{Use of Tech} Equations of tangent lines
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = −3x²+2; a=1

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First, understand that the equation of a tangent line to a curve at a given point is determined by the slope of the curve at that point and the coordinates of the point itself.

To find the slope of the tangent line, calculate the derivative of the function y = -3x² + 2. The derivative, y', represents the slope of the tangent line at any point x on the curve.

Evaluate the derivative at x = 1 to find the slope of the tangent line at this specific point. Substitute x = 1 into the derivative to get the slope value.

Next, find the y-coordinate of the point on the curve where x = 1 by substituting x = 1 into the original function y = -3x² + 2.

Use the point-slope form of the equation of a line, which is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point on the curve, to write the equation of the tangent line. Substitute the slope and the coordinates of the point into this formula.

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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 of the function.

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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 defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function y = -3x² + 2, the derivative can be calculated to find the slope of the tangent line at any point.

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05:44

Derivatives

Graphing Utilities

Graphing utilities are tools, such as graphing calculators or software, that allow users to visualize mathematical functions and their properties. By inputting equations, users can generate graphs that illustrate curves and tangent lines, making it easier to analyze their behavior and relationships visually.

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