Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.5.7

Chapter 3, Problem 3.5.7

Find an equation of the line tangent to the curve y = sin x at x = 0.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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