Question

the function f(x)=1.3x^3 - 33x^2 + 270x + 553 below models the number of discharges from the military, f(x), of active - duty gay service members under the \dont ask, dont tell\ policy x years after 1994. complete parts a and b. a. find the slope of the secant line from x_1 = 0 to x_2 = 4. the slope of the secant line is . (round to the nearest whole number as needed.) b. the actual average yearly increase from 1994 to 1998 was 141. does the slope from part (a) underestimate or overestimate this value? by how much? the slope from part (a) the actual average yearly increase.

Explanation:

Step1: Calculate f(0)

Substitute \(x = 0\) into \(f(x)=1.3x^{3}-33x^{2}+270x + 553\).

$$f(0)=1.3\times0^{3}-33\times0^{2}+270\times0 + 553=553$$

Step2: Calculate f(4)

Substitute \(x = 4\) into \(f(x)=1.3x^{3}-33x^{2}+270x + 553\).

$$
LATEXBLOCK0
$$

Step3: Calculate the slope of the secant line

The slope \(m\) of the secant line between \((x_1,f(x_1))\) and \((x_2,f(x_2))\) is given by \(m=\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here \(x_1 = 0\), \(x_2=4\), \(f(x_1)=553\), \(f(x_2)=1188.2\).

$$m=\frac{1188.2 - 553}{4-0}=\frac{635.2}{4}=158.8\approx159$$

Step4: Compare with the actual value

The actual average - yearly increase from 1994 to 1998 was 141. Since \(159>141\), the slope from part (a) overestimates the actual value. The difference is \(159 - 141=18\).

Answer:

a. 159
b. overestimates; 18