Derivatives Day 8 - Tangents and Linearizations
Learning Objectives Write an equation for the tangent line to a graph Write an equation for the normal line to a graph Find the locations of horizontal and vertical tangents Write a linearization for a function at a point Use a linearization to estimate the value of a function
Overview
Tangent Lines A line is tangent to a graph at a point, P, if and only if it intersects the graph at point P and has the same slope (derivative) of the graph at point P. Because they have the same slope, the tangent can be said to be parallel to the graph. Tangent line There is only one tangent line at any given point, but it can have many equations. y + 2 = 1 x - 2 ( ) and y = x - 4 Represent the same tangent line.
Normal Lines A normal line is perpendicular to another line. Normal line Normal lines have opposite, reciprocal slopes. Tangent line
Horizontal Tangents Horizontal tangents have slopes of zero.
Vertical Tangents Vertical tangents have undefined slopes
Linearizations If a graph has local linearity at a point, then the derivative and tangent line exist at that point. A linearization is another term for the equation of that tangent line. Near the point of tangency, the linearization can be used to approximate the function. This can be useful when the function is too complex to analyze directly.
Writing Equations of Tangent and Normal Lines
Writing an equation for a tangent line 1. Use the derivative to find the slope 2. Use the original function to find the point 3. Write a point-slope equation Tangent line
Writing the equation of the normal line Normal line 1. Write an equation for the tangent line in point-slope form 2. Change the slope to be the opposite and reciprocal Tangent line
Example Write equations of the tangent and normal line of At x = 3 f ( x) = x3 12 f '( x) = x2 4 f ( 3) = 27 12 f '( 3) = 9 4 slope Tangent: y - 27 12 = 9 4 x - 3 ( ) æ point 3, 27 ö è ç 12 ø Normal: y - 27 12 = - 4 9 x - 3 ( )
Do textbook p.87-89, #9-12
Horizontal and Vertical Tangents
Horizontal Tangents Horizontal tangents have slopes of zero.
Slope = zero where the derivative is zero. 4 2 y x 2x 2 dy dx 3 4x 4 3 4x 4x 0 Set the derivative equal to 0 x 3 2 x x x 0 1 0 x x 1 x 1 0 x 0, 1, 1 These are the x coordinates of the tangents Use the x-values with the original equation to find the y-values y 2, y 1, y 1 Equations of the horizontal x tangents are y 2, y 1
Find the equation of the horizontal tangent 3 2 f x x 6x 12x 5 f '( x) = 3x 2-12x +12 3x 2-12x +12 = 0 set = 0 x = 2 f ( 2) = 3 find the y value Horizontal tangent: y = 3
Practice Write the equations of the horizontal tangents. 3 2 1. f x x 6x 9x 5 3 2. g x x 12x 36x 4
Practice Write the equations of the horizontal tangents. 3 2 1. f x x 6x 9x 5 f x x x 2 ' 3 12 9 f ' x 0 at x 1,3 f 1 9; f 3 5 Tangents: y 9; y 5
Practice Write the equations of the horizontal tangents. 3 2 2. g x x 12x 36x 4 2 f f ' x 3x 24x 36 f ' x 0 at x 2, 6 f 2 36; 6 4 Tangents: y 36; y 4
Vertical Tangents Same as horizontal tangents, but find where the derivative does not exist.
Finding zeroes and DNE For rational functions Find zeroes by setting the numerator (dy) = 0 Find DNE by setting the denominator (dx) = 0 f '( x) = 3x4-2x 3 +10 2x 5 + 4x 2-5 Set 3x 4-2x 3 +10 = 0 to find the horizontal tangents Set 2x 5 + 4x 2-5 = 0 the find the vertical tangents
Linearizations
From earlier If a graph has local linearity at a point, then the derivative and tangent line exist at that point. A linearization is another term for the equation of that tangent line. Near the point of tangency, the linearization can be used to approximate the function. This can be useful when the function is too complex to analyze directly.
What does a linearization look like? A linearization is simply the tangent line equation, with some minor changes. 1 1 1 1 ' y y m x x y m x x y Solve for y L x f x x x f x Linearization formula: 1 1 1 ' L x f x x x f x 1 1 1 Starting with a tangent line Change notation: y x to L for "linearization" 1 1 m changes to f ' x y becomes f x 1
Example Linearization formula: ' L x f x x x f x 1 1 1 3 2 Write a linearization for f x x x at x 4 First, find the derivative 2 f ' x 3x 2x f ' 4 56 f 4 80 x L x 56 4 80 That s it. It s really just the tangent line, written a little differently.
Who cares? Linearizations are used to approximate functions near the point of tangency. The idea is that we can use to approximate the behavior of x L x 3 2 f x x x 56 4 80 near x 4 Since the linearization is a simple, linear equation it is easier to work with than the original function. Example: Use the linearization to approximate f 4.2 f L 4.2 L 4.2 4.2 56 4.2 4 80 91.2 Conclusion: f 4.2 is approximately 91.2
Wouldn t it be better to just use the original function? It depends. Linearizations are useful when the original function is too complex to analyze directly. Linearization are useful when studying general behaviors of graphs and derivatives.
Practice 2 1. Write the linearization for f x x 2x 5 at x 3 and use it to approximate f 3.1. 4 2. Write the linearization for f x x 2 x at x 2 and use it to approximate f 2.3. 3 3. Write the linearization for f x x x 2 and use it to approximate f 3.8. at x 4
Answers 2 1. Write the linearization for f x x 2x 5 at x 3 and use it to approximate f 3.1. x Answer : f ' x 2x 2; f ' 3 8 f L x L 3 20 8 3 20 3.1 8 3.1 3 20 20.8
Answers 4 2. Write the linearization for f x x 2 x at x 2 and use it to approximate f 2.3. 3 2 12 x Answer : f ' x 4x 2; f ' 2 30 f L x L 30 2 12 2.3 30 2.3 2 12 21
Answers 3 3. Write the linearization for f x x x 2 at x 4 and use it to approximate f 3.8. Answer f x x f x 2 : ' 3 1; ' 4 49 f L x L 4 66 49 4 66 3.8 49 3.8 4 66 56.2