#### To determine

**(a)**

**To estimate:** Values of F’ (1950), F’ (1965) and F’ (1987)

**Solution:**F’ (1950) = 0.11, F’ (1965) = -0.16 and F’ (1987) = 0.02

#### Answer

In the baby-boom era, there was optimism about the economy and family size was rising.

In the baby-bust era, there was less economic optimism and it was considered less socially responsible to have a large family.

In the baby boomlet era, there was increased economic optimism and a return to more conservative attitudes.

**Explanation:**

From the graph it observed that rate of change of average number of children born to each women is decreases with time.

In the baby-boom era, there was optimism about the economy and family size was rising.

In the baby-bust era, there was less economic optimism and it was considered less socially responsible to have a large family.

In the baby boomlet era, there was increased economic optimism and a return to more conservative attitudes.

#### Explanation

**1) Formula:**An estimate of F’(t) is

F't=F(t2)-F(t1)t2-t1

**2) Calculation:**

F'1950=F(1960)-F(1950)1960-1950=3.7-2.610=1.110=0.11

F'1965=F(1975)-F(1965)1975-1965=1.7-3.310=-1.610=-0.16

F'1987=F(1997)-F(1987)1997-1987=2-1.810=0.210=0.02

**Conclusion:**F’ (1950) = 0.11, F’ (1965) = -0.16 and F’ (1987) = 0.02

**(b)**

**To explain:**The meaning of derivatives in part (a)

**Solution:**The derivatives in the part (a) means a rate of change of average number of children born to each women was increasing by 0.11 in 1950, decreasing by 0.16 in 1965 and increasing by 0.02 in 1987.

From part (a), F’ (1950) = 0.11, F’ (1965) = -0.16 and F’ (1987) = 0.02

F’ (1950) = 0.11 means a rate of change of average number of children born to each women increases by 0.11 in 1950

F’ (1965) = -0.16 means a rate of change of average number of children born to each women is decreases by 0.16 in 1965

F’ (1987) = 0.02 means a rate of change of average number of children born to each women increases by 0.02 in 1987

**Conclusion:** The derivatives in the part (a) means a rate of change of average number of children born to each women was increasing by 0.11 in 1950, decreasing by 0.16 in 1965 and increasing by 0.02 in 1987.

**(c)**

**To suggest:** Reasons for values of derivatives