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Get Started Now- Intro Lesson: a7:08
- Intro Lesson: b11:40
- Intro Lesson: c16:37
- Lesson: 1a24:23
- Lesson: 1b12:38
- Lesson: 2a11:57
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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.

26.

Limits and Derivatives

26.1

Finding limits from graphs

26.2

Definition of derivative

26.3

Power rule

26.4

Slope and equation of tangent line

26.5

Chain rule

26.6

Derivative of trigonometric functions

26.7

Derivative of exponential functions

26.8

Product rule

26.9

Quotient rule

26.10

Implicit differentiation

26.11

Derivative of inverse trigonometric functions

26.12

Derivative of logarithmic functions

26.13

Higher order derivatives

We have over 1270 practice questions in AU Maths Methods for you to master.

Get Started Now26.1

Finding limits from graphs

26.2

Definition of derivative

26.3

Power rule

26.4

Slope and equation of tangent line

26.5

Chain rule

26.6

Derivative of trigonometric functions

26.7

Derivative of exponential functions

26.8

Product rule

26.9

Quotient rule

26.10

Implicit differentiation

26.11

Derivative of inverse trigonometric functions

26.12

Derivative of logarithmic functions

26.13

Higher order derivatives