<< 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!
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