Find the equation of the tangent line to each curve when x has the given value. Verify your answ

step 1 So we want to find the equation of our tangent line given x. So this time we're given that our x

Find the equation of the tangent line to each curve when x...

Key Concepts

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its independent variable, and it gives the slope of the tangent line at any point on the curve. In calculus, finding the derivative is essential for understanding how the function behaves locally and for determining the exact slope where the tangent touches the curve.

Tangent Line

A tangent line to a curve at a particular point is a straight line that touches the curve at that point and has the same slope as the curve does at that point. It provides a linear approximation of the function near that point, making it useful for analyzing the local behavior of the function.

Point-Slope Form

The point-slope form of a linear equation is used to write the equation of a line when a point on the line and the slope are known. This form is given by the equation y - y? = m(x - x?), where (x?, y?) is a point on the line and m is the slope, making it a convenient tool for finding the equation of a tangent line once the derivative and the point of tangency are determined.

Graphical Verification

Graphical verification involves plotting both the original function and the tangent line on the same set of axes to visually confirm that the line accurately touches the curve at the point of tangency. This practice reinforces the analytical solution by providing a visual check that the computed tangent line has the correct position and slope relative to the function.

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