Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.





b. Estimate home prices at the end of
i) 2007 ii) 2012 iii) 2015

b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.

Theory and Examples

In Exercises 51–54,

b. Graph y = f(x) and y = f'(x) side by side using separate sets of coordinate axes, and answer the following questions.

y = x⁴/4

Motion Along a Coordinate Line

Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.

b. Find the body’s speed and acceleration at the endpoints of the interval.

s = 25/t² − 5/t, 1 ≤ t ≤ 5

By computing the first few derivatives and looking for a pattern, find the following derivatives.

b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

Temperature The given graph shows the outside temperature T in °F, between 6 a.m. and 6 p.m.

b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times?

Common linear approximations at x = 0 Find the linearizations of the following functions at x = 0.

b. cos x