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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.

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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 ?

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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.

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

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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 ) .

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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

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f ( x ) = x 2 2 x + 1

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f ( x ) = 2 x 2 + x 3

f ( x ) = 4 x + 1

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f ( x ) = 1 x 2

f ( x ) = 1 ( x 2 ) 2

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

16 ( 3 + 2 x ) 2

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

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

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f ( x ) = 5 π

f ( x ) = 0

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

( −2 , 0 ) and ( −4 , 5 )

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( 4 , −3 ) and ( −2 , −1 )

1 3

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( 0 , 5 ) and ( 6 , 5 )

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( 7 , −2 ) and ( 7 , 10 )

undefined

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For the following polynomial functions, find the derivatives.

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

f ( x ) = 6 x 7

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

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

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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

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

y = 12 x 15

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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

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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.

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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.

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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

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f ( 1 )

f ( 1 ) = 3

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f ( 3 )

f ( 3 ) = 9

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Sketch the function based on the information below:

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

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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.

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Questions & Answers

it is the relatively stable flow of income
Chidubem Reply
what is circular flow of income
Divine Reply
branches of macroeconomics
SHEDRACK Reply
what is Flexible exchang rate?
poudel Reply
is gdp a reliable measurement of wealth
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introduction to econometrics
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Tom
Why is unemployment rate never zero at full employment?
Priyanka Reply
bcoz of existence of frictional unemployment in our economy.
Umashankar
what is flexible exchang rate?
poudel
due to existence of the pple with disabilities
Abdulraufu
the demand of a good rises, causing the demand for another good to fall
Rushawn Reply
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.
Geetha Reply
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Waeth
discus major problems of macroeconomics
Alii Reply
what is the problem of macroeconomics
Yoal
Economic growth Stable prices and low unemployment
Ephraim
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Miresa Reply
what is inflation
Getu
increase in general price levels
WEETO
Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
Chisomo Reply
how to calculate actual output?
Chisomo
how to calculate the equilibrium income
Beshir
Criteria for determining money supply
Thapase Reply
who we can define macroeconomics in one line
Muhammad
Aggregate demand
Mohammed
C=k100 +9y and i=k50.calculate the equilibrium level of output
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money as unit of account means what?
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A unit of account is something that can be used to value goods and services and make calculations
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Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500
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hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.
Galge Reply
if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️
Kalombe
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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