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The derivative of a function is interpreted as the slope of the tangent line to the curve of the function at a certain given point. In this section, we will explore the meaning of a derivative of a function, as well as learning how to find the slope-point form of the equation of a tangent line, as well as normal lines, to a curve at multiple given points.

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

•__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**.

•

The

- Introduction
__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.

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

i) $x = - 1$

ii) $x = 2$

iii) $x = - 7$

iv) $x = - 4$

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

i) $x = 2$

ii) $x = - 4$

iii) $x = - 2$

- 1.
**Determining Equations of the Tangent Line and Normal Line**

Consider the function: $f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})$a)Determine an equation of the tangent line to the curve at $x=64$.b)Determine an equation of the normal line to the curve at $x=64$. - 2.
**Locating Horizontal Tangent Lines**a)Find the points on the graph of $f(x)=2x^3-3x^2-12x+8$ where the tangent is horizontal.b)Find the vertex of each quadratic function:

$f(x)=2x^2-12x+10$

$g(x)=-3x^2-60x-50$ - 3.
**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. - 4.
**Determining Lines Passing Through a Point and Tangent to a Function**

Consider the quadratic function: $f(x)=x^2-x-2$a)Draw two lines through the point (3, -5) that are tangent to the parabola.b)Find the points where these tangent lines intersect the parabola.c)Determine the equations of both tangent lines. - 5.
**Locating Lines Simultaneously Tangent to 2 Curves**

Consider the quadratic functions:

$f(x)=x^2$

$g(x)=\frac{1}{4}x^2+3$a)Sketch each parabola.b)Determine the lines that are tangent to both curves.

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