**of a line with slope m through a point $(x_1,y_1): m=\frac{y-y_1}{x-x_1}$**

*Point-Slope Form*•

__Tangent Line & Normal Line__

The

**normal line**to a curve at a particular point is the line through that point and

*perpendicular*to the

**tangent line**.

Intros

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__Connecting: Derivative & Slope & Equation of Tangent Line__

*Exercise:*The graph of the quadratic function $f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1$ is shown below.

- Find and interpret $f'\left( x \right)$.
- Find the slope of the tangent line at:

i) $x = - 1$

ii) $x = 2$

iii) $x = - 7$

iv) $x = - 4$

v) $x = - 2$ - Find an equation of the tangent line at:

i) $x = 2$

ii) $x = - 4$

iii) $x = - 2$

Examples

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**Determining Equations of the Tangent Line and Normal Line**

Consider the function: $f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})$**Locating Horizontal Tangent Lines****Locating Tangent Lines Parallel to a Linear Function**

Consider the Cubic function: $f(x)=x^3-3x^2+3x$

i) Find the points on the curve where the tangent lines are parallel to the line $12x-y-9=0$.

ii) Determine the equations of these tangent lines.**Determining Lines Passing Through a Point and Tangent to a Function**

Consider the quadratic function: $f(x)=x^2-x-2$**Locating Lines Simultaneously Tangent to 2 Curves**

Consider the quadratic functions:

$f(x)=x^2$

$g(x)=\frac{1}{4}x^2+3$