(a) find the slope of the graph of f at the given point, (b)...

(a) find the slope of the graph of $f$ at the given point, (b) find an equation of the tangent line to the graph at the point, and (c) graph the function and the tangent line.
$f(x)=x^{3}-2 x, \quad(1,-1)$

Key Concepts

Derivative

The derivative of a function at a given point provides the instantaneous rate of change with respect to its independent variable. It is a fundamental concept in calculus as it is used to compute the slope of the tangent line at a specific point on the graph. By differentiating the function, one can substitute the point's x-coordinate to obtain the slope, which encapsulates how the function is changing locally.

Tangent Line

The tangent line to a function at a specific point is the straight line that just touches the curve at that point and has the same slope as the function does there. This concept is critical in understanding local linear approximations and can be used to estimate the value of the function near that point.

Point-Slope Form

The point-slope form is a method for writing the equation of a straight line when a point on the line and its slope are known. This form, expressed as y - y? = m(x - x?), is particularly useful when deriving the equation of the tangent line since it directly utilizes the slope from the derivative and the coordinates of the point of tangency.

Graphical Representation

Graphing both the function and its tangent line serves as an important visual aid in understanding how the tangent line approximates the function near the point of tangency. This concept reinforces the idea that, in a small neighborhood around the point, the function and the tangent line behave in a similar manner.

Transcript

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