Wup Get out note paper Find 12 3

W-up �Get out note paper �Find 12. 3 notes �Write the 3 steps to determine if a function is continuous

14. 1 Horizontal and Vertical Tangent Lines; Continuity and Differentiability • • Find horizontal and vertical tangent lines Discuss the graph of a function f where the derivative of f does not exist

Horizontal tangent line �If function is differentiable at point c, horizontal tangent line occurs when f ‘ (c) = 0 �Ex 1: At what points is the tangent line of f(x)=x 3 + 3 x 2 – 24 x horizontal? �Find derivative: f ‘(x) = 3 x 2 + 6 x – 24 �Set f ‘(x) = 0 and solve Plug x values into ORIGINAL 2 � 0 = 3 x + 6 x – 24 factor equation to find the y-value where � 3(x 2 + 2 x – 8) = 0 � (x + 4 ) (x – 2) = 0 � x = -4 and x = 2 tangent line is horizontal • f(-4) = (-4)3 + 3(-4)2 – 24(-4) = 80 • f(2) = (2)3 + 3(2)2 – 24(2) = -28 � Horizontal tangent lines will occur at (-4, 80) and (2, -28)

Conditions for Vertical Tangent Lines �If a vertical tangent line is present at a point (c, f(c)) on a continuous function then � 1) x = c is in the domain � 2) f’(x) must be unbounded at x = c � Unbounded will be where the denominator of the derivative = 0

Ex 1: find horizontal and vertical tangent lines of f(x) = -3 x 2 + 12 x �Step 1 state domain – all real numbers �Step 2 - find derivative �f ‘(x) = -6 x + 12 �To find horizontal set f ‘(x) = 0 and solve � 0 = -6 x + 12 �x = 2 �Plug 2 into original to find y �f(2) = -3(2)2 + 12(2) = 12 �Horizontal tangent line at (2, 12) When will there be a vertical tangent line? NO VERTICAL TAN LINE – there is no denominator (no unbounded points)

Ex 2: find horizontal and vertical tangent lines of � Get common denominator

�Find horizontal tangent line �To find vertical tangent line set derivative = 0 and solve set denominator of derivative = 0 and solve �A fraction will be zero when �Set each factor = 0 and solve numerator is zero �x = 0 and x = 1 �x=-2 �Throw out 1 NOT IN �Find y by plugging x into DOMAIN original equation �Find y by plugging x into original equation �Horizontal tangent line at (-2, . 53) �Vertical tangent line a (0, 0)

Continuity and Differentiability �If c is a number in the domain of f and f is differentiable at the number c, then f is continuous at c �If it is not continuous it is not differentiable. �Converse is not always true – if a graph is continuous at c, it may not be differentiable

Given: a) determine if it is continuous at x = 0 �Is it defined � find left and right side limits �Yes continuous all equal zero �B) does f’(0) exist? �Plug in zero Will this be a VERTICAL tangent line or NO tangent line?

Given: a) determine if it is continuous at x = 1 �Is it defined f(1) = 12 + 5 = 6 � find left and right side limits �Yes continuous all equal six

B) does f’(c) exist? GRAPH f(x) = 6 x; x< 1 1 0 -1 6 0 -6 f(x) = x 2+5; x> 1 1 2 3 • f ‘(1) does not exist, there are two different slopes at 1 • Find the derivative of each part • f’ (x) = 6 f’(x) = 2 x f’(1) = 6 f’ (1) = 2 6 9 14

Homework � 14. 1 # 1 – 33 odds �Don’t forget to graph # 27 -33