# Linear approximation Median home prices in Austin 160,000

Linear approximation Median home prices in Austin 160,000 Year t Median home price M(t) 1994 \$96,000 1995 \$100,500 1996 \$108,800 1997 \$112,200 1998 \$117,700 110,000 1999 \$126,700 100,000 2000 \$143,900 90,000 2001 \$150,000 2002 \$156,500 150,000 140,000 130,000 120,000 80,000

1992 1994 1996 1998 2000 2002 2004 Tangent lines often give better linear approximations than secant lines 160,000 150,000 140,000 130,000 secant line approximation tangent line approximation 1995 to 1996 at 1996 120,000 110,000 100,000 90,000 80,000 1992 1994 1996 1998 2000 2002 2004 Recall that we defined M'(t) by averaging the two adjacent secant lines at each data point Years from: to:

Average rate of change 1994 1995 \$4,500 1995 1996 \$8,300 1996 1997 \$3,400 1997 1998 \$5,500 1998 1999 \$9,000 1999 2000 \$17,200 2000 2001 \$6,100 2001 2002 \$6,500 Year t Median home price growth M'(t)

1994 \$4,500 1995 \$6,400 1996 \$5,850 1997 \$4,450 1998 \$7,250 1999 \$13,100 2000 \$11,650 2001 \$6,300 2002 \$6,500 For the endpoints 1994 and 2002, there is only one interval to use so no average is taken Use both linear methods to estimate the population in 1997 Secant line method (the old way): y 8300 1997 1996 108,800 117,100 1996 pop 1995 pop 1996 1995 Actual value 112,200 Tangent line method (the new way): y 5850 1997 1996 108,800 114,650 Average of secant slopes from 1995 to 1996 and from 1996 to 1997

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