Question

A company determines that monthly sales​ S(t), in thousands of​ dollars, after t months of marketing a product is given by ​S(t)equals2 t cubed minus 45 t squared plus 180 t plus 130. ​a) Find Upper S prime​(1​), Upper S prime​(2​), and Upper S prime​(4​). ​b) Find Upper S double prime​(1​), Upper S double prime​(2​), and Upper S double prime​(4​). ​c) Interpret the meaning of your answers to parts​ (a) and​ (b).

Answer 1

Answer: a) S'(1) = 136; S'(2) = 104; S'(4) = 76;

b) S''(1) = -38; S''(2) = -26; S''(4) = -2

Step-by-step explanation:

a) S' means first derivative;

$\frac{d}{dt}$(6t³ - 45t² +180t +130) = 6t² - 50t + 180

S'(1) = 6.1² - 50.1 + 180

S'(1) = 136

S'(2) = 6.2² - 50.2 + 180

S'(2) = 104

S'(4) = 6.4² - 50.4 + 180

S'(4) = 76

b) S'' is the second derivative of S:

$\frac{d^2}{dt^2}$(6t² - 50t + 180) = 12t - 50

S''(1) = 12.1 - 50

S''(1) = -38

S''(2) = 12.2 - 50

S"(2) = -26

S"(4) = 12.6 - 50

S"(4) = -2

c) Derivative is the rate of change of a function. The first derivative is the slope of the tangent line to the graph if the function in a determined point, while the second derivative measures how the rate of change is changing.

Analysing the values, we can conclude that the sales of the product after t months is decreasing at a rate of 12.