<< Chapter < Page Chapter >> Page >

Section exercises

Verbal

How is the slope of a linear function similar to the derivative?

The slope of a linear function stays the same. The derivative of a general function varies according to x . Both the slope of a line and the derivative at a point measure the rate of change of the function.

Got questions? Get instant answers now!

What is the difference between the average rate of change of a function on the interval [ x , x + h ] and the derivative of the function at x ?

Got questions? Get instant answers now!

A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car's average velocity? At exactly 2:30 P.M. , the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M. ? Why does this speed differ from the average velocity?

Average velocity is 55 miles per hour. The instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of the car over an interval.

Got questions? Get instant answers now!

Explain the concept of the slope of a curve at point x .

Got questions? Get instant answers now!

Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.

The average rate of change of the amount of water in the tank is 45 gallons per minute. If f ( x ) is the function giving the amount of water in the tank at any time t , then the average rate of change of f ( x ) between t = a and t = b is f ( a ) + 45 ( b a ) .

Got questions? Get instant answers now!

Algebraic

For the following exercises, use the definition of derivative lim h 0 f ( x + h ) f ( x ) h to calculate the derivative of each function.

f ( x ) = 2 x + 1

f ( x ) = 2

Got questions? Get instant answers now!

f ( x ) = x 2 2 x + 1

Got questions? Get instant answers now!

f ( x ) = 2 x 2 + x 3

f ( x ) = 4 x + 1

Got questions? Get instant answers now!

f ( x ) = 1 x 2

f ( x ) = 1 ( x 2 ) 2

Got questions? Get instant answers now!

f ( x ) = 5 2 x 3 + 2 x

16 ( 3 + 2 x ) 2

Got questions? Get instant answers now!

f ( x ) = 3 x 3 x 2 + 2 x + 5

f ( x ) = 9 x 2 2 x + 2

Got questions? Get instant answers now!

f ( x ) = 5 π

f ( x ) = 0

Got questions? Get instant answers now!

For the following exercises, find the average rate of change between the two points.

( −2 , 0 ) and ( −4 , 5 )

Got questions? Get instant answers now!

( 4 , −3 ) and ( −2 , −1 )

1 3

Got questions? Get instant answers now!

( 0 , 5 ) and ( 6 , 5 )

Got questions? Get instant answers now!

( 7 , −2 ) and ( 7 , 10 )

undefined

Got questions? Get instant answers now!

For the following polynomial functions, find the derivatives.

f ( x ) = 3 x 2 7 x = 6

f ( x ) = 6 x 7

Got questions? Get instant answers now!

f ( x ) = 3 x 3 + 2 x 2 + x 26

f ( x ) = 9 x 2 + 4 x + 1

Got questions? Get instant answers now!

For the following functions, find the equation of the tangent line to the curve at the given point x on the curve.

f ( x ) = 2 x 2 3 x x = 3

Got questions? Get instant answers now!

f ( x ) = x 3 + 1 x = 2

y = 12 x 15

Got questions? Get instant answers now!

For the following exercise, find k such that the given line is tangent to the graph of the function.

f ( x ) = x 2 k x , y = 4 x 9

k = 10 or k = 2

Got questions? Get instant answers now!

Graphical

For the following exercises, consider the graph of the function f and determine where the function is continuous/discontinuous and differentiable/not differentiable.


Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.

Discontinuous at x = 2 and x = 0. Not differentiable at –2, 0, 2.

Got questions? Get instant answers now!
Graph of a piecewise function with two segments. The first segment goes from (-4, 0), an open point to (5, -2), and the final segment goes from (5, 3), an open point, to positive infinity.

Discontinuous at x = 5. Not differentiable at -4, –2, 0, 1, 3, 4, 5.

Got questions? Get instant answers now!

For the following exercises, use [link] to estimate either the function at a given value of x or the derivative at a given value of x , as indicated.

Graph of an odd function with multiplicity of 2 with a turning point at (0, -2) and (2, -6).

f ( 1 )

f ( 1 ) = 9

Got questions? Get instant answers now!

f ( 1 )

f ( 1 ) = 3

Got questions? Get instant answers now!

f ( 3 )

f ( 3 ) = 9

Got questions? Get instant answers now!

Sketch the function based on the information below:

f ( x ) = 2 x , f ( 2 ) = 4

Got questions? Get instant answers now!

Technology

Numerically evaluate the derivative. Explore the behavior of the graph of f ( x ) = x 2 around x = 1 by graphing the function on the following domains: [ 0.9 , 1.1 ] , [ 0.99 , 1.01 ] , [ 0.999 , 1.001 ] , and [ 0.9999 , 1.0001 ] . We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the corresponding range values for this viewing window, approximate how the curve changes at x = 1 , that is, approximate the derivative at x = 1.

Answers vary. The slope of the tangent line near x = 1 is 2.

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

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