CK-12-Calculus

2.1. Tangent Lines and Rates of Change http://www.ck12.org

Example 4:
Suppose thaty=x^2 − 3.

1. Find the average rate of change ofywith respect toxover the interval[ 0 , 2 ].


2. Find the instantaneous rate of change ofywith respect toxat the pointx=− 1.

Solution:

1. Applying the formula for Average Rate of Change withf(x) =x^2 −3 andx 0 =0 andx 1 =2 yields

msec= f(xx^11 )−−xf 0 (x^0 )

= f(^22 )−− 0 f(^0 )

=^1 −( 2 −^3 )

= 2

This means that the average rate of change ofyis 2 units per unit increase inxover the interval[ 0 , 2 ].

2. From the example above, we found thatf′(x) = 2 x, so

mtan=f′(x 0 )

=f′(− 1 )

= 2 (− 1 )

=− 2

This means that the instantaneous rate of change is negative. That is,yis decreasing atx=− 1 .It is decreasing at a
rate of 2 units per unit increase inx.

Multimedia Links

For a video explaining instantaneous rates of change(4.2), see Slopes of Tangents and Instantaneous Rates of Ch
ange (9:26).

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/533

For a video with an application regarding velocity(4.2), see Calculus Help: Instantaneous Rates of Change (9:03).