72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

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To find the equation of the tangent line, we need to determine the derivative of the function y = 3x³ + sin(x) to find the slope of the tangent line at the given point (0, 0).

Differentiate the function y = 3x³ + sin(x) with respect to x. The derivative of 3x³ is 9x², and the derivative of sin(x) is cos(x). Therefore, the derivative y' = 9x² + cos(x).

Evaluate the derivative at the given point x = 0 to find the slope of the tangent line. Substitute x = 0 into y' = 9x² + cos(x) to get the slope m = 9(0)² + cos(0).

The slope m at x = 0 is cos(0), which simplifies to 1. Therefore, the slope of the tangent line at the point (0, 0) is 1.

Use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point (0, 0). Substitute m = 1, x₁ = 0, and y₁ = 0 into the equation to find 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 and has the same slope as the curve at that point. The equation of the tangent line can be found using the point-slope form, which requires the slope of the curve at the point of tangency and the coordinates of that point.

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Slopes of Tangent Lines

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is the slope of the tangent line to the curve at that point. To find the equation of the tangent line, we first compute the derivative of the function and evaluate it at the given point to obtain the slope.

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Derivatives

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 a tangent line once the slope and the point of tangency are known, allowing for a straightforward construction of the line's equation.

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