1. Home
  2. Precalculus
  3. Introduction to calculus
  4. Derivatives

12.4 Derivatives  (Page 7/18)

<< Chapter < Page
  Precalculus     Page 7 / 18
Chapter >> Page >
Graph of f(x) = x^(1/3) with a viewing window of [-3, 3] by [-3, 3].
The graph of   f ( x ) = x 1 3   has a vertical tangent at   x = 0.

Differentiability

A function   f ( x )   is differentiable at   x = a   if the derivative exists at   x = a , which means that   f ′ ( a )   exists.

There are four cases for which a function   f ( x )   is not differentiable at a point   x = a .

  1. When there is a discontinuity at   x = a .
  2. When there is a corner point at   x = a .
  3. When there is a cusp at   x = a .
  4. Any other time when there is a vertical tangent at   x = a .

Determining where a function is continuous and differentiable from a graph

Using [link] , determine where the function is

  1. continuous
  2. discontinuous
  3. differentiable
  4. not differentiable

At the points where the graph is discontinuous or not differentiable, state why.

Graph of a piecewise function that has a removable discontinuity at (-2, -1) and is discontinuous when x =1.

The graph of f ( x ) is continuous on ( −∞, −2 ) ∪ ( −2 , 1 ) ∪ ( 1, ∞ ). The graph of f ( x ) has a removable discontinuity at x = −2 and a jump discontinuity at x = 1. See [link] .

Graph of the previous function that shows the intervals of continuity.
Three intervals where the function is continuous

The graph of is differentiable on ( −∞, −2 ) ∪ ( −2, −1 ) ∪ ( −1, 1 ) ∪ ( 1, 2 ) ∪ ( 2, ∞ ) . The graph of f ( x ) is not differentiable at x = −2 because it is a point of discontinuity, at x = −1 because of a sharp corner, at x = 1 because it is a point of discontinuity, and at x = 2 because of a sharp corner. See [link] .

Graph of the previous function that not only shows the intervals of continuity but also labels the parts of the graph that has sharp corners and discontinuities. The sharp corners are at (-1, -1) and (2, 3), and the discontinuities are at (-2, -1) and (1, 1).
Five intervals where the function is differentiable
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Determine where the function   y = f ( x )   shown in [link] is continuous and differentiable from the graph.

Graph of a piecewise function with three pieces.

The graph of   f   is continuous on   ( − ∞ , 1 ) ∪ ( 1 , 3 ) ∪ ( 3 , ∞ ) .   The graph of   f   is discontinuous at   x = 1   and   x = 3.   The graph of   f   is differentiable on   ( − ∞ , 1 ) ∪ ( 1 , 3 ) ∪ ( 3 , ∞ ) .   The graph of   f   is not differentiable at   x = 1   and   x = 3.

Got questions? Get instant answers now!

Finding an equation of a line tangent to the graph of a function

The equation of a tangent line to a curve of the function   f ( x )   at   x = a   is derived from the point-slope form of a line,   y = m ( x − x 1 ) + y 1 .   The slope of the line is the slope of the curve at   x = a   and is therefore equal to   f ′ ( a ) , the derivative of   f ( x )   at   x = a .   The coordinate pair of the point on the line at   x = a   is   ( a , f ( a ) ) .

If we substitute into the point-slope form, we have

The point-slope formula that demonstrates that m = f(a), x1 = a, and y_1 = f(a).

The equation of the tangent line is

y = f ' ( a ) ( x − a ) + f ( a )

The equation of a line tangent to a curve of the function f

The equation of a line tangent to the curve of a function   f   at a point   x = a   is

y = f ' ( a ) ( x − a ) + f ( a )

Given a function   f ,   find the equation of a line tangent to the function at   x = a .

  1. Find the derivative of   f ( x )   at   x = a   using   f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h .
  2. Evaluate the function at   x = a .   This is   f ( a ) .
  3. Substitute   ( a , f ( a ) )   and   f ′ ( a )   into   y = f ' ( a ) ( x − a ) + f ( a ) .
  4. Write the equation of the tangent line in the form   y = m x + b .

Finding the equation of a line tangent to a function at a point

Find the equation of a line tangent to the curve   f ( x ) = x 2 − 4 x   at   x = 3.

Using:

f ' ( a ) = lim h → 0 f ( a + h ) − f ( a ) h

Substitute   f ( a + h ) = ( a + h ) 2 − 4 ( a + h )   and   f ( a ) = a 2 − 4 a .

  f ′ ( a ) = lim h → 0 ( a + h ) ( a + h ) − 4 ( a + h ) − ( a 2 − 4 a ) h           = lim h → 0 a 2 + 2 a h + h 2 − 4 a − 4 h − a 2 + 4 a h Remove parentheses .           = lim h → 0 a 2 + 2 a h + h 2 − 4 a − 4 h − a 2 + 4 a h Combine like terms .           = lim h → 0 2 a h + h 2 − 4 h h           = lim h → 0 h ( 2 a + h − 4 ) h Factor out  h .           = 2 a + 0 − 4 f ′ ( a ) = 2 a − 4 Evaluate the limit .   f ′ ( 3 ) = 2 ( 3 ) − 4 = 2

Equation of tangent line at   x = 3:

y = f ' ( a ) ( x − a ) + f ( a ) y = f ' ( 3 ) ( x − 3 ) + f ( 3 ) y = 2 ( x − 3 ) + ( − 3 ) y = 2 x − 9
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 7

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask