁ = sin(0).
Substitute the values of m, x₁, and y₁ into the tangent line equation y - y₁ = m(x - x₁) to find the equation of the tangent line at x = 0.
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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 represents the instantaneous rate of change of the function at that point, which can be found using the derivative.
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Percorso guidato
05:13
Slopes of Tangent Lines
Derivative
The derivative of a function at a point quantifies how the function's output changes as its input changes. For the function y = sin x, the derivative, denoted as y', gives the slope of the tangent line at any point x, which is crucial for finding the equation of the tangent line.
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Percorso guidato
05:44Derivatives
Point-Slope Form
The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is particularly useful for writing the equation of the tangent line once the slope and the point of tangency are known.
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3:56Slope-Intercept Form
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