Use a graphing calculator or a CAS. Draw the graph of...

Use a graphing calculator or a CAS.
Draw the graph of $y=f(x)=x^{3}-2 x^{2}+1 .$ Then find the slope of the tangent line at
$\begin{array}{llll}\text { (a) }-1 & \text { (b) } 0 & \text { (c) } 1 & \text { (d) } 3.2\end{array}$

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Key Concepts

Tangent Line

The tangent line to a curve at a given point is a line that touches the curve at that point without crossing it locally. Its slope is the value of the derivative of the function at that point, providing a linear approximation of the function's behavior around the point.

Graphing Functions

Graphing functions involves plotting the equation on a coordinate plane, often with the help of technology, to visualize the behavior, intercepts, turning points, and overall shape of the function. This helps in understanding the function’s characteristics and providing a reference for further analysis such as finding tangents.

Differentiation

Differentiation is a fundamental tool in calculus used to calculate the derivative of a function, which represents its instantaneous rate of change. This process allows us to determine how the function’s output changes with respect to changes in the input.

Power Rule

The power rule is a basic technique in differentiation that simplifies the process of finding derivatives for polynomial functions. It states that for any term of the form x^n, the derivative is n*x^(n-1), enabling quick and systematic differentiation of polynomial expressions.

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