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The derivative of a function represents the rate of change or the slope of the function at any given point. You can think of it as the function's "speed" at a particular moment. This is extremely useful because it allows us to find the slope of the tangent line to the curve. For the function \[ f(x) = \frac{1}{2}x^2 - 2, \]to find the derivative, we apply the power rule. The power rule states that if you have a term like \[ ax^n, \]then its derivative is \[ nax^{n-1}. \]Using this rule, we differentiate each term of our function separately.

• The derivative of \( \frac{1}{2}x^2 \) is \( x \), because the power 2 multiplies by \( \frac{1}{2} \) to make 1, and the power of \( x \) drops from 2 to 1.
• The derivative of a constant like \(-2\) is 0, since constants do not change.

Thus, the derivative of the function is \( f'(x) = x \). This derivative helps us find the slope at any point \( x \) on the function curve.

Point-slope form is an essential tool to find the equation of a tangent line to a curve at a specific point. It is especially useful when you already know a point on the line and the slope, which in our scenario is the derivative evaluated at that point. The point-slope formula is:\[ y - y_1 = m(x - x_1) \]where:

• \((x_1, y_1)\) is a known point on the line.
• \( m \) is the slope of the line, obtained from the derivative.

For the point \((2, 0)\) on the function \( f(x) = \frac{1}{2}x^2 - 2 \), evaluation of the derivative gave us a slope of 2. Substituting these into the point-slope form gives \[ y - 0 = 2(x - 2). \]This simplifies to \[ y = 2x - 4, \]the equation of the tangent line. This line touches the function at exactly one point, forming the tangent.

Graphing functions is a powerful way to visualize the relationship described by a function and to see the behavior of its tangent line. To graph the function \[ f(x) = \frac{1}{2}x^2 - 2 \],recognize it is a parabola, which is a common shape for a quadratic function. This specific parabola opens upwards because the coefficient of \(x^2\) (\(\frac{1}{2}\)) is positive. The \(-2\) term shifts the whole graph down.

• The vertex of the parabola is at the point of minimum value since it opens upwards, which happens at \((0, -2)\).
• The line of symmetry of the parabola is the y-axis.

When graphing the tangent line \[ y = 2x - 4, \] it appears as a straight line intersecting the parabola at the point \((2, 0)\). This intersection is the point of tangency, where the slope agrees for both the function and the straight line. Graphs reveal how the tangent just kisses the function at one single point, more easily than using algebra alone.